Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.47.0-wmf.12 first-letter Media Special Talk User User talk Wikiversity Wikiversity talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk School School talk Portal Portal talk Topic Topic talk Collection Collection talk Draft Draft talk TimedText TimedText talk Module Module talk Event Event talk 4 28 2819252 2818749 2026-07-24T12:25:21Z MediaWiki message delivery 983498 /* Request for comment (the future of Abstract Wikipedia) */ new section 2819252 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:14, 9 June 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) * {{oppose}} per above. Wikiversity<math>\not=</math> Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC) * {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC) * {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC) *{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on "Media & Democracy" syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that "rehost Wikinews here" would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}} ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) *::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a "regular" journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:43, 5 June 2026 (UTC) *:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC) == Create an autopatrolled user group? == {{tracked|T428269|resolved}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) : @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC) ::Thanks. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:13, 9 June 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) :Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC) ::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC) :::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC) ::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) : Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC) : Thanks for creating this draft. I've made some changes (including moving back to a one month response timeframe) and moved it from draft to proposed policy. : Based on the discussion on the talk page, I think it is close to ready (or ready) to be formally proposed as a policy (by adding it to the sitenotice) to allow wider discussion and hopefully adoption. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 06:45, 21 July 2026 (UTC) == Proposed user group and/or possible policy changes == {{tracked|T430416|fixed}} I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) :#Yes, I agree. :#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos. :#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''"II ...then you will be approved as a probationary custodian for a period of at least four weeks"''. :::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to. :[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC) == New user what do I do here == I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful. (One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}} :Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC) :Thank you very much :) hope to contribute a lot. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC) == Towards an Ethics policy == In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects. However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC) :One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:39, 9 June 2026 (UTC) ::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism. ::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted. ::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC) :::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:06, 9 June 2026 (UTC) ::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC) :AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC) ::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC) :::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC) ::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC) == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == Hello community, The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon. Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC) :Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:24, 9 June 2026 (UTC) == Image not displaying == Can anyone work out why this image isn't displaying?<br> [[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:45, 11 June 2026 (UTC) :Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:59, 11 June 2026 (UTC) == New nomination template(s) == I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC) == June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;">[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions. #'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 04:00, 15 June 2026 (UTC) == Mobile friendly main page == Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}} :{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:30, 18 June 2026 (UTC) ::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC) :::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:42, 18 June 2026 (UTC) ::::I do know CSS as I like to maintain a blog online so I could try and fix it but I don't know if I have the access to do that, would i need to be a curator/ custodian. Alternatively i could edit a sandbox version of the main page and then send it to someone. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 20:00, 18 June 2026 (UTC) :::::Oh great. There are a lot of draft versions of the main page like [[Wikiversity:Main Page/Draft version 0.2]], so you can make [[Wikiversity:Main Page/Sandbox]] if you want and edit there. If you can tinker it to your liking, I can edit the main page. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:14, 18 June 2026 (UTC) ::::::thank you, i'll check it out [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 22:16, 18 June 2026 (UTC) == Main page titles == Currently, the title says "Wikiversity:Main Page", but in my opinion, it's too basic. I would like to propose changing it with the following options (you may only pick one): # Option 1: Set both [[MediaWiki:Mainpage-title]] and [[MediaWiki:Mainpage-title-loggedin]] to blank, giving the main page a portal-like design (as with English Wikipedia, English Wikibooks, etc.) # Option 2: Modify [[MediaWiki:Mainpage-title]] to <code>Welcome to Wikiversity</code> (for unregistered users), and [[MediaWiki:Mainpage-title-loggedin]] to <code><nowiki>Welcome to Wikiversity, $1!</nowiki></code>; the latter would display to me as <code>Welcome to Wikiversity, Codename Noreste!</code> Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 18 June 2026 (UTC) : Pinging @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]] for input above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:03, 24 June 2026 (UTC) :I'm afraid that I don't have strong feelings on this. Changing to either or staying with the status quo are all fine to me. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:10, 24 June 2026 (UTC) : I like the option of being consistent with Wikipedia and Wikibooks -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:04, 25 June 2026 (UTC) : {{done}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 30 June 2026 (UTC) : Thankyou - looks good. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:29, 2 July 2026 (UTC) == Wiki x AI preconference day @ Wikimania == There will be a preconference day at Wikimania about [[meta:Artificial_intelligence/2026_Wiki_AI | Wiki AI]]. It will be mostly offline, but there will be at least one hybrid session for demos of community-developed AI tools and workflows. * If you've built something cool, that is a chance to show it off, list it on the gallery of tools in progress, and get feedback. * If you could ask the people shaping AI on the wikis (WMF, tool builders, model trainers, GLAM and policy folks) a question, what would it be? Cheers, <span style="padding:0 2px 0 2px;background-color:white;color:#bbb;">&ndash;[[User:Sj|SJ]][[User Talk:Sj|<span style="color:#ff9900;">+</span>]]</span> 23:12, 20 June 2026 (UTC) and Alaexis<br>{{comment|1=Copied from https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FAssessment%2FUsing_generative_AI&diff=2816357&oldid=2807052}} == RFC about AI-generated content in Wikimedia Commons == You are invited to participate in a [[c:Commons:Requests for comment/Policy update for AI content|request for comment on Wikimedia Commons about a policy update for AI content]]. This may affect files that are uploaded to Wikimedia Commons for use on this project. Thank you. [[m:User:Codename Noreste|Codename Noreste]] ([[m:User talk:Codename Noreste|discuss]]) 17:12, 23 June 2026 (UTC) <!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == <section begin="Message"/> '''Legal & Safety Contacts''' Hello community, the Wikimedia Foundation has provided a [[wmf:Special:MyLanguage/Legal:Wikimedia Foundation Legal and Safety Contact Information|single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish and other wikis and we will deploy to your wiki soon. [[m:Special:MyLanguage/Wikimedia_Foundation_Legal_and_Safety_Contacts_FAQ|Please read more on the project page]] and leave any comments in this thread or on the [[m:Special:MyLanguage/Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|talk page]]. <section end="Message"/> -- [[User:Sannita (WMF)|User:Sannita (WMF)]] ([[User talk:Sannita (WMF)|talk]]) 13:31, 25 June 2026 (UTC) <!-- Message sent by User:Sannita (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Sannita_(WMF)/Mass_sending_test&oldid=30731267 --> == Preparing manuscript for submission to the ''WikiJournal of Humanities'' == I am working on preparing an article in Wikipedia for a dual goal of submitting it for a featured article candidacy in Wikipedia and submitting it to the ''[[WikiJournal of Humanities]]''. I have an open request for pre-submission peer review at [[en:Wikipedia:Wikipedia:Peer review/Rei Ayanami/archive2|Wikipedia:Peer review/Rei Ayanami/archive2]], and I am asking for someone experienced with submitting journals to WikiJournals. The article is not ready for submission, and I would like to know where I can get assistance from users who submitted articles to the journal, but did not necessarily review them. Furthermore, I said there that submitting to the ''WikiJournal of Humanities'' depends on whether the article attains featured article status in Wikipedia, as I would like to use the featured article as a manuscript for a journal article. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 12:04, 30 June 2026 (UTC) == I could probably save many kilobytes by compressing my LLM chat history == I could probably save many kilobytes by compressing my LLM chat history. Would that be ok? I like how my new "method" looks: [[User:ThinkingScience/All General AI Prompt History Archive]] very compressed and neat. I like how to find new ways to make people using LLMs not become "secondary citizens". [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 05:05, 7 July 2026 (UTC) :{{replyto|ThinkingScience}} I do not see why not. People should be able to look in your page's history for LLM chats before your compression. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 02:34, 14 July 2026 (UTC) ::That's good. Then an admin if they have been instructed to compress...then they can delete the edit history perhaps and save space, then it's up to the admin and I don't need to worry about taking up too many resources. That's good! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 07:50, 14 July 2026 (UTC) == July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;">[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities in July. Both sessions will focus on Wikimedia governance, including possible follow-ups to the [https://meta.wikimedia.org/wiki/Movement_Charter Movement Charter] and options for reform. Participants may attend either or both Café sessions. This month, to deconflict the Café meetups from Wikimania, the meetups will be held one day later than usual. #'''26 July 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1785078000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''27 July 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1785121200 timestamp converter]), at a time friendly to Asia and the Pacific Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 03:51, 13 July 2026 (UTC) :@[[User:Pine|Pine]], Some wikimedians who may contribute less popular, but hopefully productive, comments, are those who are currently blocked on Meta. I myself was blocked infinitely on Meta several years ago, but was allowed to contInue contributing on other wikimedia projects, such as this one. There are many long term contributors, some who have contrIbuted tens of thousands of edits to wikimedia projects who are are now blocked on specific projects. :Those who happen to be blocked on Meta have, to all intents and purposes, been removed from the wikimedia movement. :Ottawahitech [[Special:Contributions/&#126;2026-40514-81|&#126;2026-40514-81]] ([[User talk:&#126;2026-40514-81|talk]]) 21:41, 18 July 2026 (UTC) == Request for comment (the future of Abstract Wikipedia) == <bdi lang="en" dir="ltr" class="mw-content-ltr"> You are invited to voice your opinions in a [[:m:Requests for comment/The future of Abstract Wikipedia|request for comment about the future of Abstract Wikipedia]]. {{Int:Feedback-thanks-title}} [[:m:User:Kowal2701|Kowal2701]] ([[:m:User talk:Kowal2701|talk]]) 12:25, 24 July 2026 (UTC) </bdi> <!-- Message sent by User:DreamRimmer@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> oo1pa7t0xisjh7hsv2guwq42d60wfw1 4 1558 2819323 2818226 2026-07-25T00:17:36Z Evan Mercer 3071189 2819323 wikitext text/x-wiki {{Please leave this line alone (sandbox heading)}} h bfc079nyevrtufr2ads69ogq7bir9ac 2819324 2819323 2026-07-25T00:18:51Z Evan Mercer 3071189 2819324 wikitext text/x-wiki {{Please leave this line alone (sandbox heading)}} <quiz> 1-2 </quiz> mm72ybcui15bb8mcqk5r285dtnwjuh8 2819325 2819324 2026-07-25T00:20:21Z Evan Mercer 3071189 2819325 wikitext text/x-wiki {{Please leave this line alone (sandbox heading)}} <quiz> 12-1=11 90+10=100 </quiz> 69tq6v4jljmkysfa8xmczxo3dddl3gx 0 2223 2819349 2719680 2026-07-25T09:55:57Z Д.Ильин 513564 2819349 wikitext text/x-wiki {{chemistry}} {{nav2|Wikiversity|Wikiversity:School of Chemistry}} A covalent bond is a form of chemical bonding which is characterized by the sharing of electrons between atoms. Covalent bonds are mainly formed due to the tendency of the elements to attain a completely filled outer shell, that is, attain noble gas configuration and become stable. Elements forming covalent compounds achieve noble gas configuration by sharing electrons within the atoms, unlike ionic compounds which achieve the noble gas configuration either by gaining or losing electrons from the outermost electron shell. In covalent bonds, when two atoms are in need of extra electrons to fill their outer valence electron shell, they will often share an electron. The most common example that most people are familiar with is water. The oxygen in water forms a covalent bond with the hydrogen, thus filling the hydrogen's outer shell with two electrons (this is because the outer shell of hydrogen has a maximum capacity of two electrons). While an oxygen atom, originally having six outer electrons, now has seven valence electrons, it requires another in order to fill the shell, and so it bonds with another hydrogen to form H₂O. Oxygen, thus has attained the configuration of the noble gas neon and the hydrogen atoms have obtained the configuration of the noble gas helium. Covalent bonding does not produce electrons, it simply pairs them so that each atom has access to at least one more valence electron than before the bond. Covalent bonding occurs between atoms with similar electronegativity, and thus most often occurs between non-metals. However, as there are sometimes uneven distributions of electronegativity, either one of the elements in any given compound may attract the shared electrons closer to it than the other one. Therefore, it will have more of a negative charge than the other (while the other becomes more positive). Though this is not a charge to the extent of ions when they gain/lose electrons, due to the slight charge, covalent compounds can produce ionic properties. A simple way to understand the concept of a covalent bond is, <i>Co</i> can be taken as "co-operation" or "jointly" and <i>valence</i>, so ''covalent'' means the co-operation or joining of valence electrons. ==Types of Covalent Bonds== The bonds formed by sharing of an electron pair between two atoms can typically be single, double or triple; that is, in covalent bonds, atoms may be linked together by single, double or triple bonds.However there is also quadruple covalent bond. In simplest terms, "single", "double", and "triple" refer to the number of shared electron pairs in the bond. For example, in the dioxygen (O₂) molecule, each oxygen must share two of its electrons in order to obtain the noble gas configuration of 8 valence electrons, resulting in a double bond of two shared electron pairs. The ''valency'' of an atom is the maximum number of bonds it can form, usually the number of electrons required to reach a stable noble gas configuration (there are exceptions to this rule, called the ''octet rule'' as all noble gases except helium have 8 valence electrons, discussed further down). Each double bond counts as 2 bonds for this purpose, and each triple bond counts as 3. Whether a bond is single, double, or triple is its ''bond order''. === <u>Single Covalent Bond</u> === In some molecules, a shared pair of electrons are contributed between the atoms, thus creating a "single bond". For example, in a hydrogen molecule, the two atoms of hydrogen are bonded together by a single bond. How? The ''valency'' of hydrogen is 1, that is, the number of electrons present in the outermost shell is 1 and it needs 1 more electron to attain a noble gas configuration, that is a completely filled outer shell. <big>'''H'''</big> has a valency of 1 and it needs 1 more electron to become stable. So, it forms a bond with another hydrogen atom and shares its electron with it and the second hydrogen atom shares its 1 electron with the first one, {{center top}}(<big>'''H'''</big>x) (x<big>'''H'''</big>){{center bottom}} (Here, x shows the number of electrons present in the outer shell of a hydrogen atom.) Now, the first atom shares its one electron with the second hydrogen atom and the second atom shares its one electron with the first one and thus we get one hydrogen atom. {{center top}}(<big>'''H'''</big>(xx)<big>'''H'''</big>){{center bottom}} Thus, 2 atoms of hydrogen share their electron to form a molecule of hydrogen, H₂. This allows each hydrogen atom to attain the electronic configuration of the noble gas helium. The shared pair of electrons is said to constitute a single bond within the two hydrogen atoms. It is a single bond because only one pair of electrons are shared within the hydrogen atoms. A single bond is denoted by a single line between the atoms. For example, the covalent bond in a hydrogen molecule is represented by:- {{center top}}<big>'''H—H'''</big>{{center bottom}} Such single covalent bond is also formed in chlorine molecule, Cl₂. === <u>Double covalent bonds</u> === Sometimes, two atoms share two, not just one, electron pairs to attain a noble gas configuration, with each atom contributing two electrons for a total of four bonding electrons. The minimum valency of an atom that can participate in this kind of covalent bonding must have a minimum valence of 2. <big>'''O'''</big> has a valency of 2, with 6 electrons, needing 2 more to become stable (although oxygen has 6 electrons, it has a valency of 2 because it only needs 2 more for stability). It therefore can form a ''double bond'' with another oxygen atom, with each atom keeping 4 of its electrons uninvolved in bonding and sharing 2 with the other atom. {{center top}}(xxxx<big>'''O'''</big>xx) (xx<big>'''O'''</big>xxxx){{center bottom}} thus becomes: {{center top}}(xxxx<big>'''O'''</big>(xxxx)<big>'''O'''</big>xxxx){{center bottom}} 4 kept electrons plus 4 shared electrons then equals a stable 8-electron configuration. A double bond is denoted by a doubled line between the bonding atoms: {{center top}}<big>'''O=O'''</big>{{center bottom}} Such a double bond is also found in the carbon dioxide molecule CO₂: {{center top}}(xxxx<big>'''O'''</big>(xxxx)<big>'''C'''</big>(xxxx)<big>'''O'''</big>xxxx){{center bottom}} Here, each oxygen atom double-bonds to the carbon atom. The carbon atom has 4 valence electrons, and since it needs 4 more, it has a valency of 4, allowing it to form 2 double bonds (or 4 total bonds): {{center top}}<big>'''O=C=O'''</big>{{center bottom}} === <u>Triple covalent bonds</u> === Likewise, when two atoms share three electron pairs, the bonding interaction is a ''triple bond.'' The minimum valency of an atom that can participate in this kind of bonding must be a minimum of 3. <big>'''N'''</big> has a valency of 3, with 5 electrons, needing 3 more to become stable. It can form a triple bond with another nitrogen atom, with each atom sharing a total of 6 electrons with 2 unshared, making a full octet. {{center top}}(xx<big>'''N'''</big>xxx) (xxx<big>'''N'''</big>xx){{center bottom}} bonds to form: {{center top}}(xx<big>'''N'''</big>(xxxxxx)<big>'''N'''</big>xx){{center bottom}} A triple bond is denoted by three lines connecting the bonding atoms: [[Image:Dinitrogen-2D-dimensions.png|50px|center|link=]] ==Exceptions to the octet rule== Often one or more bonding atoms in a molecule do not satisfy the octet rule, even though the atom is stable with respect to its electron configuration. For example, sulfur (<big>'''S'''</big>) has 6 valence electrons like oxygen. One might expect it to always share two to gain a noble gas configuration, but in sulfur hexafluoride (SF₆), it shares all six of its valence electrons in single bonds with fluorine, thus ''exceeding'' the octet rule by 4 electrons: [[Image:Sulfur-hexafluoride-2D-dimensions.svg|100px|center|link=]] Likewise, sometimes an atom might have ''less'' than a full octet. Most elements that are not in the main groups (IA, IIA, IIIA-VIIA) don't follow the octet rule in this manner, and group IIIA elements also tend to have less than an octet. Boron (<big>'''B'''</big>) has 3 valence electrons, needing either 5 more or 3 less to achieve a noble gas configuration. However, gaining 5 is not very easy (although it is possible, discussed later) while boron is too electronegative to fully lose all three. Therefore, boron has a valency of 3 and shares all three valence electrons. However, other atoms can only share 3 total additional electrons with boron for a total of 6 - 2 short of the octet rule. Boron compounds of this kind are nevertheless stable. One example is boron trichloride (BCl₃): [[Image:Boron-trichloride-2D.svg|75px|center|link=]] The third and least-common violation of the octet rule occurs for molecules where the total valence electron count is an odd number. This makes complete electron pairing impossible and it is not possible to achieve a full octet For example, nitrogen monoxide (NO) has a total of 11 valence electrons. Nitrogen and oxygen easily share two electrons each, but nitrogen still needs a third electron. Oxygen and nitrogen can then be said to be sharing a third electron contributed from oxygen that is unpaired. However, this is not a replacement for a full electron pair and this is not a triple bond. It is more than a double bond, though, and can be called a "2 and a half" bond, where the dashed line represents the "half-bond" (this terminology is not exactly correct but sufficient for this example): [[Image:Nitric-oxide-2D.svg|100px|center|link=]] As might be expected from the presence of an unpaired electron, odd-number-electron molecules tend to be reactive, as they are made more stable if their unpaired electron is paired. ==Dative covalent bonds== The bonding pair of electrons need not come from both atoms. In some cases a single atom supplies both electrons shared in a bond. In carbon monoxide (CO), oxygen has 6 valence electrons and carbon 4 valence electrons. If each atom shares two electrons each, then oxygen satisfies the octet rule but carbon is still in need of two. {{center top}}(xx<big>'''C'''</big>xx) (xx<big>'''O'''</big>xx{{font|color=green|xx}}){{center bottom}} {{center top}}(xx<big>'''C'''</big>(xxxx)<big>'''O'''</big>xx{{font|color=green|xx}}){{center bottom}} Oxygen thus will share two more of its electrons with carbon, forming an additional bond with the pair of electrons entirely contributed from oxygen. This bond is a ''dative'' covalent bond. The electrons in question are highlighted in green. {{center top}}(xx<big>'''C'''</big>(xxxx{{font|color=green|xx}})<big>'''O'''</big>xx){{center bottom}} This additional bond adds on to the two bonds already made, so this is a triple bond, albeit an unusual one where the atoms don't make equal contributions to the bonding. It is represented like any other triple bond: [[Image:Carbon_monoxide_2D.svg|100px|center|link=]] Dative covalent bonds can also allow boron to achieve a complete octet. Boron trifluoride, with the same structure as boron trichloride (mentioned above), can react with ammonia (NH₃), where nitrogen has 2 unshared electrons. The boron atom can link up to nitrogen in a dative covalent bond using those electrons: [[Image:NH3-BF3-adduct-bond-lengthening-2D.png|300px|center|link=]] [[Category:Chemistry]] [[Category:Secondary_Science_Lessons]] [[Category:Tertiary_Science_Lessons]] lxzxajvzih9rawb3n9gi5xeh60ybevt 10 4201 2819326 2581948 2026-07-25T00:52:54Z Evan Mercer 3071189 Better reflects Wikiversity's values 2819326 wikitext text/x-wiki <!-- PLEASE DO NOT list this template for deletion or make other changes without reading the talk page, discussing your proposed change, and understanding how the templates and the Introduction pages work together --> <div style="text-align: center;">{{Introduction}}</div> <div style="text-align:center;border-bottom:3px solid #fc0"> <div style="float:left;width:32%;font-weight:bold;background-color:#ff9;color:#000;padding:.3em 0;border:2px solid #fc0;border-bottom:0;font-size:130%">1. 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[[Wikiversity:Introduction explore|Explore Wikiversity]]</div><div style="width:0;height:0;clear:both;overflow:hidden"></div> </div> __NOEDITSECTION__ <div style="border:3px solid #fc0;padding:.5em 1em 1em 1em;border-top:none;background-color:#fff;color:#000"> {{Wikiversity:Introduction/Part 1}} <div style="width:100%"> === Make your first edit right now: === [[Image:Edit button closeup.png|200px|right|The '''edit''' button.]] [[Image:Publish changes.png|100px|right|The '''Publish changes''' button.]] #Go to the [[Wikiversity:Sandbox|sandbox]] and click the '''edit''' button #Type a message in the edit window #Click '''Publish changes''' to save your writing <div style="font-size:88%;margin-top:-.1em">...or "show preview" to test your changes</div> #Please, no copyrighted, offensive, or libelous content.</div></br> <div style="float: right; margin-top: 0.0em; background-color: #ff9; color: #000; padding: .2em .6em; font-size: 130%; border: 1px solid #fc0;">'''Next:''' [[Wikiversity:Introduction edit|'''Learn more about editing''' >>]] </div> </div> <div style="clear:both"></div> <includeonly>[[Category:Wikiversity basic information|Introduction]]</includeonly> <includeonly>[[Category:Help]]</includeonly> <includeonly>[[Category:Wikiversity tutorials]]</includeonly> nq5xbejyzsfyozxxemu2u1lg1gf6s1f 102 23500 2819332 2720127 2026-07-25T02:41:54Z Evan Mercer 3071189 Removed because of screen rendering glitch 2819332 wikitext text/x-wiki <!-- Add featured resources to the list below as a link. The displayed resource is selected randomly from this list. --> == Featured Resources == *[[Linear algebra (Osnabrück 2024-2025)/Part II/Link on portal]] *[[Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Link on portal]] *[[Geometry]] ch6v1qck5ew61o39cbbin1klna166rq 3 54428 2819344 2818648 2026-07-25T08:57:39Z OwlyKnight 2980378 /* Greetings! */ new section 2819344 wikitext text/x-wiki Thank you for making this happen: [[User:OhanaUnited/Sister Projects Interview]] - I am sure your readers will profit from the better info from all here. Below more info about Wikiversity, ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 18:51, 27 March 2008 (UTC) ==Welcome== '''Hello OhanaUnited, and [[Wikiversity:Welcome, newcomers|welcome]] to [[Wikiversity:What is Wikiversity?|Wikiversity]]!''' If you need [[Help:Contents|help]], feel free to visit my talk page, or [[Wikiversity:Contact|contact us]] and [[Wikiversity:Questions|ask questions]]. After you leave a comment on a [[Wikiversity:Talk page|talk page]], remember to [[Wikiversity:Signature|sign and date]]; it helps everyone follow the threads of the discussion. The signature icon [[Image:Signature_icon.png]] in the edit window makes it simple. To [[Wikiversity:Introduction|get started]], you may <div style="width:50.0%; float:left"> * [[Wikiversity:Guided tour|Take a guided tour]] and learn [[Help:Editing|to edit]]; * Explore our [[Portal:Learning Projects|learning projects]]; * [[Wikiversity:Browse|Browse]] our [[Wikiversity:Portals|portals]], [[Wikiversity:Schools|schools]], and [[Wikiversity:Research|research]] activities; </div> <div style="width:50.0%; float:left"> * Read and help develop our community [[Wikiversity:Policies|policies]]；or * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [irc://irc.freenode.net/wikiversity-en <kbd>#wikiversity-en</kbd>]. </div> <br clear="both"/> And don't forget to [[Wikiversity:Introduction explore|explore]] Wikiversity with the links to your left. [[Wikiversity:Be bold|Be bold]], and see you around Wikiversity! ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 18:51, 27 March 2008 (UTC) == Environmental experts needed :) == Hi OhanaUnited, There have been a number of environmental projects started here and there... a few I can think of offhand: *[[Project proposal:global warming]] -- I'm not sure where that stands now... it was one of the first proposals back in 2006 I think *[[Bloom Clock]] -- Essentially a phenology project... among other things the data collections will hopefully be handy for later projects tracking changes in bloom time as local and global temperature trends change *[[Radio Discussion/Living on Earth]] -- Something a couple of us were experimenting with this past winter, using a radio show as our "lecture" and collecting materials for further learning. I'm not by any means an expert in environmental science, but as a horticulurist and farmer I'm well-versed in managing my local ecology... let me know if you start something! --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 15:06, 28 March 2008 (UTC) :See also [[:Category:Ecology]], ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 11:18, 29 March 2008 (UTC) == Commons == Is there a page on commons somewhere with the questions? I'm sure I could round up a few interested commonists on IRC if you give me a link :). --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 14:15, 30 March 2008 (UTC) == Clarifications == Hi OhanaUnited, I've asked some questions at [[User talk:OhanaUnited/Sister Projects Interview#Voice(s)]] - I'd appreciate if you could clarify before I contribute to your initiative. Thanks, [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 13:54, 1 April 2008 (UTC) == removing == I removed the signatures after names in order to move forward summarizing the answers... and then I saw that you said to not do that... I reverted... How would be best to summarize the answers? --[[User:Remi|Remi]] 04:05, 21 April 2008 (UTC) :I voiced a related question in the "Voice(s)" section on the talk page.. [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 12:40, 21 April 2008 (UTC) == Publication date == Hi OhanaUnited, would you be able to let us know when [[User:OhanaUnited/Sister_Projects_Interview|your interview]] will be published? Perhaps either on the talk page or on the [[Wikiversity:Colloquium#User:OhanaUnited/Sister Projects Interview - the earliest publication date is April 21|Colloquium]]. Thanks. [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 12:39, 21 April 2008 (UTC) == Font Tag == The font tag is now obsolete. Please adjust your signature to something like: <blockquote> <pre> [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] </pre> </blockquote> Let me know if you have any questions. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 17:37, 29 May 2018 (UTC) == Reorganised discussion == This is to let you know that the discussion at [[Talk:WikiJournal User Group#Code of Conduct]] has been reorganised to ease constructive inputs that help in updating the [[WikiJournal User Group/Code of conduct draft|document]]. If you would like to summarily oppose implementation of any Code of Conduct, feel free to place your opposition at [[Talk:WikiJournal User Group#Discussion: Whether any Code of Conduct needs to be defined and implemented]]. For any other constructive inputs please feel free to do so at [[Talk:WikiJournal User Group#Discussion: Salient updates that need to be made to the existing draft]]. Thanks for your cooperation. <span style="font-family:Segoe script">[[w:User:Diptanshu Das|<b style="color:#f00">D</b><b style="color:#f60">ip</b><b style="color:#090">ta</b><b style="color:#00f">ns</b><b style="color:#60c">hu</b>]] [[User talk:Diptanshu Das|&#128172;]]</span> 12:20, 16 December 2018 (UTC) == Maps via Wikidata == I remember you were testing maybe plotting a map of editor locations. I've been testing [https://w.wiki/CGk generating a map in Wikidata]. If we include all journal editors on the WikiJournal's page then it's possible to find the geocoordinates of their employer. Eventually it should be automate-able via [[wikidata:Wikidata:Bot_requests#Automated_addition_of_WikiJournal_metadata_to_Wikidata|this bot request]], but would have to be done manually for now. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 06:22, 18 November 2019 (UTC) :Note, [https://w.wiki/CWP updated version] with better interface for multiple points. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 02:48, 23 November 2019 (UTC) == Query at review page == I just noticed there's a query for you at [[Talk:WikiJournal Preprints/Working with Bipolar Disorder During the COVID-19 Pandemic: Both Crisis and Opportunity|this page]] (the editor forgot to ping, or is unaware of the practice). [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 09:43, 17 May 2020 (UTC) == Re: A Phonological Analysis of Selected Nigerian Newscasters Rendition == I appreciate your consideration of my article for publication. However, you have not provided an email address where I could send the word version or preferably, I would like to be guided on how to get the article uploaded on wiki commons. Thank you. [[User:Margob28|Margob28]] ([[User talk:Margob28|discuss]] • [[Special:Contributions/Margob28|contribs]]) 07:35, 25 August 2022 (UTC) == The Validity of [[WikiJournal Preprints/The Effect of Corticosteroids on the Mortality Rate in COVID-19 Patients, v2]] == Hello Andrew, I'm coming to you to ask whether the mentioned paper's topic/objective is suitable for publication in the WikiJournal of Medicine. I was going to extensively work on it this summer, but I wanted to get written confirmation that this paper would be suited for my time in developing it. I also wanted to see if a Wikijournal of Humanities paper on Meditation would be suitable. I'm not sure if you're familiar with that wikijournal's guidelines, but I figured it was worth asking. Thank you, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:20, 27 August 2022 (UTC) == Request == Please, I do not know whether you could help upload the article if I send its soft copy as MS word document or pdf to you. Thanks. [[User:Margob28|Margob28]] ([[User talk:Margob28|discuss]] • [[Special:Contributions/Margob28|contribs]]) 03:44, 5 September 2022 (UTC) == Volunteering to help with WikiJournal of Humanities == I kinf of forgot about WikiJournals for a few years, and I am amazed at the progress made. Well, as a real-life professor of sociology, I'd be happy to help with WikiJournal of Humanities which seems to be closed to my field. Do let me know how I can help, assuming of course you need any assistance. (If you reply here, please ping me back, TIA). [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 03:31, 8 November 2022 (UTC) == In other news == I am a strict believer in learning from the bottoms up (as a teacher who tells students to edit Wikipedia, for example, I never ask them to do things I haven't done myself before). And it so happens, I have a publication that I think is within the scope of WikiJournal Medicine, and now that I know it is indexed in SCOPUS, it meets my university's requirements too. As I am not yet on the board or such, I think I have no COI, so I decided to went ahead and submit my work at [[WikiJournal Preprints/Where experts and amateurs meet: the ideological hobby of medical volunteering on Wikipedia]] . Before I finish copyediting it (I think I need to upload images to Wikimedia Commons and reformat references to footnotes) and finish the rest of the submission procedure, can I ask you to confirm that this topic is within the scope of WJMED and our previous conversation does not create any COI for me to submit it (I am fine putting my editorial application fpr WJHUM from yesterday on hold for the duration of the review process, if necessary)? Oh, to confirm, WikiJournals allows and prefers non-anonymous submissions, right? So I don't need to anonymize citations to my own work, etc.? [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 05:08, 10 November 2022 (UTC) :{{re|Piotrus}} Each WikiJournal (Medicine, Science, Humanities) has separate editorial boards, similar to how "Nature Medicine" and "Nature Chemistry" are two different journals, have different editor-in-chief and different ISSN/DOI even though they are both owned and published by Springer Nature. Each WikiJournal operates and makes article decisions independently from each other while sharing same pool of resources (hired contractors, H/R, overhead cost). Therefore, whether or not you are on the Humanities board will not cause a COI when submitting to Medicine. I am the managing editor for Science, so our conversations won't cause any COI. I will defer your question on whether your preprint falls into the scope of Medicine to [[User:Rwatson1955]], who is the managing editor for the Medicine journal. And yes, we [[WikiJournal of Medicine/Publishing#Duties_of_authors|ask that "authors should be given by real names in their articles"]] so there is no need to anonymize. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:58, 10 November 2022 (UTC) ::I submitted [[WikiJournal Preprints/Where experts and amateurs meet: the ideological hobby of medical volunteering on Wikipedia|my article]] two days ago and filled in a Google Form, which suggested I'd receive confirmation email, but nothing happened and the article still has a notice that it is not submitted for review. Any chance you could check from your end if things are fine or ping someone who can, as maybe I haven't clicked something correctly or such? [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 14:09, 16 November 2022 (UTC) :::{{re|Piotrus}} That's my fault. Been busy with work. I'll process the new submissions today and update the status. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:14, 16 November 2022 (UTC) == Concerning an article == Hello, I'm not sure if you are aware that I have written a new article on Wikiversity, entitled: [[WikiJournal Preprints/Orhan Gazi, the first statesman|Orhan Gazi, the first Statesman]], I started it in September 2022 and finished it in March of the same year, and I was hoping that finding some peer reviewers wouldn't take much time. However, the article remained as it was for more than a year, and I had to ask two professors I know personally to check my work, which they did and their notes were sent in pdf format and added [[Talk:WikiJournal Preprints/Orhan Gazi, the first statesman|here]]. Now the article still needs an editor, before it can be finalized and published, and a fellow Wikipedian, [[User:علاء|Alaa]], suggested your name. I hope that perhaps you could check it. Please let me know what you think, best wishes-- [[User:باسم|باسم]] ([[User talk:باسم|discuss]] • [[Special:Contributions/باسم|contribs]]) 20:17, 7 May 2023 (UTC) == Files Missing Information == Thanks for uploading files to Wikiversity. All files must have source and license information to stay at Wikiversity. The following files are missing {{tlx|Information}} and/or [[Wikiversity:License tags]], and will be deleted if the missing information is not added. See [[Wikiversity:Uploading files]] for more information. {{colbegin|3}} * [[:File:WikiJournal Bioclogging - ES.pdf]] {{colend}} [[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 15:41, 19 December 2023 (UTC) ==Japanese rendering== Thanks to your help, I could make [[WikiJournal_of_Science/Bioclogging/ja|Japanese translation of bioclogging article]]. I feel that display style of Japanese sentense is wierd, because breakline is restricted to some characters such as "、". Japanese does not break words with spaces, as normal in western languages, and therefore we break lines anywhere. For example, see [[w:ja:バイオクロッギング|Japanese edition of bioclogging article in Wikipedia]]. It can be fixed by using css. For example, in this paragraph バイオクロッギングは、水が浸透する様々な現場で観察される。たとえば、[[w:ja:ため池|ため池]]、浸透トレンチ、[[w:ja:灌漑|灌漑]]水路、[[w:ja:下水処理場|下水処理場]]、人工湿地、廃棄物処分場における遮水ライナー、川床や土壌のような自然環境などである。また、透過反応壁 ([[:w:Permeable reactive barrier|PRB]]) や微生物利用石油増進回収法 ([[:w:Microbial enhanced oil recovery|MEOR]]) などにおいて、[[w:ja:帯水層|帯水層]]における[[w:ja:地下水|地下水]]の流れにも影響を及ぼす。適度な水の浸透速度を保つことが必要とされるような現場では、バイオクロッギングが問題となり、定期的に水を抜くなどの対策が取られることがある。一方で、たとえば、難透水層を作って浸透速度を低下させたり、地盤工学的性質を改善させたりするなど、バイオクロッギングが有効に活用されることもある。 We can set word-break: break-all, and then <span style="word-break: break-all">バイオクロッギングは、水が浸透する様々な現場で観察される。たとえば、[[w:ja:ため池|ため池]]、浸透トレンチ、[[w:ja:灌漑|灌漑]]水路、[[w:ja:下水処理場|下水処理場]]、人工湿地、廃棄物処分場における遮水ライナー、川床や土壌のような自然環境などである。また、透過反応壁 ([[:w:Permeable reactive barrier|PRB]]) や微生物利用石油増進回収法 ([[:w:Microbial enhanced oil recovery|MEOR]]) などにおいて、[[w:ja:帯水層|帯水層]]における[[w:ja:地下水|地下水]]の流れにも影響を及ぼす。適度な水の浸透速度を保つことが必要とされるような現場では、バイオクロッギングが問題となり、定期的に水を抜くなどの対策が取られることがある。一方で、たとえば、難透水層を作って浸透速度を低下させたり、地盤工学的性質を改善させたりするなど、バイオクロッギングが有効に活用されることもある。</span> Setting this to all paragraphs may be a solution. I would like to know if there is a smarter way to do the same thing. [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 08:55, 16 February 2024 (UTC) :@[[User:Katsutoshi Seki|Katsutoshi Seki]] Thanks for raising this issue. I can read and write in Chinese (and therefore I can read Japanese Kanji) so I understand what you're describing about the software not finding spaces to break up words to the next line. I have [https://en.wikiversity.org/w/index.php?title=WikiJournal_of_Science%2FBioclogging%2Fja&diff=2606145&oldid=2605982 forced] the software to consider appropriate line break locations. I'm confident with the line breaks in Kanji but less so in Katakana and Hiragana. And I don't know how it may look like under different computer screens (or mobile phone). Please review and see if the line breaks are done accurately. Also, can you please provide a Japanese translation for the phrases "For the English translation, please see this link." and "For the Japanese translation, please see this link."? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:49, 16 February 2024 (UTC) :: Unfortunately, giving <nowiki>{{wbr}}</nowiki> to some places does not help much, because appropriate place for breaking line changes to various width of windows. Therefore, using <nowiki><span style="word-break: break-all"></nowiki> to all paragraphs, as I showed above, is necessary. I would like to know if there is an appropriate way to change the stylesheet in the page at once. For the translation, "For the English translation, please see '''this link'''."　to "英語版は'''このリンク'''参照", and "For the Japanese translation, please see '''this link'''." to "日本語版は'''このリンク'''参照" [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 01:39, 17 February 2024 (UTC) :::Thanks for verifying. I have removed {{tl|wbr}} and added <nowiki><span style="word-break: break-all"></nowiki>. It doesn't seem very effective to bulleted items. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:35, 17 February 2024 (UTC) :::: I also added css to bulleted items. Now it works find. [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 04:50, 17 February 2024 (UTC) :::: I created [[Template:BreakAll]] and applied. ChatGPT was helpful for creating the LUA module. [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 12:55, 17 February 2024 (UTC) == Article progress == Hi Ohana, it was great to meet you at the conference in November. I finally got around to finishing the revisions for [[WikiJournal Preprints/The Holocaust in Slovakia]]. As we discussed, I didn't expand the scope of the article to include Romani people, and I was unable to implement some of reviewer #2's comments because the information that would clarify is not in the cited source, or any other source that I'm aware of. Sorry for the very long delay on this article and I apologize if this is not the right forum to report progress. [[User:Buidhe|Buidhe]] ([[User talk:Buidhe|discuss]] • [[Special:Contributions/Buidhe|contribs]]) 03:45, 21 February 2024 (UTC) :Hi @[[User:Buidhe|Buidhe]], our apologies for the very long delay in replying to you. [[User:Fransplace|Fransplace]], the editor-in-chief for WikiJournal of Humanities, will be looking at your submission shortly. Since we already received two reviewers' comments and you have completed your revisions, are you ok with continuing with the submission process? I think we are on the home stretch with very few items remaining. Can you add your comments to the reviews to mark which items you have completed and which ones you cannot implement? This will speed up the review process. It probably will not take long for Fransplaces to render her publication decision once she has gone through the comments and your rebuttals. Many thanks for your patience! [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 23:06, 26 March 2025 (UTC) == Mail == {{ygm}} [[User:Serial Number 54129|Serial Number 54129]] ([[User talk:Serial Number 54129|discuss]] • [[Special:Contributions/Serial Number 54129|contribs]]) 12:04, 26 March 2024 (UTC) ==new submissions/need to be imported== Hi, I noticed there are two new submissions (from new editors) at https://en.wikipedia.org/wiki/Wikipedia:WikiJournal_article_nominations, thank you --[[User:Ozzie10aaaa|Ozzie10aaaa]] ([[User talk:Ozzie10aaaa|discuss]] • [[Special:Contributions/Ozzie10aaaa|contribs]]) 11:59, 1 April 2024 (UTC) :I don't have the required permission to import articles from Wikipedia to Wikiversity. I will need the "transwiki importer" permission, presumably to preserve article history and proper copyright attribution. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:40, 15 April 2024 (UTC) ==A message from Guy vandegrift== Hi. I am so-called "founder" of the WikiJournal of Science (although dozens of people contributed much more than I ever did.) I was wondering if the WikiJournal project needs help. If so, let me know.----[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:25, 13 April 2024 (UTC) :Yes, I'll email you with the details. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:58, 15 April 2024 (UTC) ::@[[User:Guy vandegrift|Guy vandegrift]] Did you receive the email that I sent last week? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 18:17, 22 April 2024 (UTC) :::I will look for it.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:17, 22 April 2024 (UTC). ::::My guess is that you used the google wikijournal system and it went to a google email I rarely check. I just sent you an email through Wikiversity. Meanwhile I will lookup my google email password and probably find your message.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:35, 22 April 2024 (UTC) == [[WikiJournal_Preprints/Induced_stem_cells]] == Hello, I assume that you are involved in the management of Wikijournals and their preprints. Thank you for your contributions. I'm sending this message to alert you that a preprint is currently subject to copyright-related investigations, this may affect the preprint review procedure and I thought someone who knows more about Wikijournals should be contacted. The background information can be seen at [[Wikiversity:Request_custodian_action#Induced_stem_cells_copyright_issues]]. In your opinion, what should be done by the custodians for this preprint? I look forward to hearing from you. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:02, 6 June 2024 (UTC) :Thanks for bringing this to our attention. What you described is very concerning. We did [[Talk:WikiJournal Preprints/Induced stem cells#Plagiarism check|conduct a plagiarism check]] 3 years ago when the preprint was submitted and it was determined that the similarities were deemed to be common phases in that field. Right now the tool is timing out due to high request volume so I can't do another check now. I'm going to ping @[[User:Evolution and evolvability|Evolution and evolvability]] since he's the handling editor for this submission and he knows more about cells & proteins than me. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:15, 6 June 2024 (UTC) == Question about the WikiJournal license status == Hello. At [[Special:Diff/2639304]], [[User:MGA73]] asked about the Wikijournal license status, so I'm forwarding the question here. Do you know anything about this? Should we contact [[User:Evolution and evolvability]]? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 08:02, 30 July 2024 (UTC) == Preprint related to Wikidata == Hello! I have written an article titled "[[WikiJournal Preprints/Is there a relationship between volcanoes and earthquakes based on Wikidata?|Is there a relationship between volcanoes and earthquakes based on Wikidata?]]". Could you please include this preprint in the list of [[WikiJournal of Science/Potential upcoming articles|Potential upcoming articles]]? -- [[User:AKA MBG|Andrew Krizhanovsky]] ([[User talk:AKA MBG|discuss]] • [[Special:Contributions/AKA MBG|contribs]]) 14:17, 17 February 2025 (UTC) :@[[User:AKA MBG|AKA MBG]] Hello, not sure why I didn't get a notification when you leave this message. I have taken a look at your preprint. Unfortunately I don't think we have the expertise in our editorial board to take on the role for potential publication of your submission. As a general and personal comment, I think you need to tighten up the paper by drawing comparison with existing literature around SPARQL and Wikidata, such as [https://link.springer.com/chapter/10.1007/978-3-319-46547-0_10] and [https://link.springer.com/chapter/10.1007/978-3-031-33455-9_40] [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 02:00, 24 March 2025 (UTC) == Status of WikiJournals == Good morning, I have had an article submitted to WikiJournal PrePrints since October 2024. It seems that the chair of the WikiJournal Usergroup (E&E) is entirely inactive, and I'm not sure what your status is as editor-in-chief of the science journal. If these projects are not currently working, then there should be some kind of alert given so people don't submit articles that will never be reviewed. If they are currently working, please let me know what the next steps in the process are for my submitted article. If there is any way I can help with other articles as well, I am happy to do so. [[User:Fritzmann2002|Fritzmann2002]] ([[User talk:Fritzmann2002|discuss]] • [[Special:Contributions/Fritzmann2002|contribs]]) 14:00, 6 March 2025 (UTC) :@[[User:Fritzmann2002|Fritzmann2002]] Hello, it has been busy for many of us at the board over the past few months focusing on the grant request and sustainability of the user group, and all of us serving in volunteer capacity with a daytime job. I should have a handling editor for your submission ([[WikiJournal Preprints/Hypericum sechmenii]]) within 2 weeks. Thanks. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 01:45, 24 March 2025 (UTC) ::@[[User:OhanaUnited|OhanaUnited]], thanks for your response, and apologies for the brusque nature of my original message. I appreciate the work that you do, and want to reiterate my desire to assist in any way that I can! [[User:Fritzmann2002|Fritzmann2002]] ([[User talk:Fritzmann2002|discuss]] • [[Special:Contributions/Fritzmann2002|contribs]]) 01:45, 25 March 2025 (UTC) == [[WikiJournal of Psychology, Psychiatry and Behavioral Sciences]] == Hi OhanaUnited, I'm planning on working on a paper for the WikiJournal of PPB regarding mental health in Sri Lanka (which does not seem to have a corresponding Wikipedia article, so I think this would be a very good start; especially as an aspiring clinical PhD student). I wanted to double check and make sure that this WikiJournal has personnel that can peer-review the article for submission, as there seems to be [[WikiJournal of PPB/Editors|no associate editors]] and the social medias (FB & X accounts) for this specific WikiJournal do not exist [anymore?]. Is this WikiJournal still active and can editors be assigned to my paper once its ready for peer-review? Thank you & thank you to the team for all the work you guys do! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:57, 8 May 2025 (UTC) :Hi, unfortunately I don't have any updates for WikiJournal of PPB on its launch date since the person in charge is on extended absence. I would recommend that you select either WikiJournal of Medicine (since it's mental health) or select another journal with compatible copyright license to publish. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:16, 8 May 2025 (UTC) ::I'll work on this paper through the WikiJournal of Medicine then, thanks! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:51, 8 May 2025 (UTC) :::No problem. Thanks for your ongoing support of the journal. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:30, 9 May 2025 (UTC) == WikiJournal article nominations == Hi OhanaUnited. More than 5 months ago I have nominated the page [[w:Diffeology|Diffeology]] for submission at the Wikijournal of Science, adding a line at the bottome of the page [[w:Wikipedia:WikiJournal article nominations|Wikipedia:WikiJournal article nominations]]. Unfortunately, nobody has created the corresponding preprint at [[WikiJournal Preprints|Wikijournal Preprints]], hence I cannot proceed yet with the formal submission. Since I had already a very positive experience publishing another paper ([[WikiJournal of Science/Poisson manifold|Poisson manifold]]) in the Wikijournal of Science, in the past months I tried, without success, to contact by email the editors who took care of it. I am therefore trying to reach you here. As I wrote also to them, I noticed that at [[w:Wikipedia:WikiJournal article nominations|Wikipedia:WikiJournal article nominations]] there are links to several other wikipedia pages which have not been converted to a preprint, despite being many months old. I am therefore wondering if that page is still maintained and with which frequency. This issue was also discussed on [[Talk:WikiJournal User Group#Wikipedia:WikiJournal article nominations is dead]]. I understand that you and the rest of the editorial board has a lot to do and therefore it might be just a matter of waiting. As another user pointed out ([[User talk:OhanaUnited#Status of WikiJournals]]), if there is anything I could do in order to speed up the review process, e.g. creating the preprint page myself, please let me know. In that case (i.e. if the author is allowed to import the page directly from wikipedia), I would suggest to clarify it in [[WikiJournal User Group/Editorial guidelines#Importing from Wikipedia]], since these instructions do not specify exactly who is in charge of importing the page. Thanks a lot in advance! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 10:08, 16 September 2025 (UTC) :Hi, an update. @[[User:Marshallsumter|Marshallsumter]] has suggested me in the nomination page to proceed with the import myself. As per our discussion in [[wikipedia:User_talk:Marshallsumter#Importing_Wikipedia_articles_to_Wikipreprints|User_talk:Marshallsumter#Importing_Wikipedia_articles_to_Wikipreprints]], I did attempt to import the page manually at [[WikiJournal Preprints/Diffeology]] and filled in the Authorship declaration form (providing the authors information, suggesting reviewers, etc. and mentioning also that I did the import manually). :One issue is that [[Template:Convert links]] has been deactivated just a few days ago, preventing all the links to other Wikipedia pages to be automatically converted. Since this was the only method written in [[WikiJournal User Group/Editorial guidelines#Importing from Wikipedia]], do you know if there are some alternatives, in order to avoid to do it manually? Besides that, I'm also not sure how to make the line "Additional contributors: Wikipedia community" appear under the two names of the authors. :I would appreciate if you or somebody from the editorial board could have a look at these minor issues, so that the review process could start soon. Thanks again! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 22:41, 21 September 2025 (UTC) ::@[[User:Francesco Cattafi|Francesco Cattafi]] Sorry for the late reply. Did the {{tl|Convert links}} end up working again? I see that the links are present. These functions were created long before I joined so I wouldn't be able to troubleshoot them. Sometimes I find that the bugs end up being caused by the most innocent changes in the back end, just like what I encountered [[Wikiversity:Colloquium#Figure numbers are always 1|two weeks ago]]. In the future, if you have some templates or links that aren't working, post a message on [[Wikiversity:Colloquium]] and someone with more knowledge than me may have a solution ready. In related news, there are now two peer review comments which are posted on [[Talk:WikiJournal Preprints/Diffeology]]. I think {{u|Marshallsumter}} is still looking for at least one more peer reviewer. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:01, 9 January 2026 (UTC) :::Hi @[[User:OhanaUnited|OhanaUnited]], thanks for the reply, I didn't know about this Colloquium page. Anyways, the Convert link issue was fixed; I have simply asked the user who deleted that tool to undelete it ([[User_talk:Koavf#Deleting_all_unused_templates]]), so I could use it properly. :::In the coming weeks my coauthor and I will address the two reviewers' comment! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 15:53, 10 January 2026 (UTC) ::::Thanks for your diligence and troubleshoot why it didn't work! [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:12, 2 February 2026 (UTC) :::::Hi, just to let you know that we have addressed all three reviewers' comments. Please let us know if any further revisions are needed or if the article will proceed to the next stage of the editorial process. [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 23:09, 3 March 2026 (UTC) ::::::In case you missed it, your article has been published last week and DOI has been issued. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:47, 28 April 2026 (UTC) == Found a potential reviewer == Hello @[[User:OhanaUnited|OhanaUnited]] I hope you are doing well. I write you because some weeks ago, I found a potential reviewer for [[WikiJournal Preprints/Kinematics of the cuboctahedron]] (as we talk about [[Talk:WikiJournal of Science#c-OhanaUnited-20260109204800-Regliste-20260106112200|here]]) and I sent you a mail about it. I'd like to be sure that you indeed received it.<br> On another topic, do you know if there is any progresses on [[WikiJournal Preprints/Pentagram map|my preprint]] ? Best regards, [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 16:51, 1 February 2026 (UTC) :Thanks for the reminder. I have emailed you about it. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:17, 2 February 2026 (UTC) ::Hello @[[User:OhanaUnited|OhanaUnited]], ::If you have the time, could you answer to my email about the subjects mentioned above, please ? ::Best regards, [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 17:21, 3 May 2026 (UTC) :::(Gentle reminder of my previous message.) [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 12:43, 15 May 2026 (UTC) ::::@[[User:Regliste|Regliste]] Only one reviewer accepted my invite to peer review Kinematics of the cuboctahedron. I will send another batch of review invitations later this week. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 14:55, 16 June 2026 (UTC) == Answered to reviewers == Hello @[[User:OhanaUnited|OhanaUnited]], just to inform you that I replied to the three reviewers comments. I don't know if other reviews are on the way, but in any case I remain available for the continuation of the editorial process {{=)}}. [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 14:46, 16 June 2026 (UTC) :I have one more review for Pentagram map that I'm expecting, but the review is not due for another 6 days. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 14:56, 16 June 2026 (UTC) ::Alright, there is no hurry at all. Thank you for the information. [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 14:58, 16 June 2026 (UTC) :::Hi {{ping|OhanaUnited}}, I'm the author of [[WikiJournal_of_Science/Affine_symmetric_group]], and have been thinking of submitting an article based on my recent work on [https://en.wikipedia.org/wiki/Hyperoctahedral_group]. It seems like you have a bit of a backlog, with two mathematics articles pending ([[Talk:WikiJournal_Preprints/Kinematics_of_the_cuboctahedron]] and [[Talk:WikiJournal_Preprints/Pentagram_map]]). Suppose I had a small amount of time to spend to help move one of those two articles towards resolution, so you aren't too over-burdened with math articles; what would be the best way to do that? [[User:JayBeeEll|Joel Brewster Lewis]] ([[User talk:JayBeeEll|discuss]] • [[Special:Contributions/JayBeeEll|contribs]]) 00:48, 21 July 2026 (UTC) == Greetings! == Hi Andrew! I am OwlyKnight <!-- Also known as NikolasKHF --> We met at Wikimania 2026. I am who interested in developing [[betawikiversity:Halaman Utama|Indonesian Wikiversity]], and, hopefully next, Indonesian WikiJournals. Nice to meet you! [[User:OwlyKnight|OwlyKnight]] ([[User talk:OwlyKnight|discuss]] • [[Special:Contributions/OwlyKnight|contribs]]) 08:57, 25 July 2026 (UTC) 0dh6jsxwruhx7u3u8oh75vxiufl0pnf 0 121359 2819293 2818565 2026-07-24T16:45:52Z Young1lim 21186 /* Data */ 2819293 wikitext text/x-wiki <!----------------------------------------------------------------------> == Flip Flop and Latch == * FFLatch.Overview.1.A ([[Media:FFLatch.Overview.1.A.20111103.pdf|pdf]]) * Counter.74LS193.1.A ([[Media:Counter.74LS193.1.A.20111108.pdf|pdf]]) * Clock.Overview.1.A ([[Media:Clock.Overview.1.A.20111108.pdf|pdf]]) * Function.Overview.1.A ([[Media:Function.Overview.1.A.20111201.pdf|pdf]]) <br> == Versions of VHDL == * VHDL Versions ([[Media:VHDL.1.A.Versions.20120619.pdf|pdf]]) * VHDL Libraries ([[Media:VHDL.1.A.Libraries.20140219.pdf|pdf]]) <br> == Basic Features of VHDL == ==== Data ==== * Data Objects ([[Media:Data.Object.1A.20260721.pdf|A]], [[Media:Data.Object.1B.20260602.pdf|B]]) * Data Types ([[Media:Data.Type.2A.20260602.pdf|A]], [[Media:Data.Type.2B.20260602.pdf|B]]) * Packages ([[Media:Data.Package.3A.20251206.pdf|pdf]]) * Signal Types ([[Media:Signal.Type.1A.20250614.pdf|pdf]]) * Attributes ([[Media:Data.4.A.Attribute.20251021.pdf|pdf]]) <br> ==== Signals & Variables ==== * Signals & Variables ([[Media:Signal.1A.SigVar.20250614.pdf|pdf]]) * Sequential Signal Assignments ([[Media:Signal.4A.Sequential.20250612.pdf|pdf]]) * Concurrent & Sequential Signal Assignments ([[Media:Signal.1.A.ConSeq.20120611.pdf|pdf]]) * Inertial & Transport Delay Models ([[Media:Signal.2.A.InertTrans.20120704.pdf|pdf]]) * Simulation & Synthesis ([[Media:Signal.3.A.SimSyn.20120504.pdf|pdf]]) <br> ==== Structure ==== * Component ([[Media:Struct.1.A.Component.20120804.pdf|pdf]]) * Configuration ([[Media:Struct.1.A.Configuration.20121003.pdf|pdf]]) * Generic ([[Media:Struct.1.A.Generic.20120802.pdf|pdf]]) </br> ==== Entity and Architecture ==== <br> ==== Block Statement ==== <br> ==== Process Statement ==== <br> ==== Operators ==== <br> ==== Assignment Statement ==== <br> ==== Concurrent Statement ==== <br> ==== Sequential Control Statement ==== <br> ==== Function ==== * Function.1.A Usage ([[Media:Function.1.A.Usage.20120611.pdf|pdf]]) * Function.2.A Conversion Function ([[Media:Function.2.A.Conversion.pdf|pdf]]) * Function.3.A Resolution Function ([[Media:Function.3.A.Resolution.pdf|pdf]]) <br> ==== Procedure ==== <br> ==== Package ==== </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:VHDL]] [[Category:FPGA]] 8rfig60ydopbvc63klq41e2fzdbcdfx 0 139384 2819271 2819137 2026-07-24T13:52:09Z Young1lim 21186 /* Adder */ 2819271 wikitext text/x-wiki == Adder == * Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]] ) {| class="wikitable" |- ! Adder type !! Overview !! Analysis !! VHDL Level Design !! CMOS Level Design |- | '''1. Ripple Carry Adder''' || [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]|| || [[Media:Adder.rca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]] |- | '''2. Carry Lookahead Adder''' || [[Media:VLSI.Arith.2A.CLA.20260722.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260724.pdf|B]], [[Media:VLSI.Arith.2C.CLA.20260724.pdf|C]], [[Media:VLSI.Arith.2D.CLA.20260720.pdf|D]] || || [[Media:Adder.cla.20140313.pdf|pdf]]|| |- | '''3. Carry Save Adder''' || [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]|| || || |- || '''4. Carry Select Adder''' || [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]|| || || |- || '''5. Carry Skip Adder''' || [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]|| || || [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]] |- || '''6. Carry Chain Adder''' || [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]|| || [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]], [[Media:Adder.cca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]] |- || '''7. Kogge-Stone Adder''' || [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]|| || [[Media:Adder.ksa.20140409.pdf|pdf]]|| |- || '''8. Prefix Adder''' || [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]|| || || |- || '''9.1 Variable Block Adder''' || [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]|| || || |- || '''9.2 Multi-Level Variable Block Adder''' || [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]|| || || |} </br> === Adder Architectures Suitable for FPGA === * FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]]) * FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]]) * FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]]) * FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]]) * Carry-Skip Adder </br> == Barrel Shifter == * Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]]) </br> '''Mux Based Barrel Shifter''' * Analysis ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) * Implementation </br> == Multiplier == === Array Multipliers === * Analysis ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]]) </br> === Tree Mulltipliers === * Lattice Multiplication ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]]) * Wallace Tree ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]]) * Dadda Tree ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]]) </br> === Booth Multipliers === * [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]] * Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]]) </br> == Divider == * Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])</br> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Digital Circuit Design]] [[Category:FPGA]] 4giewlovyjkdgfznninx4ywm9tx7atn 2 145726 2819289 2818587 2026-07-24T16:10:43Z Atcovi 276019 {{done}} 2819289 wikitext text/x-wiki ==Atcovi/to do== === Current Projects (2026) === * [[User:Atcovi/Journey to Clinical PhD]] - figuring this out; current life goal. * [[WikiJournal Preprints/Mental health in Sri Lanka]] (and later in August: [[User:Atcovi/APA2026 Abstract]]) {{Done}} ** [[User:Atcovi/WikiJournal Preprints/Mental health in Sri Lanka/Future Outlook]]. ====Suicidology/Psychopathology Works==== * Develop resources related to [[suicidology]] (3 stress response systems? effects of catecholamines on suicidal ideation? neurobiology of suicidal ideation? relation between autobiographical memory and suicide?), expand [[wikipedia:Suicidology#Theories_of_suicide|Suicidology#Theories_of_suicide]] either through [[WikiJournal of Science]] or WP editing. * [[WikiJournal Preprints/Suicide amongst refugees in Sweden]] [https://scholar.google.com/scholar?hl=en&as_sdt=0%2C47&as_ylo=2020&as_yhi=2025&q=Suicide+in+Sweden+refugees&btnG=] * Get [[User:Atcovi/Spring2024]] & [[User:Atcovi/Psychopathology]] into the mainspace. Develop [[Child psychology]] & [[User:Atcovi/PSYC318W]] into a complete course. Merge [[Validity]] into [[User:Atcovi/PSYC318W|PSYC318W]]. =====Wikiversity-Related Works===== * Promote [[Help:Project boxes]], something very useful and unique to Wikiversity. Focus on trying to not only create more project boxes, but to define resource types used in project boxes. **Ex, what is a [[:Category:Workshops|workshop]]? What differentiates between an [[Help:Essay|essay]] and a [[Help:Paper|paper]]? What differentiates between a [[Template:Notes|notes resource]] (that may be ''derived'' from a homework assignment) and a [[Help:Assignment|homework assignment]] [small note: this page seems to be created by accident and may need a revamp]? * [[Wikiversity:Original research and scholarly standards]] & improvements/proposals for [[Wikiversity:Original research]] (ex, [[Template:Original research]] should be a mandatory addition to original research on WV + a notice letting readers know that the work is not established science). Develop other pages related to research ethics, including [[Wikiversity:Research]] & [[Wikiversity:Research ethics]]. ** [[Wikiversity:Review board]] - should this be Wikiversity 'crats that review original research proposals? * [[Wikiversity:Verifiability]] - start heavily scrutinizing pages that don't meet this criteria. * [[Wikiversity:Artificial intelligence]] - "substantial"? What defines "substantial"? * Expand [[Wikiversity:Differences between Wikiversity and Wikipedia]]. {{Archive box| {{center top}}'''[[User:Atcovi/to do|To do list]]'''{{center bottom}} ---- {{center top}}'''Archives'''{{center bottom}} *[[User:Atcovi/to do/Current Projects/2026]] *[[User:Atcovi/to do/Current Projects/2023]] *[[User:Atcovi/to do/Current Projects/January 4, 2022]] *[[User:Atcovi/to do/Current Projects/September 2017 - January 2018]] *[[User:Atcovi/to do/Current Projects/2015]] ---- }} [[Category:Atcovi's Work]] lxvu6ryxj79r8uflh0yn9hy9quw95ku 0 171005 2819275 2819051 2026-07-24T14:02:06Z Young1lim 21186 /* Geometric Series Examples */ 2819275 wikitext text/x-wiki Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers <math>x+iy</math>, where <math>i=\sqrt{-1}</math>, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other <math>L</math>-functions, modular forms, elliptic functions, etc. <blockquote>The shortest path between two truths in the real domain passes through the complex domain. — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]</blockquote>In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}} ==''' Complex Functions '''== * Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]]) * Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]]) * Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]]) '''Complex Function Note''' : 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]]) : 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]]) : 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]]) ==''' Complex Integrals '''== * Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]]) ==''' Complex Series '''== * Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]]) ==''' Residue Integrals '''== * Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) ==='''Residue Integrals Note'''=== * Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]]) * Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]]) * Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]]) * Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]]) === Laurent Series and the z-Transform Example Note === * Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]]) ====Geometric Series Examples==== * Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) * Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) * Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) * Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) * Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) * Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260723.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) * Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) * Double Pole Case :- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) :- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]]) ====The Case Examples==== * Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]]) * Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]]) * Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]]) * Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]]) * Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]]) * Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]]) ==''' Conformal Mapping '''== * Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]]) go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Complex analysis]] pgyngi28rfaeruqfq1oyao5as7ea08g 2819277 2819275 2026-07-24T14:08:37Z Young1lim 21186 /* Geometric Series Examples */ 2819277 wikitext text/x-wiki Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers <math>x+iy</math>, where <math>i=\sqrt{-1}</math>, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other <math>L</math>-functions, modular forms, elliptic functions, etc. <blockquote>The shortest path between two truths in the real domain passes through the complex domain. — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]</blockquote>In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}} ==''' Complex Functions '''== * Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]]) * Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]]) * Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]]) '''Complex Function Note''' : 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]]) : 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]]) : 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]]) ==''' Complex Integrals '''== * Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]]) ==''' Complex Series '''== * Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]]) ==''' Residue Integrals '''== * Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) ==='''Residue Integrals Note'''=== * Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]]) * Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]]) * Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]]) * Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]]) === Laurent Series and the z-Transform Example Note === * Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]]) ====Geometric Series Examples==== * Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) * Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) * Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) * Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) * Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) * Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260724.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) * Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) * Double Pole Case :- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) :- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]]) ====The Case Examples==== * Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]]) * Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]]) * Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]]) * Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]]) * Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]]) * Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]]) ==''' Conformal Mapping '''== * Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]]) go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Complex analysis]] pbc1ul7w60oqkgk4sl32noqek4j00j1 103 206490 2819335 2425510 2026-07-25T07:01:49Z ~2026-41380-76 3103171 /* Improvements */ Reply 2819335 wikitext text/x-wiki == Improvements == How can we improve this portal? -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 04:16, 18 January 2016 (UTC) : {{ping|Dave Braunschweig}} Featured resource for this portal seems to be broken? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:32, 13 September 2022 (UTC) ::@[[User:Jtneill|Jtneill]] It looks like [[User:Evolution and evolvability]] replaced [[Module:Portal]] with a Wikipedia version that doesn't support the lead (or more accurately, lede) paragraph function. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:37, 14 September 2022 (UTC) ::Also, the WikiJournal of Medicine isn't a standard article and can't be imported as a dynamic lede paragraph. I've replaced it with a link. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:58, 14 September 2022 (UTC) :improve is can dealyed made a massh [[Special:Contributions/&#126;2026-41380-76|&#126;2026-41380-76]] ([[User talk:&#126;2026-41380-76|talk]]) 07:01, 25 July 2026 (UTC) == medicine ( corona virus and rota virus) == Can we please check the symptoms of a rota virus and of a corona virus . If there are any similarities maybe we may try to make a vaccine out of the medicine that cures rota virus. [[User:Nelo calieque|Nelo calieque]] ([[User talk:Nelo calieque|discuss]] • [[Special:Contributions/Nelo calieque|contribs]]) 20:10, 18 June 2020 (UTC) I only know the syptomps of corona virus but not of rota virus.Are the symptoms of rota virus the same or different from corona virus [[User:Kaylishka|Kaylishka]] ([[User talk:Kaylishka|discuss]] • [[Special:Contributions/Kaylishka|contribs]]) 15:03, 12 September 2021 (UTC) Rotavirus:is a contagious virus that cause inflammation of the stomach and intestines leading to diarrhea. Mostly in children younger than 5 years old. [[User:Indilomwene|Indilomwene]] ([[User talk:Indilomwene|discuss]] • [[Special:Contributions/Indilomwene|contribs]]) 05:03, 26 May 2022 (UTC) == COVID-19 == Does the vaccine cure the corona virus or does it make it to not spread.And how can we be sure that if a person vaccinated won't get infected again? {{unsigned|41.113.91.179|30 January 2021‎}} :See [[COVID-19]] for this subject. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 18:40, 2 February 2021 (UTC) I think it only stops it from spreading but not curing it,because if it cured covid-19 then many people would have recovered now. [[User:Kaylishka|Kaylishka]] ([[User talk:Kaylishka|discuss]] • [[Special:Contributions/Kaylishka|contribs]]) 15:06, 12 September 2021 (UTC) The vaccine as far as I know does not cure or stop the reinfection of a vaccinated person. It reduces the severity of the disease when a vaccinated person get infected and reduces hospitalisation. [[User:Indilomwene|Indilomwene]] ([[User talk:Indilomwene|discuss]] • [[Special:Contributions/Indilomwene|contribs]]) 04:20, 26 May 2022 (UTC) == Epilepsy == SEIZURES During a seizure you may move, see ,feel or do other things whether you want to or not! Also in some seizures, parts of the brain can still function normally while others cant. Seizures have a beginning, middle and the end, but sometimes not all the parts of a seizure is visible or easy to tell apart . [[User:Retsepile|Retsepile]] ([[User talk:Retsepile|discuss]] • [[Special:Contributions/Retsepile|contribs]]) 11:30, 11 November 2021 (UTC) ej4fv3wbcdbpzzbnpl8nvf7d447o83x 2819336 2819335 2026-07-25T07:09:53Z ~2026-41380-76 3103171 /* COVID-19 */ Reply 2819336 wikitext text/x-wiki == Improvements == How can we improve this portal? -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 04:16, 18 January 2016 (UTC) : {{ping|Dave Braunschweig}} Featured resource for this portal seems to be broken? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:32, 13 September 2022 (UTC) ::@[[User:Jtneill|Jtneill]] It looks like [[User:Evolution and evolvability]] replaced [[Module:Portal]] with a Wikipedia version that doesn't support the lead (or more accurately, lede) paragraph function. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:37, 14 September 2022 (UTC) ::Also, the WikiJournal of Medicine isn't a standard article and can't be imported as a dynamic lede paragraph. I've replaced it with a link. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:58, 14 September 2022 (UTC) :improve is can dealyed made a massh [[Special:Contributions/&#126;2026-41380-76|&#126;2026-41380-76]] ([[User talk:&#126;2026-41380-76|talk]]) 07:01, 25 July 2026 (UTC) == medicine ( corona virus and rota virus) == Can we please check the symptoms of a rota virus and of a corona virus . If there are any similarities maybe we may try to make a vaccine out of the medicine that cures rota virus. [[User:Nelo calieque|Nelo calieque]] ([[User talk:Nelo calieque|discuss]] • [[Special:Contributions/Nelo calieque|contribs]]) 20:10, 18 June 2020 (UTC) I only know the syptomps of corona virus but not of rota virus.Are the symptoms of rota virus the same or different from corona virus [[User:Kaylishka|Kaylishka]] ([[User talk:Kaylishka|discuss]] • [[Special:Contributions/Kaylishka|contribs]]) 15:03, 12 September 2021 (UTC) Rotavirus:is a contagious virus that cause inflammation of the stomach and intestines leading to diarrhea. Mostly in children younger than 5 years old. [[User:Indilomwene|Indilomwene]] ([[User talk:Indilomwene|discuss]] • [[Special:Contributions/Indilomwene|contribs]]) 05:03, 26 May 2022 (UTC) == COVID-19 == Does the vaccine cure the corona virus or does it make it to not spread.And how can we be sure that if a person vaccinated won't get infected again? {{unsigned|41.113.91.179|30 January 2021‎}} :See [[COVID-19]] for this subject. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 18:40, 2 February 2021 (UTC) I think it only stops it from spreading but not curing it,because if it cured covid-19 then many people would have recovered now. [[User:Kaylishka|Kaylishka]] ([[User talk:Kaylishka|discuss]] • [[Special:Contributions/Kaylishka|contribs]]) 15:06, 12 September 2021 (UTC) The vaccine as far as I know does not cure or stop the reinfection of a vaccinated person. It reduces the severity of the disease when a vaccinated person get infected and reduces hospitalisation. [[User:Indilomwene|Indilomwene]] ([[User talk:Indilomwene|discuss]] • [[Special:Contributions/Indilomwene|contribs]]) 04:20, 26 May 2022 (UTC) :An made in xmple old did id i degreed pain [[Special:Contributions/&#126;2026-41380-76|&#126;2026-41380-76]] ([[User talk:&#126;2026-41380-76|talk]]) 07:09, 25 July 2026 (UTC) == Epilepsy == SEIZURES During a seizure you may move, see ,feel or do other things whether you want to or not! Also in some seizures, parts of the brain can still function normally while others cant. Seizures have a beginning, middle and the end, but sometimes not all the parts of a seizure is visible or easy to tell apart . [[User:Retsepile|Retsepile]] ([[User talk:Retsepile|discuss]] • [[Special:Contributions/Retsepile|contribs]]) 11:30, 11 November 2021 (UTC) 3krobx7q2cjz14cwxqsrdrgtq52q5km 2819337 2819336 2026-07-25T07:12:28Z ~2026-41380-76 3103171 /* Epilepsy */ Reply 2819337 wikitext text/x-wiki == Improvements == How can we improve this portal? -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 04:16, 18 January 2016 (UTC) : {{ping|Dave Braunschweig}} Featured resource for this portal seems to be broken? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:32, 13 September 2022 (UTC) ::@[[User:Jtneill|Jtneill]] It looks like [[User:Evolution and evolvability]] replaced [[Module:Portal]] with a Wikipedia version that doesn't support the lead (or more accurately, lede) paragraph function. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:37, 14 September 2022 (UTC) ::Also, the WikiJournal of Medicine isn't a standard article and can't be imported as a dynamic lede paragraph. I've replaced it with a link. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:58, 14 September 2022 (UTC) :improve is can dealyed made a massh [[Special:Contributions/&#126;2026-41380-76|&#126;2026-41380-76]] ([[User talk:&#126;2026-41380-76|talk]]) 07:01, 25 July 2026 (UTC) == medicine ( corona virus and rota virus) == Can we please check the symptoms of a rota virus and of a corona virus . If there are any similarities maybe we may try to make a vaccine out of the medicine that cures rota virus. [[User:Nelo calieque|Nelo calieque]] ([[User talk:Nelo calieque|discuss]] • [[Special:Contributions/Nelo calieque|contribs]]) 20:10, 18 June 2020 (UTC) I only know the syptomps of corona virus but not of rota virus.Are the symptoms of rota virus the same or different from corona virus [[User:Kaylishka|Kaylishka]] ([[User talk:Kaylishka|discuss]] • [[Special:Contributions/Kaylishka|contribs]]) 15:03, 12 September 2021 (UTC) Rotavirus:is a contagious virus that cause inflammation of the stomach and intestines leading to diarrhea. Mostly in children younger than 5 years old. [[User:Indilomwene|Indilomwene]] ([[User talk:Indilomwene|discuss]] • [[Special:Contributions/Indilomwene|contribs]]) 05:03, 26 May 2022 (UTC) == COVID-19 == Does the vaccine cure the corona virus or does it make it to not spread.And how can we be sure that if a person vaccinated won't get infected again? {{unsigned|41.113.91.179|30 January 2021‎}} :See [[COVID-19]] for this subject. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 18:40, 2 February 2021 (UTC) I think it only stops it from spreading but not curing it,because if it cured covid-19 then many people would have recovered now. [[User:Kaylishka|Kaylishka]] ([[User talk:Kaylishka|discuss]] • [[Special:Contributions/Kaylishka|contribs]]) 15:06, 12 September 2021 (UTC) The vaccine as far as I know does not cure or stop the reinfection of a vaccinated person. It reduces the severity of the disease when a vaccinated person get infected and reduces hospitalisation. [[User:Indilomwene|Indilomwene]] ([[User talk:Indilomwene|discuss]] • [[Special:Contributions/Indilomwene|contribs]]) 04:20, 26 May 2022 (UTC) :An made in xmple old did id i degreed pain [[Special:Contributions/&#126;2026-41380-76|&#126;2026-41380-76]] ([[User talk:&#126;2026-41380-76|talk]]) 07:09, 25 July 2026 (UTC) == Epilepsy == SEIZURES During a seizure you may move, see ,feel or do other things whether you want to or not! Also in some seizures, parts of the brain can still function normally while others cant. Seizures have a beginning, middle and the end, but sometimes not all the parts of a seizure is visible or easy to tell apart . [[User:Retsepile|Retsepile]] ([[User talk:Retsepile|discuss]] • [[Special:Contributions/Retsepile|contribs]]) 11:30, 11 November 2021 (UTC) :taxm wi on i ans you nms see the great it cute [[Special:Contributions/&#126;2026-41380-76|&#126;2026-41380-76]] ([[User talk:&#126;2026-41380-76|talk]]) 07:12, 25 July 2026 (UTC) 9z0g9wuhjtlh9mjx4psqhv6a8q6vhmk 2819345 2819337 2026-07-25T09:19:41Z MathXplore 2888076 Reverted edits by [[Special:Contributions/~2026-41380-76|~2026-41380-76]] ([[User_talk:~2026-41380-76|talk]]) to last version by [[User:Dave Braunschweig|Dave Braunschweig]] using [[Wikiversity:Rollback|rollback]] 2425510 wikitext text/x-wiki == Improvements == How can we improve this portal? -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 04:16, 18 January 2016 (UTC) : {{ping|Dave Braunschweig}} Featured resource for this portal seems to be broken? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:32, 13 September 2022 (UTC) ::@[[User:Jtneill|Jtneill]] It looks like [[User:Evolution and evolvability]] replaced [[Module:Portal]] with a Wikipedia version that doesn't support the lead (or more accurately, lede) paragraph function. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:37, 14 September 2022 (UTC) ::Also, the WikiJournal of Medicine isn't a standard article and can't be imported as a dynamic lede paragraph. I've replaced it with a link. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:58, 14 September 2022 (UTC) == medicine ( corona virus and rota virus) == Can we please check the symptoms of a rota virus and of a corona virus . If there are any similarities maybe we may try to make a vaccine out of the medicine that cures rota virus. [[User:Nelo calieque|Nelo calieque]] ([[User talk:Nelo calieque|discuss]] • [[Special:Contributions/Nelo calieque|contribs]]) 20:10, 18 June 2020 (UTC) I only know the syptomps of corona virus but not of rota virus.Are the symptoms of rota virus the same or different from corona virus [[User:Kaylishka|Kaylishka]] ([[User talk:Kaylishka|discuss]] • [[Special:Contributions/Kaylishka|contribs]]) 15:03, 12 September 2021 (UTC) Rotavirus:is a contagious virus that cause inflammation of the stomach and intestines leading to diarrhea. Mostly in children younger than 5 years old. [[User:Indilomwene|Indilomwene]] ([[User talk:Indilomwene|discuss]] • [[Special:Contributions/Indilomwene|contribs]]) 05:03, 26 May 2022 (UTC) == COVID-19 == Does the vaccine cure the corona virus or does it make it to not spread.And how can we be sure that if a person vaccinated won't get infected again? {{unsigned|41.113.91.179|30 January 2021‎}} :See [[COVID-19]] for this subject. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 18:40, 2 February 2021 (UTC) I think it only stops it from spreading but not curing it,because if it cured covid-19 then many people would have recovered now. [[User:Kaylishka|Kaylishka]] ([[User talk:Kaylishka|discuss]] • [[Special:Contributions/Kaylishka|contribs]]) 15:06, 12 September 2021 (UTC) The vaccine as far as I know does not cure or stop the reinfection of a vaccinated person. It reduces the severity of the disease when a vaccinated person get infected and reduces hospitalisation. [[User:Indilomwene|Indilomwene]] ([[User talk:Indilomwene|discuss]] • [[Special:Contributions/Indilomwene|contribs]]) 04:20, 26 May 2022 (UTC) == Epilepsy == SEIZURES During a seizure you may move, see ,feel or do other things whether you want to or not! Also in some seizures, parts of the brain can still function normally while others cant. Seizures have a beginning, middle and the end, but sometimes not all the parts of a seizure is visible or easy to tell apart . [[User:Retsepile|Retsepile]] ([[User talk:Retsepile|discuss]] • [[Special:Contributions/Retsepile|contribs]]) 11:30, 11 November 2021 (UTC) 6sdws32cztcx95xyhberaeiplipikxw 0 217152 2819327 2819240 2026-07-25T01:09:20Z Evan Mercer 3071189 2819327 wikitext text/x-wiki [[File:Quadratic formula.svg|thumb|right|An example of a Algebra formula (quadratic formula)]] {{mathematics}} {{secondary education}} {{lesson}} {{complete}} '''Algebra''' (from the Arabic word "al-jabr" (الجبر), meaning "reunion of broken parts") can feel like quite a complicated language of mathematics. However, as time goes on, completing Algebra will get easier and easier until it's a breeze. Completing Algebra takes true dedication with a worthwhile reward. This week, we will get into what Algebra is, and some warm ups (on arithmetic). Even though this may seem pointless, it is <small>IMPORTANT</small> that you review through these warm ups and get comfortable in solving them to lay a strong foundation for understanding larger topics later on. Without further do, let's dig right into this! ==Algebra== ===What is Algebra?=== [[File:AlgebraJournalWork11-14-16.jpg|thumb|left|You might have to do this much work for a small answer!]]In Algebra, we use letters to represent number or a amount of something that is not known yet. This called a '''pronumeral''' or a '''variable.''' Imagine you have a bag full of jellybeans on a table; with 10 green jelly beans and a unknown amount of blue jellybeans. Let's call the blue jellybeans x. Well done, this is a pronumeral they are that simple. Now, your friend comes over, tells you there is 20 total jellybeans in the bag. How many blue jellybeans are there? The core concept of algebra is the equal sign (=). Think of an equation as a balanced scale. Whatever you do to one side, you must do to the other side to keep it balanced. To find the unknown number, you need to get the letter completely by itself. You do this by using inverse operations (doing the opposite). * Addition (+) and Subtraction (-) are opposites. * Multiplication (times*, ×) and Division÷, /) are opposites. To work out the number of blue jellybeans follow these steps: x+10=20 # Identify the goal: We want x by itself. # See the obstacle: There is a +10 next to the x. # Now take away ten and add the opposite to the other side of the equation. x=20-10 4. Do 20-10 x=10 Good job. Checking Your Work In maths, you should always check your work. You can do this by working if the original equation equals the same number now that you know the pronumeral or variable. Does (10 + x (10) = 20)? Yes! Your answer is correct. '''Example two''' In algebra, a fraction line means division. So, this equation means "x (unknown number) divided by 4 equals 3." <math>\tfrac{x}{4}</math> = 3 # Identify the goal: Get x by itself. # See the obstacle: The x is being divided by 4. # Do the opposite: The opposite of division is multiplication. Multiply both sides by 4, because whatever you do to one side you must do to the other to balance the equation x = 3*4 x=12 Check Your Work Put 12 back into the original equation: * Does 12 / 4 = 3? Yes! The answer is correct. ===== '''Important notes:''' ===== * Can be called variable OR pronumeral. * And can be any letter from a to z. * Solving for (pronumeral here) e.g., "solving for x" means finding the pronumeral. ====== '''Fun Fact''' ====== The letter that is most commonly used for variables is x and the reason for this dates back to the origin of Algebra itself; Muhammad ibn Musa al-Khwarizmi, often called one of the main "founders" of Algebra you could say, used to call the unknown pronumeral "'''shay'''". "'''Shay"''' comes from the Arabic word '''شَيْء''', which essentially means "thing". When Al-Khwarizmi's works were translated to Latin in medieval Spain, "shay" was translated as '''"xay",''' since the letter x was pronounced as "sh" in Spain. Later on, this word "'''xay"''' got abbreviated to "'''x"''' to represent the symbol of the unknown, so we normally use x for standard questions. For more information, visit this [https://www.pbs.org/empires/islam/innoalgebra.html PBS] page. = Algebra problems = Solve for x.<quiz display="simple" points="1/1"> {''x'' − 9 = 20 |type="{}"} ''x''={ 29_3 } {''x'' − 3 = 6 |type="{}"} ''x''={ 9_3 } {''x'' + 5 = 15 |type="{}"} ''x''={ 10_3 } {''x'' + 17 = 23 |type="{}"} ''x''={ 6_3 } {4''x'' = 12 |type="{}"} ''x''={ 3_3 } {''x''/2 = 0.5 |type="{}"} ''x''={ 1_3 } {''x''/50 = 2 |type="{}"} ''x''={ 100_3 } {''x''/9 = 5 |type="{}"} ''x''={ 45_3 } </quiz> Seems simple, huh? Well, it will get complicated, which is why it is important for you to do some review of your arithmetic! Let's dig into that... Arithmetic was moved to [https://en.wikiversity.org/w/index.php?title=Algebra_1/Arithmetic] Not checked. {{subpage navbar}} [[Category:Speak Math Now!]] pzo8hpvsqvknq5zq6pc23cgcxr421e2 2819328 2819327 2026-07-25T01:11:11Z Evan Mercer 3071189 2819328 wikitext text/x-wiki [[File:Quadratic formula.svg|thumb|right|An example of a Algebra formula (quadratic formula)]] {{mathematics}} {{secondary education}} {{lesson}} {{complete}} '''Algebra''' (from the Arabic word "al-jabr" (الجبر), meaning "reunion of broken parts") can feel like quite a complicated language of mathematics. However, as time goes on, completing Algebra will get easier and easier until it's a breeze. Completing Algebra takes true dedication with a worthwhile reward. This week, we will get into what Algebra is, and some warm ups (on arithmetic). Even though this may seem pointless, it is <small>IMPORTANT</small> that you review through these warm ups and get comfortable in solving them to lay a strong foundation for understanding larger topics later on. Without further do, let's dig right into this! ==Algebra== ===What is Algebra?=== [[File:AlgebraJournalWork11-14-16.jpg|thumb|left|You might have to do this much work for a small answer!]]In Algebra, we use letters to represent number or a amount of something that is not known yet. This called a '''pronumeral''' or a '''variable.''' Imagine you have a bag full of jellybeans on a table; with 10 green jelly beans and a unknown amount of blue jellybeans. Let's call the blue jellybeans x. Well done, this is a pronumeral they are that simple. Now, your friend comes over, tells you there is 20 total jellybeans in the bag. How many blue jellybeans are there? The core concept of algebra is the equal sign (=). Think of an equation as a balanced scale. Whatever you do to one side, you must do to the other side to keep it balanced. To find the unknown number, you need to get the letter completely by itself. You do this by using inverse operations (doing the opposite). * Addition (+) and Subtraction (-) are opposites. * Multiplication (times*, ×) and Division÷, /) are opposites. To work out the number of blue jellybeans follow these steps: x+10=20 # Identify the goal: We want x by itself. # See the obstacle: There is a +10 next to the x. # Now take away ten and add the opposite to the other side of the equation. x=20-10 4. Do 20-10 x=10 Good job. Checking Your Work In maths, you should always check your work. You can do this by working if the original equation equals the same number now that you know the pronumeral or variable. Does (10 + x (10) = 20)? Yes! Your answer is correct. '''Example two''' In algebra, a fraction line means division. So, this equation means "x (unknown number) divided by 4 equals 3." <math>\tfrac{x}{4}</math> = 3 # Identify the goal: Get x by itself. # See the obstacle: The x is being divided by 4. # Do the opposite: The opposite of division is multiplication. Multiply both sides by 4, because whatever you do to one side you must do to the other to balance the equation x = 3*4 x=12 Check Your Work Put 12 back into the original equation: * Does 12 / 4 = 3? Yes! The answer is correct. ===== '''Important notes:''' ===== * Can be called variable OR pronumeral. * And can be any letter from a to z. * Solving for (pronumeral here) e.g., "solving for x" means finding the pronumeral. ====== '''Fun Fact''' ====== The letter that is most commonly used for variables is x and the reason for this dates back to the origin of Algebra itself; Muhammad ibn Musa al-Khwarizmi, often called one of the main "founders" of Algebra you could say, used to call the unknown pronumeral "'''shay'''". "'''Shay"''' comes from the Arabic word '''شَيْء''', which essentially means "thing". When Al-Khwarizmi's works were translated to Latin in medieval Spain, "shay" was translated as '''"xay",''' since the letter x was pronounced as "sh" in Spain. Later on, this word "'''xay"''' got abbreviated to "'''x"''' to represent the symbol of the unknown, so we normally use x for standard questions. For more information, visit this [https://www.pbs.org/empires/islam/innoalgebra.html PBS] page. = Algebra problems = Solve for x.<quiz display="simple" points="1/1"> {''x'' − 9 = 20 |type="{}"} ''x''={ 29_3 } {''x'' − 3 = 6 |type="{}"} ''x''={ 9_3 } {''x'' + 5 = 15 |type="{}"} ''x''={ 10_3 } {''x'' + 17 = 23 |type="{}"} ''x''={ 6_3 } {4''x'' = 12 |type="{}"} ''x''={ 3_3 } {''x''/2 = 0.5 |type="{}"} ''x''={ 1_3 } {''x''/50 = 2 |type="{}"} ''x''={ 100_3 } {''x''/9 = 5 |type="{}"} ''x''={ 45_3 } </quiz> Seems simple, huh? Well, it will get complicated, which is why it is important for you to do some review of your arithmetic! Let's dig into that... Arithmetic was moved to [https://en.wikiversity.org/w/index.php?title=Algebra_1/Arithmetic] Not checked. {{subpage navbar}} [[Category:Speak Math Now!]] ea6fo6kb6liymbndfql0wur7fxksib8 0 238631 2819248 2819245 2026-07-24T12:02:52Z Andy?yes 3006471 2819248 wikitext text/x-wiki '''OpenStax''' (formerly OpenStax College) is a nonprofit ed-tech initiative based at Rice University. Since 2012, OpenStax has created peer-reviewed, openly licensed textbooks, which are available as free downloadable PDFs, web versions, audiobooks<ref>{{Cite web|url=https://openstax.org/blog/guest-post-how-audio-technology-is-creating-more-inclusive-learning|title=OpenStax {{!}} How audio technology is creating more inclusive learning|website=openstax.org|language=en-US|access-date=2025-10-20}}</ref> and for a low cost in print. 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Although some versions found on Wikiversity are out-of-date, some might find them more convenient to access. *[[OpenStax American Government 3e]] *[[OpenStax American Government 4e]] *[[OpenStax University Physics|OpenStax University Physics (click to visit)]] *[[OpenStax College Physics|OpenStax College Physics (click to visit)]] *[[OpenStax Astronomy|OpenStax Astronomy (click to visit)]] *[[OpenStax Astronomy 2e|OpenStax Astronomy 2e (click to visit)]] *[[OpenStax Anatomy and Physiology 2e|OpenStax Anatomy & Physiology 2e]] *[[OpenStax Biology 2e]] *[[OpenStax Business Ethics]] *[[OpenStax Clinical Nursing Skills]] *[[OpenStax College Success Concise]] *[[OpenStax Concepts of Biology]] *[[OpenStax Introduction to Anthropology]] *[[OpenStax Introduction to Business]] *[[OpenStax Introduction to Business 2e]] *[[OpenStax Introduction to Political Science]] *[[OpenStax Introduction to Sociology 3e]] *[[OpenStax Lifespan Development]] *[[OpenStax Nutrition for Nurses]] *[[OpenStax Organizational Behavior]] *[[OpenStax Principles of Economics 3e]] *[[OpenStax Principles of Macroeconomics 3e]] *[[OpenStax Principles of Microeconomics 3e]] *[[OpenStax Psychology 2e]] *[[OpenStax US History]] *[[OpenStax world history volume 1 to 1500|OpenStax World History, Volume 1: to 1500]] *'''OpenStax Calculus: ''' No resources have been developed, but (out-of-date) pdf versions of the three volume textbook are posted on Wikiversity at: '''[[:File:CalculusVolume1-OP.pdf|V1]]''' | '''[[:File:CalculusVolume2-OP.pdf|V2]]''' | '''[[:File:CalculusVolume3-LR.pdf|V3]]''' == See Also == * [[Wikipedia: OpenStax]] * [https://openstax.org/ OpenStax.org] * [https://audileo.com/ Official OpenStax Audio Textbooks] *[[:Category:openstax textbook]] *[https://www.facebook.com/openstax/ OpenStax Facebook page] * [https://www.ted.com/talks/richard_baraniuk_the_birth_of_the_open_source_learning_revolution TED Talk dated 2006-02] Founder Richard Baraniuk discussing Connexions *[[Quizbank]] * [https://www.youtube.com/watch?v=Xog2X2SnjvQ YouTube: Importing OpenStax content into Pressbooks] == References == {{reflist}} {{subpages/List}} [[category:openstax file]] [[Category:Quizbank]] dol19e08ozfls4vcejownxe6fzr2r5b 2 250295 2819343 2818182 2026-07-25T08:17:13Z Platos Cave (physics) 2562653 2819343 wikitext text/x-wiki {{Original research}} α, Ω, π, e '''Simulation universe modelling at the Planck scale''' The model is constructed around geometrical Planck objects, the following is a handy reference guide. {{main|User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_units_(geometrical)}} <math>\alpha^{-1} = 137.0359931388</math>, ... <math>\Omega = \sqrt{ \left(\pi^e e^{(1-e)}\right)} = 2.0071349543... </math> <math>i = \Omega^{15}</math>, ... <math>k = \frac{i\;u^{15}\; r^4}{v}</math>, ... <math>t = \frac{i^{-2}\; u^{-30}\; r^9}{v^6}</math> <math>v = 11843707.84994 ...,\; units = \frac{m}{s}</math>, (θ = 17) <math>r = 0.71256251971257 ...,\; units = (\frac{kg.m}{s})^{1/4}</math>, (θ = 8) <math>\psi_{electron} = \frac{(2^7 3 \pi^3 \alpha^{-1} \Omega^5)^3}{2\pi}</math>, (θ = 0) {| class="wikitable" |+Geometrical objects (reference table) ! Attribute ! Formula ! Object ! <math>u^{\theta}</math> ! Units |- | M (mass) | <math>(1) k i^{-1}</math> | <math>(1) \dfrac{r^4}{v}</math> | <math>u^{15}</math> | <math>kg</math> |- | T (time) | <math>(\pi) t i^{2}</math> | <math>(\pi) \dfrac{r^9}{v^6}</math> | <math>u^{-30}</math> | <math>s</math> |- | P (sqrt of momentum) | <math>\dfrac{k^{(4/5)}}{t^{(2/15)}} i^{-1}</math> | <math>(\Omega) r^2</math> | <math>u^{16}</math> | <math>\sqrt{\frac{kg\;m}{s}}</math> |- | V (velocity) | <math>\dfrac{2\pi P^2}{M}</math> | <math>(2\pi\Omega^2) v</math> | <math>u^{17}</math> | <math>\frac{m}{s}</math> |- | L (length) | <math>VT </math> | <math>(2\pi^2\Omega^2) \dfrac{r^9}{v^5}</math> | <math>u^{-13}</math> | <math>m</math> |- | A (ampere) | <math>\dfrac{2^4 V^3 \alpha}{P^3}</math> | <math>(2^7\pi^3 \alpha \Omega^3) \dfrac{v^3}{r^6}</math> | <math>u^{3}</math> | <math>\frac{m^{3/2}}{kg^{3/2} s^{3/2}}</math> |- | K (ampere) | <math>\dfrac{AV}{2\pi}</math> | <math>(2^7 \pi^3 \alpha \Omega^5) \dfrac{v^4}{r^6}</math> | <math>u^{20}</math> | <math>\left(\frac{A \; m}{s}\right)</math> |} {| class="wikitable" |+Geometrical constants (deviation from CODATA 2014) ! Constant ! Formula ! Calculated ! Deviation |- | [[w:Elementary charge | Elementary charge]] | <math>e^* = A T</math> | <math>1.6021766516 \times 10^{-19}</math> | <math>1.92\times10^{-8}</math> |- | [[w:Planck constant | Planck constant]] | <math>h^* = 2 \pi M V L</math> | <math>6.6260700046\times10^{-34}</math> | <math>-5.34\times10^{-9}</math> |- | [[w:Electron mass | Electron mass]] | <math>m_e^* = \frac{M}{\psi_{electron}}</math> | <math>9.109383509\times10^{-31}</math> | <math>-5.60\times10^{-9}</math> |} |- | [[w:Vacuum permeability | Vacuum permeability]] | <math>\mu_0^* = \frac{4 \pi V^2 M}{a L A^2}</math> | 56 | <math>r^7</math> | 4π/10^7 | <math>\frac{kg m}{A^2 s^2}</math> |- | [[w:Electron mass | Electron mass]] | <math>m_e^* = \frac{M}{\psi_{electron}}</math> | 15 | <math>\frac{r^4}{v}</math> | 9.10938231256 x 10^-31 | <math>kg</math> |- | [[w:Rydberg constant | Rydberg constant]] | <math>R^* = (\frac{m_e^*}{4 \pi L a^2 M})</math> | 13 | <math>\frac{v^5}{r^9}</math> | 10973731.568508 | <math>\frac{1}{m}</math> |- | [[w:Planck constant | Planck constant]] | <math>h^* = 2 \pi M V L</math> | 19 | <math>\frac{r^{13}}{v^5}</math> | 6.626069134 x 10^-34 | <math>\frac{kg m^2}{s}</math> |} ==Article series== * [[https://simulationuniverse.org/ simulationuniverse.org]]: Home page ===Wiki series=== * [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_units_(geometrical)]]: Planck units MLTPA as geometrical objects * [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Physical_constant_(anomaly)]]: Anomalies within the physical constants * [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Gravity_via_Atomic_orbitals]]: Gravity as a function of atomic orbitals * [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Relativity]]: Relativity as a translation between 2 co-ordinate systems * [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_unit_scaffolding]]: CMB and a Planck unit universe scaffolding * [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Sqrt_Planck_momentum]]: Link between charge and mass * [[User:Platos_Cave_(physics)/Simulation_Hypothesis/God_(programmer)]]: Introduction to a Planck scale Programmer God Simulation Hypothesis model ===General articles=== Articles have been transcribed to HTML format for ease of reference. * [[https://simulationuniverse.org/1-Planck-unit-CMB.html 1-Planck-unit-CMB.html]]: Constructs the universe frame from Planck units * [[https://simulationuniverse.org/2-Relativity-hypersphere.html 2-Relativity-hypersphere.html]]: Relativity as the mathematics of perspective * [[https://simulationuniverse.org/3-Gravitational-orbitals.html 3-Gravitational-orbitals.html]]: Gravity as sum of n-body rotating orbital particle-particle pairs * [[https://simulationuniverse.org/4-Atomic-orbitals.html 4-Atomic-orbitals.html]]: Atomic orbitals as single rotating orbital particle-particle pairs * [[https://simulationuniverse.org/5-w_axis.html 5-w_axis.html]]: Imaginary number axis (radiation domain) * [[https://simulationuniverse.org/6-Physical-constant-anomalies.html 6-Physical-constant-anomalies.html]]: Statistical analysis of physical constant anomalies * [[https://simulationuniverse.org/7-Monopole-quarks.html 7-Monopole-quarks.html]]: Quarks as monopoles * [[https://simulationuniverse.org/8-Holographic-universe.html 8-Holographic-universe.html]]: Hypersphere surface as 2-D analogue ==References== {{Reflist}} [[Category:Physics| ]] [[Category:Philosophy of science| ]] cnraj1ugb99ogu35wk9s97d7ssugw83 2 266516 2819249 2819242 2026-07-24T12:15:43Z Alandmanson 1669821 /* Wings dark fuscous */ 2819249 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' Head with golden pubescence - ''Liris nugax'' Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings hyaline === Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin <gallery mode=packed heights=200> </gallery> === Wings flavo-hyaline, often with a fusous (darker) apical margin === ''Liris haemorrhoidalis'', ''Liris bembesianus'', ''Liris solstitialis'' <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == 2gavdixwi1p1x3ehxbvyetbno624fuh 2819250 2819249 2026-07-24T12:16:14Z Alandmanson 1669821 /* Wings dark fuscous */ 2819250 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections *Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' *Head with golden pubescence - ''Liris nugax'' *Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' *Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' *Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings hyaline === Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin <gallery mode=packed heights=200> </gallery> === Wings flavo-hyaline, often with a fusous (darker) apical margin === ''Liris haemorrhoidalis'', ''Liris bembesianus'', ''Liris solstitialis'' <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == 6tu9zrwui16q3rnaauvs105h4wg5o9q 2819251 2819250 2026-07-24T12:23:41Z Alandmanson 1669821 /* Wings dark fuscous */ 2819251 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections *Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' *Head with golden pubescence - ''Liris nugax'' Kohl (descr. Arnold 1923c:232) *Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' *Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' *Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings hyaline === Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin <gallery mode=packed heights=200> </gallery> === Wings flavo-hyaline, often with a fusous (darker) apical margin === ''Liris haemorrhoidalis'', ''Liris bembesianus'', ''Liris solstitialis'' <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == qgj1l57upl7nkxyxlmjqm0a36ffhrpt 2819256 2819251 2026-07-24T12:33:54Z Alandmanson 1669821 /* Wings dark fuscous */ 2819256 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections *Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' (descr. Arnold 1923c:252) *Head with golden pubescence - ''Liris nugax'' Kohl (descr. Arnold 1923c:232) *Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' (descr. Arnold 1923c:253) *Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' (descr. Arnold 1923c:233) *Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' (descr. Arnold 1923c:234) <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings hyaline === Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin <gallery mode=packed heights=200> </gallery> === Wings flavo-hyaline, often with a fusous (darker) apical margin === ''Liris haemorrhoidalis'', ''Liris bembesianus'', ''Liris solstitialis'' <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == 0kaitd2vrkixjcvg2pt0dohl81rscjg 2819261 2819256 2026-07-24T12:45:06Z Alandmanson 1669821 /* Wings flavo-hyaline, often with a fusous (darker) apical margin */ 2819261 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections *Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' (descr. Arnold 1923c:252) *Head with golden pubescence - ''Liris nugax'' Kohl (descr. Arnold 1923c:232) *Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' (descr. Arnold 1923c:253) *Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' (descr. Arnold 1923c:233) *Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' (descr. Arnold 1923c:234) <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings hyaline === Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin <gallery mode=packed heights=200> </gallery> === Wings flavo-hyaline, often with a fusous (darker) apical margin === *''Liris haemorrhoidalis'' *''Liris bembesianus'' *''Liris solstitialis'' *Black; the tarsi and all the spines on the legs dark ferruginous; apical abdominal segment piceous - ''Liris croesus'' <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == 71bcmswzlm18ihbl3l7xtnr1hr3blgs 2819263 2819261 2026-07-24T12:52:36Z Alandmanson 1669821 /* Rough guide to Liris species of South Africa */ 2819263 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections *Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' (descr. Arnold 1923c:252) *Head with golden pubescence - ''Liris nugax'' Kohl (descr. Arnold 1923c:232) *Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' (descr. Arnold 1923c:253) *Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' (descr. Arnold 1923c:233) *Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' (descr. Arnold 1923c:234) <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings hyaline === Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin <gallery mode=packed heights=200> </gallery> === Wings flavo-hyaline, often with a fusous (darker) apical margin === *Black; mandibles except their apices, scape and first three joints of the flagellum, apical half of the fifth and the whole of the sixth abdominal segments above (and more or less also at the sides and below), and all the legs, ferruginous; the coxae and trochanters, and basal inner half of the middle and hind femora, black. Head, pro-mesonotum, scutellum, metanotum and dorsal surface of the abdomen clothed with a very dense, adpressed and short pubescence, of a brassy golden colour, somewhat darker on the thorax than on the abdomen - ''Liris haemorrhoidalis'' (descr. Arnold 1923c:251) *Black; the scape and first two joints of the flagellum, the legs, the fifth and sixth tergites and sometimes also the apical margin of the fourth, the fifth and sixth and greater part of the fourth sternites, bright ferruginous (pale burnt sienna); the apical margins of the first three abdominal segments more or less reddish brown. The middle and hind coxae and trochanters sometimes entirely red, or more or less marked with black. Spines on the legs ferruginous, the posterior calcaria piceous. - ''Liris bembesianus'' (descr. Arnold 1923c:236) *Black; the posterior femora ferruginous. Clypeus and face with short and dense silvery pubescence, the rest of the body and legs covered with a dense and exceedingly fine pruinose bloom, somewhat yellowish on the mesonotum and epinotum, and forming more conspicuous transverse and silvery grey fasciae on the apical margins of the abdominal segments. ''Liris solstitialis'' (descr. Arnold 1923c:238) *Black; the tarsi and all the spines on the legs dark ferruginous; apical abdominal segment piceous - ''Liris croesus'' (descr. Arnold 1923c:235) <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == 5joxsszw9cftd4oi1nzr1p05cip8dn8 2819264 2819263 2026-07-24T12:53:10Z Alandmanson 1669821 /* Wings flavo-hyaline, often with a fusous (darker) apical margin */ 2819264 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections *Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' (descr. Arnold 1923c:252) *Head with golden pubescence - ''Liris nugax'' Kohl (descr. Arnold 1923c:232) *Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' (descr. Arnold 1923c:253) *Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' (descr. Arnold 1923c:233) *Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' (descr. Arnold 1923c:234) <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings hyaline === Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin <gallery mode=packed heights=200> </gallery> === Wings flavo-hyaline, often with a fusous (darker) apical margin === *Black; mandibles except their apices, scape and first three joints of the flagellum, apical half of the fifth and the whole of the sixth abdominal segments above (and more or less also at the sides and below), and all the legs, ferruginous; the coxae and trochanters, and basal inner half of the middle and hind femora, black. Head, pro-mesonotum, scutellum, metanotum and dorsal surface of the abdomen clothed with a very dense, adpressed and short pubescence, of a brassy golden colour, somewhat darker on the thorax than on the abdomen - ''Liris haemorrhoidalis'' (descr. Arnold 1923c:251) *Black; the scape and first two joints of the flagellum, the legs, the fifth and sixth tergites and sometimes also the apical margin of the fourth, the fifth and sixth and greater part of the fourth sternites, bright ferruginous (pale burnt sienna); the apical margins of the first three abdominal segments more or less reddish brown. The middle and hind coxae and trochanters sometimes entirely red, or more or less marked with black. Spines on the legs ferruginous, the posterior calcaria piceous. - ''Liris bembesianus'' (descr. Arnold 1923c:236) *Black; the posterior femora ferruginous. Clypeus and face with short and dense silvery pubescence, the rest of the body and legs covered with a dense and exceedingly fine pruinose bloom, somewhat yellowish on the mesonotum and epinotum, and forming more conspicuous transverse and silvery grey fasciae on the apical margins of the abdominal segments. ''Liris solstitialis'' (descr. Arnold 1923c:238) *Black; the tarsi and all the spines on the legs dark ferruginous; apical abdominal segment piceous - ''Liris croesus'' (descr. Arnold 1923c:235) <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == rguox0rj6zmp6y2sy3pdxrgruj8hx83 2819297 2819264 2026-07-24T17:47:14Z Alandmanson 1669821 /* Wings hyaline */ 2819297 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections *Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' (descr. Arnold 1923c:252) *Head with golden pubescence - ''Liris nugax'' Kohl (descr. Arnold 1923c:232) *Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' (descr. Arnold 1923c:253) *Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' (descr. Arnold 1923c:233) *Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' (descr. Arnold 1923c:234) <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings hyaline === *Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin Black; the tegulae reddish yellow, '''all the legs (excluding the coxae and trochanters) bright ferruginous'''; the lower surface of the scapes reddish piceous. Wings hyaline, with a narrow fuscous apical border, the veins brown. Clypeus and lower half of the face with bright silvery pubescence. The rest of the head and thorax with a very fine and inconspicuous pubescence, yellowish grey on the mesonotum, dull white elsewhere. Abdomen with a pruinose pubescence, more silvery on the apical halves of the segments, where it forms transverse fasciae, more conspicuous when viewed from behind. Pygidial area with short and dull golden hairs, the apical margin with a row of reddish yellow setae. - ''Liris thysanomerus'' <gallery mode=packed heights=200> </gallery> === Wings flavo-hyaline, often with a fusous (darker) apical margin === *Black; mandibles except their apices, scape and first three joints of the flagellum, apical half of the fifth and the whole of the sixth abdominal segments above (and more or less also at the sides and below), and all the legs, ferruginous; the coxae and trochanters, and basal inner half of the middle and hind femora, black. Head, pro-mesonotum, scutellum, metanotum and dorsal surface of the abdomen clothed with a very dense, adpressed and short pubescence, of a brassy golden colour, somewhat darker on the thorax than on the abdomen - ''Liris haemorrhoidalis'' (descr. Arnold 1923c:251) *Black; the scape and first two joints of the flagellum, the legs, the fifth and sixth tergites and sometimes also the apical margin of the fourth, the fifth and sixth and greater part of the fourth sternites, bright ferruginous (pale burnt sienna); the apical margins of the first three abdominal segments more or less reddish brown. The middle and hind coxae and trochanters sometimes entirely red, or more or less marked with black. Spines on the legs ferruginous, the posterior calcaria piceous. - ''Liris bembesianus'' (descr. Arnold 1923c:236) *Black; the posterior femora ferruginous. Clypeus and face with short and dense silvery pubescence, the rest of the body and legs covered with a dense and exceedingly fine pruinose bloom, somewhat yellowish on the mesonotum and epinotum, and forming more conspicuous transverse and silvery grey fasciae on the apical margins of the abdominal segments. ''Liris solstitialis'' (descr. Arnold 1923c:238) *Black; the tarsi and all the spines on the legs dark ferruginous; apical abdominal segment piceous - ''Liris croesus'' (descr. Arnold 1923c:235) <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == qtk7notiqmolslol1777a08k0ai28ek 2819299 2819297 2026-07-24T17:48:15Z Alandmanson 1669821 /* Wings hyaline */ 2819299 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections *Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' (descr. Arnold 1923c:252) *Head with golden pubescence - ''Liris nugax'' Kohl (descr. Arnold 1923c:232) *Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' (descr. Arnold 1923c:253) *Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' (descr. Arnold 1923c:233) *Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' (descr. Arnold 1923c:234) <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings hyaline === Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin *Black; the tegulae reddish yellow, '''all the legs (excluding the coxae and trochanters) bright ferruginous'''; the lower surface of the scapes reddish piceous. Wings hyaline, with a narrow fuscous apical border, the veins brown. Clypeus and lower half of the face with bright silvery pubescence. The rest of the head and thorax with a very fine and inconspicuous pubescence, yellowish grey on the mesonotum, dull white elsewhere. Abdomen with a pruinose pubescence, more silvery on the apical halves of the segments, where it forms transverse fasciae, more conspicuous when viewed from behind. Pygidial area with short and dull golden hairs, the apical margin with a row of reddish yellow setae. - ''Liris thysanomerus'' <gallery mode=packed heights=200> </gallery> === Wings flavo-hyaline, often with a fusous (darker) apical margin === *Black; mandibles except their apices, scape and first three joints of the flagellum, apical half of the fifth and the whole of the sixth abdominal segments above (and more or less also at the sides and below), and all the legs, ferruginous; the coxae and trochanters, and basal inner half of the middle and hind femora, black. Head, pro-mesonotum, scutellum, metanotum and dorsal surface of the abdomen clothed with a very dense, adpressed and short pubescence, of a brassy golden colour, somewhat darker on the thorax than on the abdomen - ''Liris haemorrhoidalis'' (descr. Arnold 1923c:251) *Black; the scape and first two joints of the flagellum, the legs, the fifth and sixth tergites and sometimes also the apical margin of the fourth, the fifth and sixth and greater part of the fourth sternites, bright ferruginous (pale burnt sienna); the apical margins of the first three abdominal segments more or less reddish brown. The middle and hind coxae and trochanters sometimes entirely red, or more or less marked with black. Spines on the legs ferruginous, the posterior calcaria piceous. - ''Liris bembesianus'' (descr. Arnold 1923c:236) *Black; the posterior femora ferruginous. Clypeus and face with short and dense silvery pubescence, the rest of the body and legs covered with a dense and exceedingly fine pruinose bloom, somewhat yellowish on the mesonotum and epinotum, and forming more conspicuous transverse and silvery grey fasciae on the apical margins of the abdominal segments. ''Liris solstitialis'' (descr. Arnold 1923c:238) *Black; the tarsi and all the spines on the legs dark ferruginous; apical abdominal segment piceous - ''Liris croesus'' (descr. Arnold 1923c:235) <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == 8lb6m5shd5j5gi9lu7j1ce5hoousbf3 2819300 2819299 2026-07-24T17:50:01Z Alandmanson 1669821 /* Rough guide to Liris species of South Africa */ 2819300 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections *Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' (descr. Arnold 1923c:252) *Head with golden pubescence - ''Liris nugax'' Kohl (descr. Arnold 1923c:232) *Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' (descr. Arnold 1923c:253) *Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' (descr. Arnold 1923c:233) *Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' (descr. Arnold 1923c:234) <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings hyaline === Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin *Black; the tegulae reddish yellow, '''all the legs (excluding the coxae and trochanters) bright ferruginous'''; the lower surface of the scapes reddish piceous. Wings hyaline, with a narrow fuscous apical border, the veins brown. Clypeus and lower half of the face with bright silvery pubescence. The rest of the head and thorax with a very fine and inconspicuous pubescence, yellowish grey on the mesonotum, dull white elsewhere. Abdomen with a pruinose pubescence, more silvery on the apical halves of the segments, where it forms transverse fasciae, more conspicuous when viewed from behind. Pygidial area with short and dull golden hairs, the apical margin with a row of reddish yellow setae. - ''Liris thysanomerus'' (descr. Arnold 1923c:239) <gallery mode=packed heights=200> </gallery> === Wings flavo-hyaline, often with a fusous (darker) apical margin === *Black; mandibles except their apices, scape and first three joints of the flagellum, apical half of the fifth and the whole of the sixth abdominal segments above (and more or less also at the sides and below), and all the legs, ferruginous; the coxae and trochanters, and basal inner half of the middle and hind femora, black. Head, pro-mesonotum, scutellum, metanotum and dorsal surface of the abdomen clothed with a very dense, adpressed and short pubescence, of a brassy golden colour, somewhat darker on the thorax than on the abdomen - ''Liris haemorrhoidalis'' (descr. Arnold 1923c:251) *Black; the scape and first two joints of the flagellum, the legs, the fifth and sixth tergites and sometimes also the apical margin of the fourth, the fifth and sixth and greater part of the fourth sternites, bright ferruginous (pale burnt sienna); the apical margins of the first three abdominal segments more or less reddish brown. The middle and hind coxae and trochanters sometimes entirely red, or more or less marked with black. Spines on the legs ferruginous, the posterior calcaria piceous. - ''Liris bembesianus'' (descr. Arnold 1923c:236) *Black; the posterior femora ferruginous. Clypeus and face with short and dense silvery pubescence, the rest of the body and legs covered with a dense and exceedingly fine pruinose bloom, somewhat yellowish on the mesonotum and epinotum, and forming more conspicuous transverse and silvery grey fasciae on the apical margins of the abdominal segments. ''Liris solstitialis'' (descr. Arnold 1923c:238) *Black; the tarsi and all the spines on the legs dark ferruginous; apical abdominal segment piceous - ''Liris croesus'' (descr. Arnold 1923c:235) <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == r1dpa8i18u2o445a9xkbo1b1jm1uq75 2819313 2819300 2026-07-24T19:13:11Z Alandmanson 1669821 /* Wings flavo-hyaline, often with a fusous (darker) apical margin */ 2819313 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections *Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' (descr. Arnold 1923c:252) *Head with golden pubescence - ''Liris nugax'' Kohl (descr. Arnold 1923c:232) *Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' (descr. Arnold 1923c:253) *Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' (descr. Arnold 1923c:233) *Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' (descr. Arnold 1923c:234) <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings hyaline === Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin *Black; the tegulae reddish yellow, '''all the legs (excluding the coxae and trochanters) bright ferruginous'''; the lower surface of the scapes reddish piceous. Wings hyaline, with a narrow fuscous apical border, the veins brown. Clypeus and lower half of the face with bright silvery pubescence. The rest of the head and thorax with a very fine and inconspicuous pubescence, yellowish grey on the mesonotum, dull white elsewhere. Abdomen with a pruinose pubescence, more silvery on the apical halves of the segments, where it forms transverse fasciae, more conspicuous when viewed from behind. Pygidial area with short and dull golden hairs, the apical margin with a row of reddish yellow setae. - ''Liris thysanomerus'' (descr. Arnold 1923c:239) <gallery mode=packed heights=200> </gallery> === Wings flavo-hyaline, often with a fusous (darker) apical margin === *Black; mandibles except their apices, scape and first three joints of the flagellum, apical half of the fifth and the whole of the sixth abdominal segments above (and more or less also at the sides and below), and all the legs, ferruginous; the coxae and trochanters, and basal inner half of the middle and hind femora, black. Head, pro-mesonotum, scutellum, metanotum and dorsal surface of the abdomen clothed with a very dense, adpressed and short pubescence, of a brassy golden colour, somewhat darker on the thorax than on the abdomen - ''Liris haemorrhoidalis'' (descr. Arnold 1923c:251) *Black; the scape and first two joints of the flagellum, the legs, the fifth and sixth tergites and sometimes also the apical margin of the fourth, the fifth and sixth and greater part of the fourth sternites, bright ferruginous (pale burnt sienna); the apical margins of the first three abdominal segments more or less reddish brown. The middle and hind coxae and trochanters sometimes entirely red, or more or less marked with black. Spines on the legs ferruginous, the posterior calcaria piceous. - ''Liris bembesianus'' (descr. Arnold 1923c:236) *Black; the '''posterior femora ferruginous'''. Clypeus and face with short and dense silvery pubescence, the rest of the body and legs covered with a dense and exceedingly fine pruinose bloom, somewhat yellowish on the mesonotum and epinotum, and forming more conspicuous transverse and silvery grey fasciae on the apical margins of the abdominal segments. ''Liris solstitialis'' (descr. Arnold 1923c:238) *Black; the tarsi and all the spines on the legs dark ferruginous; apical abdominal segment piceous - ''Liris croesus'' (descr. Arnold 1923c:235) <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == qy1lfn7zd2hgnzu949seomx1a7htd2i 2819317 2819313 2026-07-24T20:21:24Z Alandmanson 1669821 /* Wings flavo-hyaline, often with a fusous (darker) apical margin */ 2819317 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections *Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' (descr. Arnold 1923c:252) *Head with golden pubescence - ''Liris nugax'' Kohl (descr. Arnold 1923c:232) *Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' (descr. Arnold 1923c:253) *Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' (descr. Arnold 1923c:233) *Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' (descr. Arnold 1923c:234) <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings hyaline === Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin *Black; the tegulae reddish yellow, '''all the legs (excluding the coxae and trochanters) bright ferruginous'''; the lower surface of the scapes reddish piceous. Wings hyaline, with a narrow fuscous apical border, the veins brown. Clypeus and lower half of the face with bright silvery pubescence. The rest of the head and thorax with a very fine and inconspicuous pubescence, yellowish grey on the mesonotum, dull white elsewhere. Abdomen with a pruinose pubescence, more silvery on the apical halves of the segments, where it forms transverse fasciae, more conspicuous when viewed from behind. Pygidial area with short and dull golden hairs, the apical margin with a row of reddish yellow setae. - ''Liris thysanomerus'' (descr. Arnold 1923c:239) <gallery mode=packed heights=200> </gallery> === Wings flavo-hyaline, often with a fusous (darker) apical margin === *Black; mandibles except their apices, scape and first three joints of the flagellum, apical half of the fifth and the whole of the sixth abdominal segments above (and more or less also at the sides and below), and all the legs, ferruginous; the coxae and trochanters, and basal inner half of the middle and hind femora, black. Head, pro-mesonotum, scutellum, metanotum and dorsal surface of the abdomen clothed with a very dense, adpressed and short pubescence, of a brassy golden colour, somewhat darker on the thorax than on the abdomen - ''Liris haemorrhoidalis'' (descr. Arnold 1923c:251) *Black; the scape and first two joints of the flagellum, the legs, the fifth and sixth tergites and sometimes also the apical margin of the fourth, the fifth and sixth and greater part of the fourth sternites, bright ferruginous (pale burnt sienna); the apical margins of the first three abdominal segments more or less reddish brown. The middle and hind coxae and trochanters sometimes entirely red, or more or less marked with black. Spines on the legs ferruginous, the posterior calcaria piceous. - ''Liris bembesianus'' (descr. Arnold 1923c:236) *Black; the '''posterior femora ferruginous'''. Clypeus and face with short and dense silvery pubescence, the rest of the body and legs covered with a dense and exceedingly fine pruinose bloom, somewhat yellowish on the mesonotum and epinotum, and forming more conspicuous transverse and silvery grey fasciae on the apical margins of the abdominal segments. ''Liris solstitialis'' (descr. Arnold 1923c:238) *Black; the tarsi and all the spines on the legs dark ferruginous; apical abdominal segment piceous - ''Liris croesus'' (descr. Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:238.) <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == 70aljvrbsvayrsndvfx5dn2dsrv81bh 2819318 2819317 2026-07-24T20:22:22Z Alandmanson 1669821 Undid revision [[Special:Diff/2819317|2819317]] by [[Special:Contributions/Alandmanson|Alandmanson]] ([[User talk:Alandmanson|talk]]) 2819318 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections *Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' (descr. Arnold 1923c:252) *Head with golden pubescence - ''Liris nugax'' Kohl (descr. Arnold 1923c:232) *Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' (descr. Arnold 1923c:253) *Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' (descr. Arnold 1923c:233) *Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' (descr. Arnold 1923c:234) <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings hyaline === Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin *Black; the tegulae reddish yellow, '''all the legs (excluding the coxae and trochanters) bright ferruginous'''; the lower surface of the scapes reddish piceous. Wings hyaline, with a narrow fuscous apical border, the veins brown. Clypeus and lower half of the face with bright silvery pubescence. The rest of the head and thorax with a very fine and inconspicuous pubescence, yellowish grey on the mesonotum, dull white elsewhere. Abdomen with a pruinose pubescence, more silvery on the apical halves of the segments, where it forms transverse fasciae, more conspicuous when viewed from behind. Pygidial area with short and dull golden hairs, the apical margin with a row of reddish yellow setae. - ''Liris thysanomerus'' (descr. Arnold 1923c:239) <gallery mode=packed heights=200> </gallery> === Wings flavo-hyaline, often with a fusous (darker) apical margin === *Black; mandibles except their apices, scape and first three joints of the flagellum, apical half of the fifth and the whole of the sixth abdominal segments above (and more or less also at the sides and below), and all the legs, ferruginous; the coxae and trochanters, and basal inner half of the middle and hind femora, black. Head, pro-mesonotum, scutellum, metanotum and dorsal surface of the abdomen clothed with a very dense, adpressed and short pubescence, of a brassy golden colour, somewhat darker on the thorax than on the abdomen - ''Liris haemorrhoidalis'' (descr. Arnold 1923c:251) *Black; the scape and first two joints of the flagellum, the legs, the fifth and sixth tergites and sometimes also the apical margin of the fourth, the fifth and sixth and greater part of the fourth sternites, bright ferruginous (pale burnt sienna); the apical margins of the first three abdominal segments more or less reddish brown. The middle and hind coxae and trochanters sometimes entirely red, or more or less marked with black. Spines on the legs ferruginous, the posterior calcaria piceous. - ''Liris bembesianus'' (descr. Arnold 1923c:236) *Black; the '''posterior femora ferruginous'''. Clypeus and face with short and dense silvery pubescence, the rest of the body and legs covered with a dense and exceedingly fine pruinose bloom, somewhat yellowish on the mesonotum and epinotum, and forming more conspicuous transverse and silvery grey fasciae on the apical margins of the abdominal segments. ''Liris solstitialis'' (descr. Arnold 1923c:238) *Black; the tarsi and all the spines on the legs dark ferruginous; apical abdominal segment piceous - ''Liris croesus'' (descr. Arnold 1923c:235) <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == qy1lfn7zd2hgnzu949seomx1a7htd2i 2819319 2819318 2026-07-24T20:22:57Z Alandmanson 1669821 /* Wings flavo-hyaline, often with a fusous (darker) apical margin */ 2819319 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections *Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' (descr. Arnold 1923c:252) *Head with golden pubescence - ''Liris nugax'' Kohl (descr. Arnold 1923c:232) *Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' (descr. Arnold 1923c:253) *Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' (descr. Arnold 1923c:233) *Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' (descr. Arnold 1923c:234) <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings hyaline === Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin *Black; the tegulae reddish yellow, '''all the legs (excluding the coxae and trochanters) bright ferruginous'''; the lower surface of the scapes reddish piceous. Wings hyaline, with a narrow fuscous apical border, the veins brown. Clypeus and lower half of the face with bright silvery pubescence. The rest of the head and thorax with a very fine and inconspicuous pubescence, yellowish grey on the mesonotum, dull white elsewhere. Abdomen with a pruinose pubescence, more silvery on the apical halves of the segments, where it forms transverse fasciae, more conspicuous when viewed from behind. Pygidial area with short and dull golden hairs, the apical margin with a row of reddish yellow setae. - ''Liris thysanomerus'' (descr. Arnold 1923c:239) <gallery mode=packed heights=200> </gallery> === Wings flavo-hyaline, often with a fusous (darker) apical margin === *Black; mandibles except their apices, scape and first three joints of the flagellum, apical half of the fifth and the whole of the sixth abdominal segments above (and more or less also at the sides and below), and all the legs, ferruginous; the coxae and trochanters, and basal inner half of the middle and hind femora, black. Head, pro-mesonotum, scutellum, metanotum and dorsal surface of the abdomen clothed with a very dense, adpressed and short pubescence, of a brassy golden colour, somewhat darker on the thorax than on the abdomen - ''Liris haemorrhoidalis'' (descr. Arnold 1923c:251) *Black; the scape and first two joints of the flagellum, the legs, the fifth and sixth tergites and sometimes also the apical margin of the fourth, the fifth and sixth and greater part of the fourth sternites, bright ferruginous (pale burnt sienna); the apical margins of the first three abdominal segments more or less reddish brown. The middle and hind coxae and trochanters sometimes entirely red, or more or less marked with black. Spines on the legs ferruginous, the posterior calcaria piceous. - ''Liris bembesianus'' (descr. Arnold 1923c:236) *Black; the '''posterior femora ferruginous'''. Clypeus and face with short and dense silvery pubescence, the rest of the body and legs covered with a dense and exceedingly fine pruinose bloom, somewhat yellowish on the mesonotum and epinotum, and forming more conspicuous transverse and silvery grey fasciae on the apical margins of the abdominal segments. ''Liris solstitialis'' (Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:238.) *Black; the tarsi and all the spines on the legs dark ferruginous; apical abdominal segment piceous - ''Liris croesus'' (descr. Arnold 1923c:235) <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == pxik9dhm8061uqlhgwcwjbdg8v91qs5 2819320 2819319 2026-07-24T20:23:13Z Alandmanson 1669821 /* Wings flavo-hyaline, often with a fusous (darker) apical margin */ 2819320 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections *Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' (descr. Arnold 1923c:252) *Head with golden pubescence - ''Liris nugax'' Kohl (descr. Arnold 1923c:232) *Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' (descr. Arnold 1923c:253) *Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' (descr. Arnold 1923c:233) *Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' (descr. Arnold 1923c:234) <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings hyaline === Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin *Black; the tegulae reddish yellow, '''all the legs (excluding the coxae and trochanters) bright ferruginous'''; the lower surface of the scapes reddish piceous. Wings hyaline, with a narrow fuscous apical border, the veins brown. Clypeus and lower half of the face with bright silvery pubescence. The rest of the head and thorax with a very fine and inconspicuous pubescence, yellowish grey on the mesonotum, dull white elsewhere. Abdomen with a pruinose pubescence, more silvery on the apical halves of the segments, where it forms transverse fasciae, more conspicuous when viewed from behind. Pygidial area with short and dull golden hairs, the apical margin with a row of reddish yellow setae. - ''Liris thysanomerus'' (descr. Arnold 1923c:239) <gallery mode=packed heights=200> </gallery> === Wings flavo-hyaline, often with a fuscous (darker) apical margin === *Black; mandibles except their apices, scape and first three joints of the flagellum, apical half of the fifth and the whole of the sixth abdominal segments above (and more or less also at the sides and below), and all the legs, ferruginous; the coxae and trochanters, and basal inner half of the middle and hind femora, black. Head, pro-mesonotum, scutellum, metanotum and dorsal surface of the abdomen clothed with a very dense, adpressed and short pubescence, of a brassy golden colour, somewhat darker on the thorax than on the abdomen - ''Liris haemorrhoidalis'' (descr. Arnold 1923c:251) *Black; the scape and first two joints of the flagellum, the legs, the fifth and sixth tergites and sometimes also the apical margin of the fourth, the fifth and sixth and greater part of the fourth sternites, bright ferruginous (pale burnt sienna); the apical margins of the first three abdominal segments more or less reddish brown. The middle and hind coxae and trochanters sometimes entirely red, or more or less marked with black. Spines on the legs ferruginous, the posterior calcaria piceous. - ''Liris bembesianus'' (descr. Arnold 1923c:236) *Black; the '''posterior femora ferruginous'''. Clypeus and face with short and dense silvery pubescence, the rest of the body and legs covered with a dense and exceedingly fine pruinose bloom, somewhat yellowish on the mesonotum and epinotum, and forming more conspicuous transverse and silvery grey fasciae on the apical margins of the abdominal segments. ''Liris solstitialis'' (Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:238.) *Black; the tarsi and all the spines on the legs dark ferruginous; apical abdominal segment piceous - ''Liris croesus'' (descr. Arnold 1923c:235) <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == br28y2ps5jii9lfxh2w1bd7q5i6cgcg 2819338 2819320 2026-07-25T07:19:03Z Alandmanson 1669821 /* Wings hyaline */ 2819338 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections *Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' (descr. Arnold 1923c:252) *Head with golden pubescence - ''Liris nugax'' Kohl (descr. Arnold 1923c:232) *Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' (descr. Arnold 1923c:253) *Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' (descr. Arnold 1923c:233) *Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' (descr. Arnold 1923c:234) <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings hyaline === Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin *Black; the tegulae reddish yellow, '''all the legs (excluding the coxae and trochanters) bright ferruginous'''; the lower surface of the scapes reddish piceous. Wings hyaline, with a narrow fuscous apical border, the veins brown. Clypeus and lower half of the face with bright silvery pubescence. The rest of the head and thorax with a very fine and inconspicuous pubescence, yellowish grey on the mesonotum, dull white elsewhere. Abdomen with a pruinose pubescence, more silvery on the apical halves of the segments, where it forms transverse fasciae, more conspicuous when viewed from behind. Pygidial area with short and dull golden hairs, the apical margin with a row of reddish yellow setae. - ''Liris thysanomerus'' (descr. Arnold 1923c:239) *''Liris odontophorus'' *''Liris rubellus'' (F. Smith, 1856), description in *'' *'' *'' *'' *'' <gallery mode=packed heights=200> </gallery> === Wings flavo-hyaline, often with a fuscous (darker) apical margin === *Black; mandibles except their apices, scape and first three joints of the flagellum, apical half of the fifth and the whole of the sixth abdominal segments above (and more or less also at the sides and below), and all the legs, ferruginous; the coxae and trochanters, and basal inner half of the middle and hind femora, black. Head, pro-mesonotum, scutellum, metanotum and dorsal surface of the abdomen clothed with a very dense, adpressed and short pubescence, of a brassy golden colour, somewhat darker on the thorax than on the abdomen - ''Liris haemorrhoidalis'' (descr. Arnold 1923c:251) *Black; the scape and first two joints of the flagellum, the legs, the fifth and sixth tergites and sometimes also the apical margin of the fourth, the fifth and sixth and greater part of the fourth sternites, bright ferruginous (pale burnt sienna); the apical margins of the first three abdominal segments more or less reddish brown. The middle and hind coxae and trochanters sometimes entirely red, or more or less marked with black. Spines on the legs ferruginous, the posterior calcaria piceous. - ''Liris bembesianus'' (descr. Arnold 1923c:236) *Black; the '''posterior femora ferruginous'''. Clypeus and face with short and dense silvery pubescence, the rest of the body and legs covered with a dense and exceedingly fine pruinose bloom, somewhat yellowish on the mesonotum and epinotum, and forming more conspicuous transverse and silvery grey fasciae on the apical margins of the abdominal segments. ''Liris solstitialis'' (Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:238.) *Black; the tarsi and all the spines on the legs dark ferruginous; apical abdominal segment piceous - ''Liris croesus'' (descr. Arnold 1923c:235) <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == obc7430io61c8f41g3yf7coup7hmu41 2819339 2819338 2026-07-25T07:31:58Z Alandmanson 1669821 /* Rough guide to Liris species of South Africa */ 2819339 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections *Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' (descr. Arnold 1923c:252) *Head with golden pubescence - ''Liris nugax'' Kohl (descr. Arnold 1923c:232) *Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' (descr. Arnold 1923c:253) *Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' (descr. Arnold 1923c:233) *Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' (descr. Arnold 1923c:234) <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings flavo-hyaline, often with a fuscous (darker) apical margin === *Black; mandibles except their apices, scape and first three joints of the flagellum, apical half of the fifth and the whole of the sixth abdominal segments above (and more or less also at the sides and below), and all the legs, ferruginous; the coxae and trochanters, and basal inner half of the middle and hind femora, black. Head, pro-mesonotum, scutellum, metanotum and dorsal surface of the abdomen clothed with a very dense, adpressed and short pubescence, of a brassy golden colour, somewhat darker on the thorax than on the abdomen - ''Liris haemorrhoidalis'' (descr. Arnold 1923c:251) *Black; the scape and first two joints of the flagellum, the legs, the fifth and sixth tergites and sometimes also the apical margin of the fourth, the fifth and sixth and greater part of the fourth sternites, bright ferruginous (pale burnt sienna); the apical margins of the first three abdominal segments more or less reddish brown. The middle and hind coxae and trochanters sometimes entirely red, or more or less marked with black. Spines on the legs ferruginous, the posterior calcaria piceous. - ''Liris bembesianus'' (descr. Arnold 1923c:236) *Black; the '''posterior femora ferruginous'''. Clypeus and face with short and dense silvery pubescence, the rest of the body and legs covered with a dense and exceedingly fine pruinose bloom, somewhat yellowish on the mesonotum and epinotum, and forming more conspicuous transverse and silvery grey fasciae on the apical margins of the abdominal segments. ''Liris solstitialis'' (Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:238.) *Black; the tarsi and all the spines on the legs dark ferruginous; apical abdominal segment piceous - ''Liris croesus'' (descr. Arnold 1923c:235) <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> === Wings hyaline === Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin *''Liris thysanomerus'' (Kohl, 1894), description as ''Notogonidea thysanomera'' in Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:239: > "Black; the tegulae reddish yellow, '''all the legs (excluding the coxae and trochanters) bright ferruginous'''; the lower surface of the scapes reddish piceous. Wings hyaline, with a narrow fuscous apical border, the veins brown. Clypeus and lower half of the face with bright silvery pubescence. The rest of the head and thorax with a very fine and inconspicuous pubescence, yellowish grey on the mesonotum, dull white elsewhere. Abdomen with a pruinose pubescence, more silvery on the apical halves of the segments, where it forms transverse fasciae, more conspicuous when viewed from behind. Pygidial area with short and dull golden hairs, the apical margin with a row of reddish yellow setae." *''Liris odontophorus'' (Kohl, 1894), description as ''Notogonidea odontophora'' in Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:240. *''Liris rubellus'' (F. Smith, 1856), description as ''Notogonidea cyphononyx'' in Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:241. *'' *'' *'' *'' *'' <gallery mode=packed heights=200> </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == q3s3xflp2he9q9bpw21p7haptzeyyyx 2819340 2819339 2026-07-25T07:41:03Z Alandmanson 1669821 /* Rough guide to Liris species of South Africa */ 2819340 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections *Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' (descr. Arnold 1923c:252) *Head with golden pubescence - ''Liris nugax'' Kohl (descr. Arnold 1923c:232) *Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' (descr. Arnold 1923c:253) *Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' (descr. Arnold 1923c:233) *Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' (descr. Arnold 1923c:234) <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings flavo-hyaline, often with a fuscous (darker) apical margin === *Black; mandibles except their apices, scape and first three joints of the flagellum, apical half of the fifth and the whole of the sixth abdominal segments above (and more or less also at the sides and below), and all the legs, ferruginous; the coxae and trochanters, and basal inner half of the middle and hind femora, black. Head, pro-mesonotum, scutellum, metanotum and dorsal surface of the abdomen clothed with a very dense, adpressed and short pubescence, of a brassy golden colour, somewhat darker on the thorax than on the abdomen - ''Liris haemorrhoidalis'' (descr. Arnold 1923c:251) *Black; the scape and first two joints of the flagellum, the legs, the fifth and sixth tergites and sometimes also the apical margin of the fourth, the fifth and sixth and greater part of the fourth sternites, bright ferruginous (pale burnt sienna); the apical margins of the first three abdominal segments more or less reddish brown. The middle and hind coxae and trochanters sometimes entirely red, or more or less marked with black. Spines on the legs ferruginous, the posterior calcaria piceous. - ''Liris bembesianus'' (descr. Arnold 1923c:236) *Black; the '''posterior femora ferruginous'''. Clypeus and face with short and dense silvery pubescence, the rest of the body and legs covered with a dense and exceedingly fine pruinose bloom, somewhat yellowish on the mesonotum and epinotum, and forming more conspicuous transverse and silvery grey fasciae on the apical margins of the abdominal segments. ''Liris solstitialis'' (Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:238.) *Black; the tarsi and all the spines on the legs dark ferruginous; apical abdominal segment piceous - ''Liris croesus'' (descr. Arnold 1923c:235) <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> === Wings hyaline === Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin *''Liris thysanomerus'' (Kohl, 1894), description as ''Notogonidea thysanomera'' in Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:239: "Black; the tegulae reddish yellow, '''all the legs (excluding the coxae and trochanters) bright ferruginous'''; the lower surface of the scapes reddish piceous. Wings hyaline, with a narrow fuscous apical border, the veins brown. Clypeus and lower half of the face with bright silvery pubescence. The rest of the head and thorax with a very fine and inconspicuous pubescence, yellowish grey on the mesonotum, dull white elsewhere. Abdomen with a pruinose pubescence, more silvery on the apical halves of the segments, where it forms transverse fasciae, more conspicuous when viewed from behind. Pygidial area with short and dull golden hairs, the apical margin with a row of reddish yellow setae." *''Liris odontophorus'' (Kohl, 1894), description as ''Notogonidea odontophora'' in Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:240. "Black; mandibles, anterior margin of the clypeus, scapes, the first four joints of the flagellum and the fifth underneath, the apex of the posterior tibiae inwardly, the apical margin of the third, and the whole of the fourth to sixth abdominal segments, ferruginous; the tarsi reddish brown, the tegulae ferruginous behind. Face and clypeus with fine silvery pubescence. Thorax and apical margins of the first three tergites with an exceedingly fine or almost pollinose pubescence, on the epinotum a little longer and outstanding. Wings very pale fusco-hyaline, a little darker beyond the second cubital cell." *''Liris rubellus'' (F. Smith, 1856), description as ''Notogonidea cyphononyx'' in Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:241. "Black; first three segments of the abdomen dark castaneous red, with an indistinct transverse band of black behind the apical margin of the first segment, and a spot on each side of the second, or, as in the type of the species, the whole abdomen red. Head and thorax clothed with a very fine, short and yellowish grey pubescence, not obscuring the sculpture, somewhat longer and silvery on the clypeus and face. Abdomen with a little whitish pubescence on the apical margins of the first three segments. Wings faintly fusco-hyaline, darker towards the apex." *'' *'' *'' *'' *'' <gallery mode=packed heights=200> </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == hz5u96msvkxa9i4s738sz68naygyvaa 2819342 2819340 2026-07-25T08:13:46Z Alandmanson 1669821 /* Wings hyaline */ 2819342 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections *Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' (descr. Arnold 1923c:252) *Head with golden pubescence - ''Liris nugax'' Kohl (descr. Arnold 1923c:232) *Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' (descr. Arnold 1923c:253) *Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' (descr. Arnold 1923c:233) *Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' (descr. Arnold 1923c:234) <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings flavo-hyaline, often with a fuscous (darker) apical margin === *Black; mandibles except their apices, scape and first three joints of the flagellum, apical half of the fifth and the whole of the sixth abdominal segments above (and more or less also at the sides and below), and all the legs, ferruginous; the coxae and trochanters, and basal inner half of the middle and hind femora, black. Head, pro-mesonotum, scutellum, metanotum and dorsal surface of the abdomen clothed with a very dense, adpressed and short pubescence, of a brassy golden colour, somewhat darker on the thorax than on the abdomen - ''Liris haemorrhoidalis'' (descr. Arnold 1923c:251) *Black; the scape and first two joints of the flagellum, the legs, the fifth and sixth tergites and sometimes also the apical margin of the fourth, the fifth and sixth and greater part of the fourth sternites, bright ferruginous (pale burnt sienna); the apical margins of the first three abdominal segments more or less reddish brown. The middle and hind coxae and trochanters sometimes entirely red, or more or less marked with black. Spines on the legs ferruginous, the posterior calcaria piceous. - ''Liris bembesianus'' (descr. Arnold 1923c:236) *Black; the '''posterior femora ferruginous'''. Clypeus and face with short and dense silvery pubescence, the rest of the body and legs covered with a dense and exceedingly fine pruinose bloom, somewhat yellowish on the mesonotum and epinotum, and forming more conspicuous transverse and silvery grey fasciae on the apical margins of the abdominal segments. ''Liris solstitialis'' (Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:238.) *Black; the tarsi and all the spines on the legs dark ferruginous; apical abdominal segment piceous - ''Liris croesus'' (descr. Arnold 1923c:235) <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> === Wings hyaline === Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin *''Liris thysanomerus'' (Kohl, 1894), description as ''Notogonidea thysanomera'' in Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:239: "Black; the tegulae reddish yellow, '''all the legs (excluding the coxae and trochanters) bright ferruginous'''; the lower surface of the scapes reddish piceous. Wings hyaline, with a narrow fuscous apical border, the veins brown. Clypeus and lower half of the face with bright silvery pubescence. The rest of the head and thorax with a very fine and inconspicuous pubescence, yellowish grey on the mesonotum, dull white elsewhere. Abdomen with a pruinose pubescence, more silvery on the apical halves of the segments, where it forms transverse fasciae, more conspicuous when viewed from behind. Pygidial area with short and dull golden hairs, the apical margin with a row of reddish yellow setae." *''Liris odontophorus'' (Kohl, 1894), description as ''Notogonidea odontophora'' in Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:240. "Black; mandibles, anterior margin of the clypeus, scapes, the first four joints of the flagellum and the fifth underneath, the apex of the posterior tibiae inwardly, the apical margin of the third, and the whole of the fourth to sixth abdominal segments, ferruginous; the tarsi reddish brown, the tegulae ferruginous behind. Face and clypeus with fine silvery pubescence. Thorax and apical margins of the first three tergites with an exceedingly fine or almost pollinose pubescence, on the epinotum a little longer and outstanding. Wings very pale fusco-hyaline, a little darker beyond the second cubital cell." *''Liris rubellus'' (F. Smith, 1856), description as ''Notogonidea cyphononyx'' in Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:241. "Black; '''first three segments of the abdomen dark castaneous red''', with an indistinct transverse band of black behind the apical margin of the first segment, and a spot on each side of the second, '''or, as in the type of the species, the whole abdomen red'''. Head and thorax clothed with a very fine, short and yellowish grey pubescence, not obscuring the sculpture, somewhat longer and silvery on the clypeus and face. Abdomen with a little whitish pubescence on the apical margins of the first three segments. Wings faintly fusco-hyaline, darker towards the apex." *''Liris denticulatus'' (R. Turner, 1920), description as ''Notogonidea denticulata'' in Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:242. *''Liris nigricans'' (Walker, 1871), description as ''Notogonidea nigricans'' in Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:243. "Black; wings hyaline, faintly fuscous over the apical third. Clypeus, face and back of the head with silvery pubescence, thorax with a very short, scanty and whitish pubescence; abdomen and legs with a thin and inconspicuous pruinose pubescence, not forming apical fasciae on the former." *''Liris niger namana'' (Fabricius, 1775) *'' *''Liris sepulchralis'' (Gerstaecker in Peters, 1858), description as ''Notogonidea sepulchralis'' (and as ''N. pompiliformis intermedia'') in Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:244. "Black; the tarsi reddish brown, the basal joint usually darker towards the base. Wings hyaline, slightly smoky, with a narrow apical fuscous border beyond the cells. Clypeus and face, back of the head and anterior femora below, with dense silvery pubescence, the thorax with a scanty and greyish pubescence, fairly short, and on the mesonotum confined to the lateral margins. Legs and abdomen pruinose, the abdomen with apical fasciae of greyish silvery pubescence on the first four tergites. Pygidial area covered with short, decumbent and yellowish grey setae, with a few long and finer hairs intermixed, the apical margin with a row of six reddish spines. <gallery mode=packed heights=200> </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == kxu9ci9g4xecyke4p8vvhjn9jvp2r1e 2819351 2819342 2026-07-25T10:50:48Z Alandmanson 1669821 /* Rough guide to Liris species of South Africa */ 2819351 wikitext text/x-wiki <!--Info--> == Rough guide to Liris species of South Africa == === Wings dark fuscous === Very dark wings, often with violet reflections *Dense dark fulvous (to golden) pubescence on head, pronotum and mesonotum - ''Liris'' ''diabolicus'' (descr. Arnold 1923c:252) *Head with golden pubescence - ''Liris nugax'' Kohl (descr. Arnold 1923c:232) *Face and clypeus with a yellowish silvery pubescence - ''Liris atropos'' (descr. Arnold 1923c:253) *Lower half of the face and the clypeus with a thin greyish silvery pubescence - ''Liris rufoscapus'' (descr. Arnold 1923c:233) *Foreleg tarsi with a comb composed of long, flattened spines; scapes black - ''Liris ciliata'' (descr. Arnold 1923c:234) <gallery mode=packed heights=200> Liris inaturalist 197329080.jpg </gallery> === Wings flavo-hyaline, often with a fuscous (darker) apical margin === *Black; mandibles except their apices, scape and first three joints of the flagellum, apical half of the fifth and the whole of the sixth abdominal segments above (and more or less also at the sides and below), and all the legs, ferruginous; the coxae and trochanters, and basal inner half of the middle and hind femora, black. Head, pro-mesonotum, scutellum, metanotum and dorsal surface of the abdomen clothed with a very dense, adpressed and short pubescence, of a brassy golden colour, somewhat darker on the thorax than on the abdomen - ''Liris haemorrhoidalis'' (descr. Arnold 1923c:251) *Black; the scape and first two joints of the flagellum, the legs, the fifth and sixth tergites and sometimes also the apical margin of the fourth, the fifth and sixth and greater part of the fourth sternites, bright ferruginous (pale burnt sienna); the apical margins of the first three abdominal segments more or less reddish brown. The middle and hind coxae and trochanters sometimes entirely red, or more or less marked with black. Spines on the legs ferruginous, the posterior calcaria piceous. - ''Liris bembesianus'' (descr. Arnold 1923c:236) *Black; the '''posterior femora ferruginous'''. Clypeus and face with short and dense silvery pubescence, the rest of the body and legs covered with a dense and exceedingly fine pruinose bloom, somewhat yellowish on the mesonotum and epinotum, and forming more conspicuous transverse and silvery grey fasciae on the apical margins of the abdominal segments. ''Liris solstitialis'' (Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:238.) *Black; the tarsi and all the spines on the legs dark ferruginous; apical abdominal segment piceous - ''Liris croesus'' (descr. Arnold 1923c:235) <gallery mode=packed heights=200> Liris on Crassula iN 42678436 03.jpg </gallery> === Wings hyaline === Wings hyaline (colorless, glassy, and transparent), or pale fusco-hyaline (slightly darker than hyaline), sometimes with a darker apical margin *''Liris thysanomerus'' (Kohl, 1894), description as ''Notogonidea thysanomera'' in Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:239: "Black; the tegulae reddish yellow, '''all the legs (excluding the coxae and trochanters) bright ferruginous'''; the lower surface of the scapes reddish piceous. Wings hyaline, with a narrow fuscous apical border, the veins brown. Clypeus and lower half of the face with bright silvery pubescence. The rest of the head and thorax with a very fine and inconspicuous pubescence, yellowish grey on the mesonotum, dull white elsewhere. Abdomen with a pruinose pubescence, more silvery on the apical halves of the segments, where it forms transverse fasciae, more conspicuous when viewed from behind. Pygidial area with short and dull golden hairs, the apical margin with a row of reddish yellow setae." *''Liris odontophorus'' (Kohl, 1894), description as ''Notogonidea odontophora'' in Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:240. "Black; mandibles, anterior margin of the clypeus, scapes, the first four joints of the flagellum and the fifth underneath, the apex of the posterior tibiae inwardly, the apical margin of the third, and the whole of the fourth to sixth abdominal segments, ferruginous; the tarsi reddish brown, the tegulae ferruginous behind. Face and clypeus with fine silvery pubescence. Thorax and apical margins of the first three tergites with an exceedingly fine or almost pollinose pubescence, on the epinotum a little longer and outstanding. Wings very pale fusco-hyaline, a little darker beyond the second cubital cell." *''Liris rubellus'' (F. Smith, 1856), description as ''Notogonidea cyphononyx'' in Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:241. "Black; '''first three segments of the abdomen dark castaneous red''', with an indistinct transverse band of black behind the apical margin of the first segment, and a spot on each side of the second, '''or, as in the type of the species, the whole abdomen red'''. Head and thorax clothed with a very fine, short and yellowish grey pubescence, not obscuring the sculpture, somewhat longer and silvery on the clypeus and face. Abdomen with a little whitish pubescence on the apical margins of the first three segments. Wings faintly fusco-hyaline, darker towards the apex." *''Liris denticulatus'' (R. Turner, 1920), description as ''Notogonidea denticulata'' in Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:242. *''Liris nigricans'' (Walker, 1871), description as ''Notogonidea nigricans'' in Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:243. "Black; wings hyaline, faintly fuscous over the apical third. Clypeus, face and back of the head with silvery pubescence, thorax with a very short, scanty and whitish pubescence; abdomen and legs with a thin and inconspicuous pruinose pubescence, not forming apical fasciae on the former." *''Liris niger namana'' (Bischoff, 1913b:121), described as "Notogonia pompiliformis namana" Bischoff, 1913 *''Liris felina'' (Arnold, 1923), described as "Notogonidea felina'' in Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:247. "Black; tarsi pale ferruginous, the basal two-thirds of the anterior metatarsus fusco-ferruginous. Wings hyaline, with a narrow fuscous apical border which is darker than in sepulchralis, the veins blackish brown. Pubescence like that of sepulchralis but only the first three tergites have apical fasciae. The coarse decumbent pubescence of the pygidial area is less dense, greyer than in sepulchralis and intermixed with more numerous fine blackish hairs; the apical margin has a row of eight black spines. The spines on the tibiae are black, on the tarsi fusco-ferruginous." *''Liris sepulchralis'' (Gerstaecker in Peters, 1858), description as ''Notogonidea sepulchralis'' (and as ''N. pompiliformis intermedia'') in Arnold, G. 1923b. The Sphegidae of South Africa. Part III. Annals of the Transvaal Museum 9:244. "Black; the tarsi reddish brown, the basal joint usually darker towards the base. Wings hyaline, slightly smoky, with a narrow apical fuscous border beyond the cells. Clypeus and face, back of the head and anterior femora below, with dense silvery pubescence, the thorax with a scanty and greyish pubescence, fairly short, and on the mesonotum confined to the lateral margins. Legs and abdomen pruinose, the abdomen with apical fasciae of greyish silvery pubescence on the first four tergites. Pygidial area covered with short, decumbent and yellowish grey setae, with a few long and finer hairs intermixed, the apical margin with a row of six reddish spines. <gallery mode=packed heights=200> </gallery> = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == 2c1rbnxm3q84txb3uokqyrmavjltfo4 0 267628 2819341 2716735 2026-07-25T07:56:12Z ~2026-41475-67 3103179 Remove image that does not support comprehension of the topic. 2819341 wikitext text/x-wiki —Exploring the basis for belief {{TOC right | limit|limit=2}} {{100%done}} == Introduction == Street Epistemology is a reasoned conversation about the basis for belief. More formally described, Street Epistemology is a conversational tool that helps people reflect on the quality of their reasons and the reliability of the methods used to derive one's confidence level in their deeply-held beliefs.<ref>StreetEpistemology.com See: https://streetepistemology.com/</ref> An [[w:Epistemology|epistemology]] is a way of knowing. Street Epistemology (SE) applies [[Socratic methods]] to explore questions of [[w:belief|belief]]. It explores the basis we have for the beliefs we hold. SE is a ''practice''. It is a collection of skills and techniques that you can apply during sincere conversations to explore important topics and discover new insights. While your objectives may differ, Street Epistemology is generally used to understand a claim (generally a [[w:Proposition|proposition]]), identify the actual reasons and reliability of the methods used to support the claim, and determine if one's confidence in the claim is justified. [[File:Street Epistemology Audio Dialogue.wav|thumb|Street Epistemology Audio Dialogue]] == Orientation == The term "Street Epistemology" (SE)<ref>This course material is largely adapted from [https://docs.google.com/document/d/1YOqUGBlTJ6cCnkfZCYN6zV-csG85b_fkIiQAi3EPXSw/pub The Complete Street Epistemology Guide How to Talk About Beliefs], licensed under a [http://creativecommons.org/licenses/by-sa/4.0/&sa=D&ust=1598873869323000&usg=AOvVaw2zEwx5fEW_5WV6wtPwXO8y Creative Commons Attribution-ShareAlike 4.0 International License].</ref> originates in Dr. Peter Boghossian's book, ''A Manual for Creating Atheists''<ref>{{cite book |last=Boghossian |first=Peter |author-link=w:Peter_Boghossian |date=November 1, 2013 |title=A Manual for Creating Atheists |publisher=Pitchstone Publishing |page=280 |isbn=978-1939578099}}</ref>(AMFCA). In the book, [[w:Peter_Boghossian|Dr. Boghossian]] describes how people often use faith as an [[w:epistemology|epistemology]] — that is, as a way of coming to knowledge and justifying their beliefs. His central theme is that unreliable epistemologies, such as faith, are used to arrive at potentially [[Facing Facts/Harmful false beliefs|harmful false beliefs]]. Because faith-based belief systems typically encourage or require adherents to spread the belief system, he uses the metaphor of "virus of the mind" to describe the effects faith has on people. Faith gains traction by presenting itself as a reliable method, akin to [[w:Trust_(social_science)|trust]], by presenting reasonable doubt as an epistemological failing. This course recognizes that people are guided by a wide range of claims and epistemologies — not always religious or faith-based. Therefore, this course does not use "virus" or "intervention" metaphors because the course applies not only to faith used as a way of knowing, but all ways of knowing and all kinds of beliefs. Some ways of knowing may indeed turn out to be reliable enough to justify the belief in question. As such, the scope has broadened beyond what Dr. Boghossian originally described in AMFCA, creating a space for this course to offer additional information and advice. This course is based on the best practices and lessons learned by active Street Epistemologists. The community welcomes every opportunity for improvement, driven by constructive criticism, study, and experience. As Dr. Boghossian writes, "It's important for your growth and for the development of the techniques to experiment and develop your own ideas and strategies".<ref>{{cite book |last=Boghossian |first=Peter |author-link=w:Peter_Boghossian |date=November 1, 2013 |title=A Manual for Creating Atheists |publisher=Pitchstone Publishing |page=280 |isbn=978-1939578099}}, p 118.</ref> SE [[Practicing Dialogue|dialogues]] work toward mutual agreement about the reliability of different ways of assessing whether or not a belief is true or likely to be true, without devolving into debate. As a Street Epistemologist, you start from a position of "[[w:Doxastic_logic|doxastic openness]]" in which you acknowledge that the other person’s position may be correct. You should be willing to revise your beliefs if this turns out to be the case. Ideally, it becomes increasingly clear to both of you whether or not the methods can be relied upon to lead one to the truth. You may use Street Epistemology [[Practicing Dialogue|in dialogue]] because you value truth and because you desire to help yourself and others use methods that are less likely to produce false beliefs. You can expect either party's confidence or beliefs to change as a result of such dialogues. Realizing you have been using an unreliable method may lead you to re-examine your beliefs, alter your confidence, or even renounce a belief having deemed it unlikely to be true. By holding [[Seeking True Beliefs|true beliefs]] about reality on matters of practical consequence, we can all make better choices for our lives and for our communities. This course is organized along the structure of a complete dialogue: preparation, initiating the dialogue, managing the dialogue, targeting epistemology, ending the dialogue, and following up. Much of the advice is generally applicable, or derivable from common experience, or at least came from other fields; very little is unique to SE. What is new is the selection, arrangement, presentation, and synthesis of these techniques to improve the way we form beliefs. This course encourages you to try the techniques in this guide yourself, test whether they work, and modify them to suit your conversations with others. The course also encourages sharing of the interactions so we may build as a community and expand our own knowledge of reliable epistemologies. As you use SE and discover new insights, please [[Talk:Street_Epistemology|leave course feedback]] so this course can continue to improve. This course has no prerequisites, and all students are welcome. Several companion courses are available that can help students gain additional background or bolster their understanding of various concepts and techniques useful to SE practitioners. *The [[Socratic Methods]] course provides general background and practice using Socratic methods. *The [[Deductive Logic/Clear Thinking curriculum|Clear Thinking Curriculum]] provides information on objective reality, deductive logic, logical fallacies, evaluating evidence, seeking true beliefs, intellectual honesty, and more. *The course on [[Knowing How You Know]] guides students in creating their personal epistemology by writing down [[Knowing How You Know/gallery|how they choose beliefs]]. *The course [[Finding Common Ground]] provides a framework for aligning concepts with reality. *The course [[Embracing Ambiguity]] helps us loosen our grip on arbitrarily chosen certainties. *The [[Emotional Competency|Emotional Competency curriculum]] includes courses that can help you stay calm and constructive before, during, and after SE session. *The [[Real Good Religion]] course describes satisfing alternatives to theism. *The [[Fostering Curiosity]] course encourages us to wonder why. Study these companion courses any time they may be helpful. Specific companion course suggestions appear in relevant sections throughout this course. This course includes a [[Street_Epistemology#Terminology|glossary of terms]] that can be useful throughout your study. The course contains many [[w:Hyperlink|hyperlinks]] to further information. Use your judgment and these [[What Matters/link following guidelines|link following guidelines]] to decide when to follow a link, and when to skip over it. ==Street Epistemology == ===What it is === Street Epistemology is a movement to apply the tools of philosophy in everyday conversations in order to encourage people to use reliable ways of forming beliefs. While professional philosophers may publish articles and books, anyone who values truth can engage friends, family, community members, etc. in respectful dialogues about how beliefs are known to be true. The goal is to encourage ourselves and others to examine the methods we use to judge the accuracy of truth claims, and ultimately to improve the reliability of our epistemology. While people may alter conclusions as a result, that is not the express goal. As Dr. Boghossian writes, "the core of the dialogue is not changing beliefs, but changing the way people form beliefs"<ref>{{cite book |last=Boghossian |first=Peter |author-link=w:Peter_Boghossian |date=November 1, 2013 |title=A Manual for Creating Atheists |publisher=Pitchstone Publishing |page=280 |isbn=978-1939578099}} p72</ref> Neither participant should fear being persuaded into holding a false belief so long as a high standard for justification is sought. If anyone realizes that they have used an unreliable method to arrive at some belief, how they use that insight is entirely up to them. They are never pressured to accept any specific belief or to act against their own best interests. ===Why we do it === Whether you realize it or not, you have arrived at your beliefs using specific criteria: modes of justification. Even when you intend to do good, you can produce bad outcomes when you act upon false beliefs. Street Epistemology addresses the root cause of such bad outcomes — not the people whose actions or inaction produce the outcomes, nor even the probably-false beliefs on which they acted, but the unreliable methods by which they tend to acquire such beliefs in the first place. By engaging in dialogue, you can put your epistemology to the test and help yourself and others more reliably arrive at true beliefs. ===What it's based on === Street Epistemology is based on [[Socratic Methods]] combined with the field of [[w:epistemology|epistemology]]. Socratic methods engage an [[w:Interlocutor_(linguistics)|interlocutor]]—your dialogue partner—in answering well-formed questions. Knowing how to formulate good questions will work in most cases to uncover unreliable justifications. However, the better you understand epistemology, the more easily you can see where an interlocutor's implicit epistemology may be unreliable, and the more you will have to offer in terms of reliable alternatives. You will also be better-equipped to engage those interlocutors who make explicit use of epistemological theories or apologetic arguments in justifying their claims. Next we will mention some specific philosophical ideas used in Street Epistemology. If you have never looked into epistemology, consider reading an introductory philosophy book or taking a class online, and watching the Wireless Philosophy<ref>YouTube video playlist, Epistemology: Introduction to Theory of Knowledge. See https://www.youtube.com/playlist?list=PLtKNX4SfKpzUxuye9OdaRfL5fbpGa3bH5</ref> and Crash Course Philosophy<ref>YouTube video playlist, Crash Course Philosophy. See: https://www.youtube.com/playlist?list=PL8dPuuaLjXtNgK6MZucdYldNkMybYIHKR</ref> epistemology videos. If we had to pick one word to sum up the epistemology that Street Epistemology is based on, it would be [[w:Reliabilism|reliabilism]]: the notion that what makes one justified in holding a belief is the truth-conduciveness of the process by which one arrived at the belief.<ref>Goldman, Alvin and Bob Beddor, "[https://plato.stanford.edu/archives/win2016/entries/reliabilism Reliabilist Epistemology]", The Stanford Encyclopedia of Philosophy (Winter 2016 Edition), Edward N. Zalta (ed.), </ref> How you get to your beliefs matters, because it affects how likely you are to have true beliefs. Reliabilism is implied in our stated goal of reducing reliance on unreliable epistemology. Since we humans are [[w:Fallibilism|fallible]] (we cannot "know that we know") we settle for processes that with limited available [[Evaluating Evidence|evidence]] lead us to an appropriate degree of confidence regarding claims being true. However, the Street Epistemologist is not dogmatic: we will use any theory of epistemology and any mode of justification as a tool, provided that for the type of claims at hand it tends to justify true claims and not false claims, with an appropriate degree of confidence. Like everyone we make use of [[w:Inductive_reasoning|inductive reasoning]] and [[w:Deductive_reasoning|deductive reasoning]], while being on guard for [[w:fallacies|fallacies]]. Being human we too acquire many of our beliefs through [[w:Testimony|testimony]]<ref>Also see: the Internet Encyclopedia of Philosophy, entry on [https://iep.utm.edu/ep-testi/ Epistemology of Testimony].</ref> (while being aware of its problems<ref>Adler, Jonathan, "[https://plato.stanford.edu/archives/win2017/entries/testimony-episprob/ Epistemological Problems of Testimony]", The Stanford Encyclopedia of Philosophy (Winter 2017 Edition), Edward N. Zalta (ed.)</ref>), and cannot help but make intuitive [[w:Coherentism|coherentist]]<ref>Also see: the Internet Encyclopedia of Philosophy, entry on [https://iep.utm.edu/coherent/ Coherentism in Epistemology]</ref> evaluations of plausibility when encountering new claims. However, with [[w:Defeasibility|Defeasibility Tests]]<ref>Koons, Robert, "[https://plato.stanford.edu/archives/win2017/entries/reasoning-defeasible/ Defeasible Reasoning]", The Stanford Encyclopedia of Philosophy (Winter 2017 Edition), Edward N. Zalta (ed.)</ref> we emphasize [[w:Falsifiability|falsifiability]] — the mode of justification that powers much of modern science. We also compare and evaluate the probabilities of competing explanations using [[w:Bayesian_inference|Bayesian inference]], and turn to [[w:Occam's_razor|Occam's razor]] to favor the simplest process that would generate observed events. We'll even dig into the justification of extraordinary claims in a somewhat [[w:Foundationalism|foundationalist]]<ref>Hasan, Ali and Richard Fumerton, "[https://plato.stanford.edu/archives/fall2018/entries/justep-foundational/ Foundationalist Theories of Epistemic Justification]", The Stanford Encyclopedia of Philosophy (Fall 2018 Edition), Edward N. Zalta (ed.)</ref> manner, by following the chain of justification until it reaches ordinary claims, and seeing whether the inference holds up. One could also say Street Epistemologists are being [[w:Pragmatism|pragmatic]]<ref>Legg, Catherine and Christopher Hookway, "[https://plato.stanford.edu/archives/fall2020/entries/pragmatism Pragmatism]", The Stanford Encyclopedia of Philosophy (Fall 2020 Edition), Edward N. Zalta (ed.)</ref> when we apply [[w:John_W._Loftus#The_Outsider_Test_for_Faith|Outsider Tests]], as we seek the practical consequences that enable us to adjudicate between competing claims. ===When to use it === You can use Street Epistemology whenever a truth claim is being made. However it is most useful for [[w:Sagan_standard|extraordinary claims]], such as miracles and supernatural phenomena, including: *Existence of one or more gods or immaterial persons (theism). *Phenomena that violate or suspend the operation of natural laws (supernaturalism, paranormal and psychic phenomena, miracles, karma). *Biological death does not end one's existence as a conscious being (afterlife, reincarnation, resurrection). *The effectiveness of healing modalities that science based medicine rejects as unproven or ineffective (quackery). *The scientific validity of an idea or system which has never been adequately researched or fails under scientific testing ([[w:Pseudoscience|pseudosciences]]). *A covert but powerful force/group is responsible for certain events or situations, where evidence of that force/group is lacking ([[w:Conspiracy_theory|conspiracy theories]]). In such cases, we often encounter the following justifications, and the Street Epistemologist asks whether they are sufficiently reliable to warrant belief in the claim. *[[w:Faith|'''Faith''']]: When given as a reason for belief, it can be understood as firm confidence in the claim in excess of what is warranted by evidence.<ref>Bishop, John, "[https://plato.stanford.edu/archives/win2016/entries/faith/ Faith]", The Stanford Encyclopedia of Philosophy (Winter 2016 Edition), Edward N. Zalta (ed.)</ref> *[[w:Numinous|'''Numinous''']], [[w:Revelation|'''revelatory''']], or [[w:Mysticism|'''mystical experiences''']]<ref>Webb, Mark, "[https://plato.stanford.edu/archives/win2017/entries/religious-experience Religious Experience]", The Stanford Encyclopedia of Philosophy (Winter 2017 Edition), Edward N. Zalta (ed.)</ref> *'''Personal experiences:''' answered prayers, "worked for me" therapies. *[[w:Testimony|'''Testimony''']]: including personal anecdotes, tradition, authorities. Testimony is particularly vulnerable to errors and omissions by the reporter, intentional or not (even if only unreliability of perception and memory), and further errors if second hand.<ref>Internet Encyclopedia of Philosophy, entry on [https://iep.utm.edu/ep-testi/ Epistemology of Testimony].</ref>^,<ref>Adler, Jonathan, "[https://plato.stanford.edu/archives/win2017/entries/testimony-episprob/ Epistemological Problems of Testimony]", The Stanford Encyclopedia of Philosophy (Winter 2017 Edition), Edward N. Zalta (ed.)</ref> === Assignment === Please answer each of the following questions in your own words. How would you convey your understanding of each of these topics to your next-door neighbor, or your barber? It is perfectly acceptable to simply reflect on each of these questions, choose a (real or imaginary) friend to discuss this with, or write your answers in a notebook or diary. *What is Street Epistemology? *Why are you interested in practicing Street Epistemology? *What is the practice based on? *When is it appropriate to use SE? When is it inappropriate to use SE? ==Preparing for dialogues == Prior to conducting Street Epistemology, take steps to prepare mentally as well as practically. This section covers mindset, materials, location, medium, and recording. ===Developing the right mindset === ====Promote good epistemology, not specific beliefs ==== Aim to improve the reliability of the methods we use to form beliefs — for your interlocutor and yourself. Stand in stark contrast to the street preachers, evangelists, debaters and others who aim to persuade others to adopt specific beliefs, by whatever means works, regardless of whether they are reliable guides to truth. Use [[Socratic Methods|Socratic dialogue]] to seek agreement on how reliable different ways of knowing are, with the long-term goal of everyone holding justified beliefs where it matters most. '''Ask yourself:''' Do you seek to understand and promote better ways of knowing what claims are true, and not to promote specific claims of your own? ====Model doxastic openness ==== [[w:Doxastic_attitudes|Doxastic attitudes]] are epistemic attitudes which a person can hold towards a proposition. Model the epistemic attitudes that you want to see everywhere. In particular, be genuinely [[w:Open-mindedness|open to revising your own beliefs]]. You cannot fairly expect the interlocutor to be willing to revise their beliefs if you are not willing to do the same. '''Ask yourself:''' If an interlocutor shows that they are using reliable epistemology that justifies belief in their proposition, would you honestly be willing to revise your own beliefs? ====Be collaborative and respectful ==== Be respectful, honest, [[Fostering Curiosity|curious]], collaborative, empathetic and non-judgmental. Seek to understand what the interlocutor believes and how they justify their belief. Form a collaborative partnership to clarify the justifications and evaluate their reliability. Remind yourself that while beliefs deserve questioning, people deserve respect. '''Ask yourself:''' Are you prepared to behave in a respectful, empathetic, and collaborative manner with your interlocutors? ====Have an optimistic growth mindset ==== Treat each dialogue as an opportunity to practice and improve your skills, no matter the outcome. Cultivate an attitude of acceptance regarding outcomes. Not every request for a dialogue will be accepted; in fact, most are not. Be aware that in many discussions, you and the interlocutor will not come to agreement on anything of substance. Also, note that even after reaching agreement that certain methods are unreliable, it is common for people to hold onto beliefs they no longer know to be true due to social and emotional factors. '''Ask yourself:''' Do you see every interaction and every mistake as an opportunity to learn and grow? Do you accept that a mixture of outcomes is perfectly normal and that there are factors at work that you have no influence over? ====Know what success looks like ==== Take care to view your interactions as a potential learning experience for all parties, and not as some sort of “conquest”. Instead, strive to “sow seeds of doubt that will blossom into ever-expanding moments of doxastic openness”<ref>{{cite book |last=Boghossian |first=Peter |author-link=w:Peter_Boghossian |date=November 1, 2013 |title=A Manual for Creating Atheists |publisher=Pitchstone Publishing |page=280 |isbn=978-1939578099}} p51</ref>. Think of your questions as a pebble in the interlocutor’s shoe that will cause them to revisit the conversation all day long. '''Ask yourself:''' In holding this dialogue, what do I want for myself, for the interlocutor, and for the relationship? What would a successful dialogue look like? ===Presentation and materials === All you really need to bring is yourself and the right mindset. However, if you wish to carry out dialogues on the street, some materials can be helpful. Bring writing materials such as a whiteboard and marker, or clipboard and pad of paper. Dialogues wander and writing down the interlocutor's key points helps bring structure and focus, helps to avoid talking in circles, and enables you to illustrate epistemology with diagrams. Being prepared also shows that you are not a random passer-by but someone who is approaching people for a reason: namely to hold dialogues with the public about how they form beliefs. Bring a timer or use the timer on your phone to limit the length of the dialogue. The timer helps to focus the dialogue and shows that you respect/value their time. When the time expires, it gives both of you an opportunity to exit the dialogue. You can always continue if both parties are comfortable doing so. If you have trouble with dialogues running on, try setting a backup timer and when it goes off tell the interlocutor that you have to leave right now and suggest following up at another time. Consider bringing contact cards or providing contact information to facilitate follow-up dialogues. ===Recording dialogues === Consider recording your dialogues in order to monitor and improve your own performance, solicit constructive feedback, and demonstrate your techniques to others. If you are interested in recording your dialogues, please study the [[/Recording dialogues/]] module. ===Choosing the medium === '''Face to face encounters.''' When speaking face-to-face, communication is enhanced through facial expressions, voice inflection, and body language. You can tell if the interlocutor is beginning to feel uncomfortable long before they express their discomfort in words, and correct misunderstandings immediately. Face-to-face is simply the highest-bandwidth communication medium possible between two people. '''Video calls/conferencing.''' You might use a video call and lose some resolution in facial expressions and body language or be forced to cope with delays between replies. Though not as good as face-to-face discussions, this can be a good approximation. '''Audio only.''' You can use an audio call although you’ll lose all of the visual information and cues. '''Real time text chat.''' With text chat you lose all visual and auditory information so the risk of misunderstandings and misinterpretations is greater. The less visual information you have, the more time you need to spend on clear communications that show the utmost respect. The cost is in time: a dialogue that takes minutes face-to-face can take hours over text. '''Email / Comment Threads.''' By far the least effective medium for Street Epistemology is email or comment threads. Over email, a short dialogue can take weeks. It is tempting to tackle multiple points and ask multiple questions in a single message, which is antithetical to the single thread that characterizes Socratic dialogue. Comment threads have the additional disadvantage of being public, making interlocutors more likely to double-down to save face. If you begin a dialogue over a text-based medium, offer to hold a face-to-face meeting, or at least a video or audio call. If you must use text, keep your replies short and to the point: a couple sentences to summarize their main idea, followed by your one most promising question. ===Choosing your interlocutors === You may not need to do much choosing; Sometimes interlocutors will come to you by offering a belief that you find worthy of examination. Your interlocutors may be friends, family, acquaintances, people you meet at social gatherings, people you meet on the street, or people you encounter over social media. Be cautious when your relationship with the interlocutor is valuable to you. This raises the stakes. It's possible for a dialogue to go sour, especially when you are just starting out. You don't want to risk that in a relationship where there could be serious consequences. On the one hand, if it is a strong relationship, like a good friend or family member, it may easily handle a challenging dialogue involving deeply-held beliefs. Take care when attempting Street Epistemology at work. A dialogue about deeply-held beliefs may go poorly and have long-term negative consequences. It may also be inappropriate or against the rules to discuss deeply-held beliefs in cases where you are expected to interact with someone on a purely professional level. ===Choosing the Location === You may not have to choose a location; Be prepared for spontaneous opportunities for Street Epistemology whenever someone expresses a belief for which you lack a reliable way to know that it's true. Such opportunities can happen anywhere: chatting in the checkout, on the bus, over lunch with a co-worker, or in social situations such as conferences, out with friends, or at a party. Like Socrates, you may wish to put the "street" into Street Epistemology by engaging the public in face-to-face dialogues about the epistemology behind their beliefs. Choose places where people are milling about and chatting in public already, where striking up a dialogue with a stranger is safe and socially acceptable. For example: public squares and markets, public gatherings of any sort, public college campuses, tourist sites, and pedestrian malls lined with cafés and shops. Avoid noisy or uncomfortable places for dialogue, such as public transport areas or streets with motorized traffic. Avoid places where your right to conduct interviews might be questioned, such as shops and private campuses, and places like quiet parks, where interacting with strangers breaks with social norms. If you are planning to ask people for a recorded interview, choose a quiet, well-lit location to minimize distracting sounds and interruptions. If striking up a dialogue with strangers seems daunting, you may wish to begin in an online location instead of out in the "real world” but remember that the quality of communication will be reduced. You may find a good balance on public video chat systems. There are some places in the world where political topics, especially criticism of the government, or criticism of certain religions, is dangerous. Never practice Street Epistemology if it places anyone's (including your own) personal safety at risk. === Assignment === Plan for your first session. Work to attain the right mindset, then gather any materials you will use, and identify likely interlocutors and locations. ==Beginning the dialogue == If you are already on good terms with the person, starting a dialogue with them should not present a problem. This section deals mainly with the more difficult case of how to start epistemological dialogues with strangers ===Approaching people on the street === Having [[Street_Epistemology#Choosing_the_Location|chosen the location]], casually observe and identify friendly-looking people who seem relaxed. Avoid anyone who looks to be in a hurry. Even if a hurried person agrees to speak with you, they may not be able to focus on the dialogue. Look for people who are sitting down or strolling about. Be aware that approaching total strangers and asking them about what may be deeply-held beliefs can be stressful. It is normal to feel nervous and shaky or to doubt your abilities. Consciously acknowledging that it is normal to be nervous in such situations can paradoxically help to calm your nerves. You can also gradually increase your engagement with the public. Start by smiling and greeting people as they walk by. Slowly increase the number of sentences with each passerby ('''“Good afternoon! Amazing weather today, yes‽”''') until you are ready to ask someone to actually stop and speak with you. After you have had a couple of dialogues, a feeling of confidence and optimism in being able to have deep discussions with a complete stranger may replace the anxiety you felt at the start. ===Initiating the dialogue === After getting someone's attention, get right to the point with a polite and simple question, such as, “Do you have five minutes to chat about how you arrived at your god belief?”, or a similar question about any other belief. A question like this is non-intrusive and interesting. Smile and look people in the eye when you ask. When a person declines your request simply wish them a nice day, and move on. Accept that the choice to decline is a perfectly valid and normal response to a request. When a person accepts your request, thank them and greet them, and try to make them as comfortable as possible. Ask their first name and provide yours, using their name throughout the dialogue. As mentioned in "[[Street_Epistemology#Presentation_and_materials|Presentation and materials]]", set a time limit on the interview. Ideally between 5 and 15 minutes. Avoid talking for more than 1 hour. It is difficult to maintain focus for much longer and you'll end up talking in circles. You also want your dialogue to be memorable, and it's very difficult to remember and reflect on all the topics covered in a marathon session. ===One person at a time === Engage only one person at a time. The interaction will be much more honest, open, and sincere. An audience or bystander may interject and knock the dialogue off-track, make the interlocutor less willing to open up, and make them self-conscious of their status so that they are more likely to double-down or attempt to save face when privately they are beginning to doubt. If you are in the middle of a dialogue and a bystander cuts in to hinder the discussion, remember that the interlocutor can see what is happening as well. Offer the bystander a future conversation once the interlocutor and you are done talking. This also applies to discussions in other mediums such as text chats. ===Building rapport === After having successfully engaged an interlocutor, establish a friendly environment and build rapport. You can ask general, small-talk questions first to make them comfortable. Asking their opinions about topics of current or local interest, can put them at ease, but also helps you build some context about them. Creating a sense of ease will help both your dialogue partner and you relax. You may also be able to build examples later in the dialogue from these early clues about your interlocutor. Adopt a collaborative stance with the interlocutor. You are there to seek truth and reliable ways of knowing what is true, not to prove yourself right and them wrong. Frame the dialogue as a partnership. For example: '''“How can we figure out whether there is a reliable way to know that this is true?”''' Questioning deeply held beliefs can be uncomfortable. Strive for a balance between giving a pass on unreliable epistemology and challenging a belief so directly that they end the dialogue. Pay attention to your tone, body language and facial expressions, and maintain a relaxed manner. Watch for nonverbal clues that rapport is being lost: crossed arms, looking around, looking at their watch, nervousness, shaky hands. Point out that they appear worried and offer to continue the dialogue another day. You can even ask if they feel comfortable continuing the conversation. The offer may put them more at ease and they may agree to continue. Reiterate your collaborative intent. Use humor carefully. Some people take their beliefs very seriously and may interpret attempts at humor as disrespectful of their beliefs or mocking them. If you do like to use humor in building rapport, use it on a subject other than the belief under discussion. ===Eliciting a controversial belief === After building rapport, elicit a belief worth discussing: one where you find the claim to be implausible, where you presently lack any reliable way to know that the claim is true, and where you think there are important consequences to holding the belief. The belief may already be on the table, in which case you can skip to the next step. If not, here are some ways to go about finding one. You might try defining the subject up front by saying something like, '''"Do you have a few minutes for a quick interview?"''' then '''"I'm interviewing people on the reliability of the methods they use to form their god beliefs - do you have any such beliefs?"''' If you're specifically targeting faith, AMFCA-style, you might say '''"Do you rely on faith to be confident about your religious beliefs?"''' Asking different opening questions can lead to very different subjects of dialogue. Religion is the most common topic of dialogues but there can be many others. You can ask whether they believe in any supernatural or paranormal phenomena such as ghosts or psychic powers. If the interlocutor lacks all supernatural beliefs, there is still plenty of [[w:pseudoscience|pseudoscience]], [[w:Quackery|quackery]] and [[w:List_of_conspiracy_theories|conspiracy theorizing]] out there. Don't use those terms, however, as the negative connotations may put the interlocutor on the defensive. Ask instead what they think about climate change (it's a hot topic), theories outside mainstream science, alternative medicines and therapies, and whether they think some powerful but covert group is secretly responsible for major events. See “[[Street_Epistemology#When_to_use_it|When to use it]]” for more suggestions. Even if the interlocutor holds no supernatural, pseudoscience or conspiracy-theory beliefs that doesn't mean all their beliefs are well-justified. You may even agree with them on something, but it turns out that the interlocutor got lucky and came to believe it by means of an unreliable method. For example, if someone relies on personal experiences of hot summers in recent years to conclude that climate change is real, there is still room to improve that person's epistemology. Finally, even people who apply high epistemological standards to claims about what's real may have more lax standards in their political or ethical beliefs. In such areas, the influence of morals, values and the complexity of human societies does make it much more difficult to judge how well their beliefs correspond to with reality. You might ask what they believe about a current political or societal topic of interest. ===Eliciting the interlocutor's confidence === Getting a sense of how confident your interlocutor is in the belief early on will help you to work out how to proceed, and work out how much their confidence has changed (if at all) as an immediate result of the dialogue. The easiest way to gauge the interlocutor's confidence is by listening to how assured they sound when talking about their belief. You can make them aware of their own level of confidence by asking directly, '''"How confident are you that the belief is true?"''' You might present a "belief scale" by asking, '''“On a scale from zero to one hundred, how confident are you that your belief is true?”''' If they are uncomfortable putting a number on their confidence, accept a qualitative strength of belief such as "absolutely", "almost certainly", "very confident", "probably", etc. If they resist or hesitate to qualify their degree of confidence, just go with it and move on. For someone absolutely confident (100%), you might first examine human fallibility. If they are fairly confident (70-99%), you may look at their major sources of confidence. If they are not very confident (<50%), you may ask what is holding them back from discarding the belief. At the end of the talk, ask about the interlocutor's confidence level again. Compare it to their confidence at the beginning to get a feeling for how far (if any) they have moved and in which direction. ===Eliciting the interlocutor's epistemology === The interlocutor's epistemology is the method of justification that they use to know the belief is true, such as faith, testimony, or personal experience. There may be many. To elicit their way of knowing, you might ask '''"How did you (originally) conclude that this belief was true?"''' By asking about how they originally formed the belief, they are less likely to fall back on answering "How do you make your belief sound reasonable to someone else?". If they offer multiple justifications, you could ask '''"What are the top three things that make you confident that your belief is true?"''' — this gives you some options to pick the most promising line first. If you suspect the interlocutor is in fact relying on a specific method (such as faith) but is reluctant to expose reliance on a method that may seem less persuasive to others, you might ask about that method directly — '''"What role does X have in your knowing that the belief is true?"''' Be aware that the first justification that the interlocutor gives is not necessarily a major contributor to their confidence. They may be repeating what they’ve heard others say, or what they think sounds reasonable to others, instead of what really gives them confidence in the belief. You may examine that justification and agree that it's unreliable, only to find that the interlocutor doesn't care. To avoid this, ask the interlocutor '''"How confident would you be in the belief without X?"''' If it turns out that X matters little to their confidence, try again by asking '''"What gives you the most confidence that your belief is true?"''', and check that they would indeed be less confident without that critical foundation. The interlocutor may give a justification for their belief that relies on an equally extraordinary claim, such as a specific miracle. In this case think of yourself as a foundation inspector. Work with the interlocutor to determine whether their beliefs are built on solid ground or shifting sand. Dig deeper into the foundations of the interlocutor's belief system by asking, '''"What gives you confidence that X is true?"''' Keep digging until you reach a justification that is not based on something extraordinary. At that point you are ready to begin inspecting the quality of the foundation, determining the reliability of the methods that the interlocutor uses to know that the foundational belief is true. ==Managing the dialogue == This section provides general-purpose techniques for managing the dialogue. Students may benefit from completing the Wikiversity course [[Practicing Dialogue]] along with this section. The Wikiversity course [[Earning Trust]] may also be helpful. ===Building respect === Respect is like air: you only notice it when it's not there. The moment mutual respect is lost between you and the interlocutor, the dialogue ceases to be about the original subject, and becomes entirely about respect until respect is restored or you part ways. As such, you can only make progress so long as there is mutual respect. Take care to affirm that you value the interlocutor as a human being, and that you believe that they are well-meaning and intelligent. In matters of social interaction "perception is often reality", meaning that if the interlocutor perceives your behavior as disrespectful, that suffices to make your behavior disrespectful. People take things personally; If you call someone's ideas stupid, they will likely think that you are calling them stupid. People also read between the lines; If you suggest that they don't truly believe something they will think you are calling them a liar. Of course, people do have bad ideas, and do tell lies. The trick is in addressing a specific behavior, pattern of behavior, or set of ideas, while simultaneously affirming the person. Some interlocutors may begin with little respect for you. For example, they may peg you as an amoral, untrustworthy, spiritually blind heathen. They may assume that you have no respect for them and treat you likewise. ''This makes it important to build rapport and respect first before getting into deep dialogue.'' Be sensitive to any indications that the interlocutor has taken offense or is becoming aggressive. If this happens, don't forge ahead with challenging questions — step out of the dialogue, and rebuild respect. Only then return to the dialogue. One way to rebuild respect is with contrasting—establishing useful distinctions. For example, after asking the interlocutor whether it's possible they are mistaken about the cause of an experience, they might take that as an insult to their intelligence and start defending their education. If this happens, continue with what you don't mean: '''"I'm not questioning your intelligence or education,"''' then contrast with a clarification of what you do mean: '''"When I asked whether you could be mistaken, I meant whether anyone—even the most intelligent and well-educated of people—could make an error in attributing the cause of an experience."''' When the interlocutor is unaware of some fact or holds a [[w:List_of_common_misconceptions|common misconception]], don't feign surprise by saying something like, "I can't believe you don't know that". People know the difference between feigned surprise as a put-down and genuine surprise at an unexpected response. ===Remaining calm === It's easier said than done to remain calm and relaxed, since interlocutors may express a myriad of emotions, including anger and disgust when they begin to doubt the underpinnings of deeply cherished beliefs. By remaining calm, you can help to calm your interlocutor as well. ''Of course you should always end the interview if confronted with hostility.'' You can help to remain calm by reminding yourself that you are not in a battle. You are collaborating with your interlocutor to find the best way of knowing what is true. Most of us have tried and failed to argue people out of seemingly unreasonable positions and found the collaborative approach more successful in moving dialogue forward — as well as more pleasant for all parties! You can also model the behavior you want to see: take time to consider your answers, and they may take time to consider your questions. If you sense discomfort, you can de-escalate in response. Slow down and recap where you are and what you are trying to achieve together. You will learn to notice when you are becoming impatient or agitated and take a step back to refocus. Redefine what "success" means. Even if you do not achieve any detectable change in your interlocutor's confidence, you may still have planted some seeds in their mind that they will ponder later. Your interaction may have dispelled previously held, negative opinions about atheists for example. If onlookers were present, some of them may have been swayed even if the person you were talking to wasn't. Review the section on “[[Street_Epistemology#Developing_the_right_mindset|Developing the right mindset]]” often. Finally, remember that nobody is perfect and we all make mistakes. Every conversation is an opportunity to learn from what went well (or not so well), and to improve your skills. See [[Street Epistemology#Reflecting_on you_ performance|Reflecting on your performance]] for a list of elements to review and assess after each interview. Focus on the ways you can improve, rather than the mistakes you’ve made. Courses from the Wikiversity [[Emotional Competency]] curriculum may provide helpful supplementary material that can help you stay calm, transcend conflict, and remain constructive. ===Answering questions about your beliefs === If asked about your own beliefs you should be prepared to answer. When doing so, model the epistemic stance you would like to see in your interlocutors: analyzing beliefs, seriously considering ways in which your belief could be false, and doxastic openness - clear and genuine willingness to revise your beliefs if warranted. The Wikiversity course on [[Knowing How You Know]] can help prepare you. You may be asked about your beliefs if you ask a good question for which your interlocutor doesn’t have a ready answer. To deal with this, offer to discuss your beliefs after the timed interview, or in a follow-up discussion where the tables are turned. You might say that you are interested in how we might know whether their claim is true, rather than promoting your own beliefs. If the interlocutor asks if you believe X when you are fairly sure you do not, you might reinforce your openness by saying, '''"Right now I lack a reliable way to know X, but I'm interested to find out whether there is one."''' If your interlocutor is truly interested in what you believe and why, it can wait until after the interview or a follow-up discussion. You may only be seconds away from the end, if the interlocutor agrees that the way of knowing they've been using is not reliable. When you do answer their questions, model reliable epistemology by apportioning your confidence to the available evidence and remaining open to new evidence. You can present things you believe about reality that seem inconsistent with the interlocutor's belief. Use "I" statements to admit the possibility that you are mistaken, instead of presenting your beliefs as definite facts. Presenting contrary claims as definite facts may get you into a debate because the interlocutor will reject them. Phrase things to avoid triggering the interlocutor's prejudices: if they seem prejudiced about "atheists", say instead that you think gods are unlikely. Also avoid referring to their belief with what they might take as a pejorative terms, for example "conspiracy theory" or "pseudoscience". This is called “unpacking” a contentious concept.<ref>See: https://wiki.lesswrong.com/wiki/Rationalist_taboo</ref> Lastly and most importantly, if you don't know something, simply answer “I don’t know”. Demonstrating the willingness to be okay with uncertainty may help them feel safe in doing the same. ===The "Spider on the ceiling" === If your interlocutor pauses for an unusually long time to answer a question they will often tilt their head and look up as if observing a spider on the ceiling. Be alert for this. When it happens they may be experiencing an "[[w:Aporia|aporia]]" — a state of puzzlement or a sudden inability to resolve an internal contradiction. This is a very good thing! Don't interrupt the pause; allow them to reflect on their answer. It provides an opportunity for them to recognize the discrepancy and consider how it affects their justification for the belief. If you speak too soon, you interrupt that important process. ''This pause for reflection may be more powerful than anything you could say in those few seconds.'' Use the pause yourself to assess the state of the dialogue and think about the next steps. Rushing them to answer will likely come across as rude, and doesn't give them time to reflect, leading them to fall back on standard responses. They will speak when they have collected their thoughts. Learning these deliberate pauses may be challenging if you are accustomed to debating, as it can be difficult to resist jumping in with your next question. Nevertheless, you should practice it at every opportunity as it is very important. Note that pausing for the interlocutor to answer is not the same as allowing the interlocutor to dominate the dialogue with verbose answers that distract from the purpose of the interview. An occasional polite interruption may be necessary to keep the dialogue on track. These pauses are often ''the best time'' to end the encounter and thank your interlocutor for their time. You should also consider a "spider moment" to be a success even if you achieve nothing more in the dialogue. It signifies that they have genuinely reflected on whether or not their beliefs are truly justified. ===Summarizing their position === Take care to avoid misunderstandings by focusing intently on what your interlocutor is telling you and clarifying the interlocutor's position before asking a challenging question. If it is helpful, write key points down so you can later revisit them. If you are having trouble making out their words, try summarizing their statements back to them and check whether you heard them correctly. What goes for the words also goes for concepts: if their point is difficult, rephrase it in a way that you find easier to grasp and ask whether you understood them correctly. Don't move on to your question if they have not agreed that you understood them. If you later discover that you have misunderstood their position, recognize and apologize for the error. When summarizing their position, talk about what you've heard rather than what they've said: '''"What I'm hearing is ..."''' or '''"If I understand you correctly, ..."'''. Don't put words into their mouth with "you" statements such as, "You mean that ..." or "What you're saying is ..." Such statements may sound accusatory and trigger defensiveness. You may however phrase your summary in a way that makes flawed reasoning more obvious, but do so free from terms that imply a negative value judgement. This gives the interlocutor a chance to spot the flaw and refine their statement. Summarize the interlocutor's beliefs with non-possessive phrasing: talk about "the belief" and "this way of knowing" rather than "your belief" or "your way of knowing". Talk about the belief as an independent thing that you are both examining. This will help them feel safer questioning it. People build part of their identities on certain core beliefs, which is partly what makes those beliefs extremely tenacious, and why people take offense when they are questioned.<ref>[https://ed.stanford.edu/sites/default/files/identity_belief_and_bias_2.pdf Geoffrey L. Cohen: Identity, Belief, and Bias], 2012</ref> By consistently referring to “the belief” not "your belief", you avoid conflating their identity with what they believe. This can put them more at ease during the interview and more willing to question the belief. ===Asking questions effectively === Asking challenging questions about our ways of knowing is at the heart of SE. What to ask is covered under [[Street_Epistemology#Examining_epistemology|Examining epistemology]], and you can find thousands of examples of questions in the Atheos App.<ref>http://www.atheos-app.com/</ref> Here we look at how you ask your questions. The delivery of your questions will have a major impact on how they are received. You should ask them in an open, neutral manner, free from assumptions. Ask yourself, "Would I consider this a fair question if it were asked about my beliefs?" If not, maybe it's not a fair question to ask about their beliefs. ====Asking open questions ==== Try to ask your questions in an [[w:Open-ended_question|open manner]]. Phrase it as an open-ended "how", "what", or "why" question. These invite explanations, which give you the opportunity to better understand your interlocutor’s position and choose the best approach to examine the foundational belief. Compare that to a yes/no question or a "who" or "where" question which needs only a short answer. For example, “Do you believe in God?” is closed (yes/no), while '''"What beliefs do you have regarding a god or gods?"''' is open and invites a nuanced explanation of their position. Sometimes there is no natural way to open up your question — because what you are offering is in fact a statement. It's okay to use such "..., do you agree?" statement-questions sparingly to move the dialogue forward by agreement. ====Using neutral language ==== Find ways to make your questions neutral. For example, if you present a [[w:Defeasibility|defeasibility]] test by asking “What evidence would lower your confidence in the belief?”, the word “lower” implies that you want to take away their belief. This may trigger a defensive response to protect their belief instead of analyze it honestly. However, if you replace "lower" with the more neutral word "change", as in “What evidence would change your confidence in the belief?”, the question becomes non-threatening by admitting answers regarding evidence that would increase their confidence. Ask your question using "we" instead of "you", as if you are colleagues working together on the problem. For example, "How might we use faith to decide which claim is correct?" instead of the more-accusatory "How might you use faith...". This is part of building a collaborative partnership with your interlocutor. You can also use "one" instead of "you" to make a question less personal and more general: "How does one being raised with this belief make it true?" versus "How does your being raised with this belief make it true?" Avoid leading questions and words that carry emotional or moral judgments. For example, "You can't use faith to determine which claim is correct, can you?" is leading, negative, and closed. It seems to imply only "No, I can't" as a valid answer. Compare that to asking "How might we use faith to determine which claim is correct?" This is neutral and open. It allows for the possibility that the interlocutor has a way to do so, even if you think there isn't one. Keep in mind that you will learn by doing. You will likely ask questions poorly at first, and only discover this after reflecting on the conversation later. Don't get discouraged though. It will definitely get easier with practice! ===Giving and receiving feedback === We all need [[w:Corrective_feedback|feedback]] to learn and grow — but it can be challenging to deliver it without raising defenses and receive it without becoming defensive. You might receive unsolicited feedback for example if you are asking too many questions, or interrupting the interlocutor. You might solicit the interlocutor's feedback after the dialogue to discover ways to improve your manner, and what you are doing well. You can accept feedback more easily by setting aside your ego: tell yourself that this is feedback on your behavior and a great opportunity to learn and improve — it is not an attack on your value or character. As you receive feedback, so should you be willing to give feedback to your interlocutors. You might give positive feedback, such as encouraging the interlocutor when they show willingness to revise beliefs, when they engage openly with difficult questions, when they become more ok with not knowing, and when they determine to act on a realization that some way of knowing is not so reliable. You might also find yourself in the uncomfortable position of giving negative feedback about difficult behaviors such as rambling, derailing the dialogue into unrelated topics, delivering lengthy monologues, evading questions, responding to every question with the same few talking points, being aggressive or using insults. How you deliver the feedback has great impact on the outcome for yourself, the interlocutor and the relationship. Most importantly, ensure that safety is in place — if you sense defensiveness, your feedback will only make them more defensive (see [[Street_Epistemology#Building_respect|Building respect]]). Assuming you are in a position where your feedback won't be perceived as an attack, deliver it by first outlining the situation where the behavior occurred, then describing what they did, and lastly by explaining the impact, positive or negative, that their behavior had. Pause to let them absorb it, and let them suggest what they might do in future, only giving your recommendation if they ask for it. ==Examining epistemology == Here we get to the heart of the matter - exactly how do you work towards mutual agreement on whether a way of knowing is sufficiently reliable to justify using it to believe the claim in question? ===The Socratic method === Socratic dialogue is the core dialectical technique used in Street Epistemology. Your questions help the interlocutor to use what they already know to see where they may have gone wrong in reaching a conclusion. Socratic Dialogue is far more powerful than presenting counter-arguments which they can find any reason to dismiss. As presented in AMFCA, the Socratic method proceeds through five steps: #'''Wonder:''' The big question. #'''Hypothesis:''' The interlocutor's proposed answer to the question. #'''Elenchus:''' Question and answer to discover what other reasonable propositions are likely to be true that refute the hypothesis. Proceed by mutual agreement at every step - elenchus is not a debate. #'''Accept or Revise:''' Refine or revise the hypothesis - either without vulnerability to that elenchus, with reduced confidence, or even rejecting it. On the other hand, if the hypothesis survives repeated attempts at elenchus, provisionally accept it. #'''Act accordingly:''' Reduce confidence in and reliance on the hypothesis that is refuted by elenchus. The contradiction that weakly or strongly refutes the hypothesis may derive from: *The hypothesis itself - a strong refutation. *Mutually agreed-on facts - a moderate refutation. *Claims agreed to be plausible or probable - a weak refutation. One way to characterize Street Epistemology is simply as the Socratic method applied to wonder-questions of the form '''"What ways of knowing are sufficiently reliable to justify holding this belief?"''', where the interlocutor's way of knowing is the hypothesis: "X is sufficiently reliable to justify holding this belief". The elenchus may show that the same justification applies equally to two or more mutually contradictory conclusions (the principle behind the Outsider Test). Or, the elenchus may show that the justification is a weak test that is passed with ease by other beliefs that you both agree to be false. Such elenchi show that the way of knowing is an unreliable process for generating true beliefs about the types of claim being considered. If the interlocutor cannot find a more reliable process leading to the claim, they should in the long run reduce their confidence in the claim. Successful elenchus may result in an [[w:Aporia|aporia]] — recognition that they do not know what they thought they knew. An aporia can provide a moment of doxastic openness in which the interlocutor is more willing to revise their beliefs. The Wikiversity course on [[Socratic Methods]] provides additional opportunities for study and practice. === 6.2 Asking the right questions === Street Epistemology applies Socratic Method to ask challenging questions about how we know what's true. You'll find that there is no simple formula or script, rather it's a skill that takes practice and patience to develop. However, we can still give some general advice on crafting good questions. First and foremost, stay focused on how they came to their beliefs rather than on whether the beliefs are true or not. Their claims may indeed be true, but we are only justified in believing them if we have a reliable way of ''knowing'' them to be true. Discovering which process they use is covered in [[Street Epistemology#Eliciting_the_interlocutor's_epistemology|Eliciting the interlocutor's epistemology]]. Once they offer some way of knowing, do not assume that their process is unreliable, lest the dialogue devolves into "It is reliable!", "No, it's not reliable!" gainsaying (see Monty Python's [[w:Argument_Clinic|Argument Clinic]]). Work only from assumptions that the both of you agree on, and work backwards from the interlocutor's starting point. This applies even if they are using faith: you are then in a dialogue where ''you'' believe faith to be unreliable, and ''they'' believe otherwise. So meet them where they are — they think faith is a reliable way of knowing, and you ask questions to explore whether or not that is true. Leave open the possibility that they have some way to show that their way of knowing is reliable. Frame it as a collaboration in that you both share the goal of agreeing on ''just how reliable this way of knowing is''. It's even possible that you could be the one moving toward their position. ===Deepities === Coined by philosopher [[w:Daniel_Dennett|Dr Daniel Dennett]], a deepity is a proposition or definition that seems profound at first glance by having meanings on multiple levels - but on closer inspection these meanings turn out to be either true but trivial or significant but false (or even nonsensical). Classic deepities include "faith is the substance of things hoped for, the evidence of things not seen", "love is just a word", "everything is connected" and "reality is created by consciousness”. You can find many more examples in the deepities test in the Atheos App<ref>http://www.atheos-app.com</ref> and a recent paper "On the reception and detection of pseudo-profound bullshit".<ref>[http://journal.sjdm.org/15/15923a/jdm15923a.html On the reception and detection of pseudo-profound bullshit], by Pennycook et al, 2015</ref> If your interlocutor uses what appears to be a deepity you should not assume that it is a deepity, nor accuse them of using a deepity. Instead, ask them to unpack the meaning(s) of their statement, in plain language. Try to rephrase plainly in a way they might agree with, and ask them what they would change about your rephrasing. Hopefully you can work towards agreeing on a non-deepity version of their definition or claim, which is tractable and won't slip away into a different meaning the moment you ask a challenging question about it. ===Faith === [[w:Faith|Faith]] mostly comes up in discussions about religious belief, but occasionally in other contexts. When it does, work with the interlocutor to agree on a straightforward definition of faith, and don't start examining the reliability of faith until you've agreed on what faith is. Often people will agree that faith has an element of "choosing to be more confident than you would be if you were relying solely on evidence," though more likely phrased along the lines of "evidence brings you only so far, faith takes you the rest of the way [to knowing]." It can work to first agree with a positively-worded definition of faith as a way of knowing, and then raise equivalent rephrasings that make the epistemological weaknesses more obvious. Don't expect to the interlocutor to agree on "pretending to know things you don't know" as a working definition. People also make use of the word "faith" to mean strong belief or trust in someone or something.<ref>[https://www.merriam-webster.com/dictionary/faith Merriam-Webster entry on faith]</ref> The "faith as trust" definition makes sense if by "faith in God" one means "trusting God to fulfill certain promises". Such a definition does not make sense in the context of "trusting God to exist" - and yet often a believer will say "evidence alone is not enough to know God is real, you also need faith". They're not talking about trusting a human being such as their religious leader or the author(s) of their holy book to be correct about God's reality: they're talking about believing anyway, without the evidence. Even so, an [[w:Equivocation|equivocation]] of faith that allows it to also mean "trust" is much of what gives faith status as a moral virtue in religious circles: to not have faith is seen as being mistrustful or even disloyal. If your interlocutor defines faith in terms of trust, you might still be able to agree that such a definition does not work well for the question of God's very existence. AMFCA also has a section entitled "Disambiguation: Faith Is Not Hope" on page 26 that may help with this. You may meet sophisticated believers who use the word "faith" differently from popular usage. For example, they may use "faith" to mean the act of committing to a belief that one arrived at through evidence and reason — faith commitment as an act of doxastic closure. In this case, they aren’t claiming to use faith as a way of knowing their beliefs are true, but rather as an end state of some other way of knowing. They may also use "faith" to mean simply acting on what you consider likely to be true, as in "you have faith when you cross the road". You might have some luck seeking to clarify the difference between "faith in God's existence" and broader uses of the word. Such interlocutors generally claim that to have sufficient (non-faith) evidence to warrant relatively high confidence, so you will have to examine the reliability of these justifications. ===Relativism === [[w:Relativism|Relativism]] in epistemology is the idea that true and false are relative to the context in which they arise .<ref>Baghramian, Maria and J. Adam Carter, "[https://plato.stanford.edu/archives/win2019/entries/relativism/ Relativism]", The Stanford Encyclopedia of Philosophy (Winter 2019 Edition), Edward N. Zalta (ed.)</ref> In dialogues, you most frequently encounter this in the refrain "My religion is true for me; and their religion is true for them", "All paths lead to God" or "All religions are equally true". In applying relativism, the interlocutor is treating religious claims as if they are subjective. A subjective claim might be "Chocolate is my favorite flavor of ice cream" or "this ice cream tastes weird to me" — it's a claim about your own preferences or perceptions; true for you, possibly false for someone else. An objective claim is one that is true or false for everyone regardless of their beliefs or opinions (provided they are defining words the same way), such as "The mass of this ice cream is 75-80 grams". Holding a claim to be both subjective (true for me) and objective (as a statement about the external world) entails a contradiction that you can use Socratic elenchus to draw out, if you can first agree on the ideas of subjective and objective claims. One increasingly common flavor of relativism is the idea that "All gods are the same god." This is subtly different from "It's true for me". It's a new claim that contradicts what followers of most religions believe about their God being the real one. It's easy enough to ask questions that make this obvious: people use faith, experiences, and holy books to come to very different and contradictory conclusions about the nature of god. You can also ask how the interlocutor knows that all gods are in fact one, and examine their ways of knowing, as usual. You can also use the anti-relativism roadmap presented in AMFCA,<ref>{{cite book |last=Boghossian |first=Peter |author-link=w:Peter_Boghossian |date=November 1, 2013 |title=A Manual for Creating Atheists |publisher=Pitchstone Publishing |page=280 |isbn=978-1939578099}}</ref> Chapter 8: Beyond Relativism, which addresses the "It's true for you" brand of relativism. Broadly, you start by addressing whether some people misconstrue reality by coming to false conclusions about it. You then introduce the idea of processes for knowing reality, and whether some are more reliable than others. For example, flipping a coin is an unreliable process for knowing reality — you'll only be right 50% of the time. Then you talk about how to discover which processes are more reliable, perhaps using ideas about evaluating reliability from the section on “[[Street Epistemology#Asking_questions_effectively|Asking questions effectively]]”. ===Outsider tests === Popularized by author John W. Loftus (a former Christian apologist turned atheist) in his book, ''The Outsider Test for Faith: How to Know Which Religion Is True''<ref>{{cite book |last=Loftus |first=John W. |author-link=w:John_W._Loftus |date=March 19, 2013 |title=The Outsider Test for Faith: How to Know Which Religion Is True |publisher= Prometheus |pages=300 |isbn=978-1616147372}}</ref>, the [[w:John_W._Loftus#The_Outsider_Test_for_Faith|Outsider Test]] for Faith helps interlocutors to see that their reasons for believing are no different from the reasons used by those from other religions, and thus not a good way to judge which religion is true. You can use an outsider test for any kind of evidence used to justify contradictory conclusions. For religions, this includes faith, [[w:numinous|numinous]] experiences, fulfilled prophecies, reported miracles, and answered prayers. For example, to apply the outsider test to a Christian who claims that faith gives them confidence that Jesus is real, first clarify what they mean by faith. Then ask '''"Does the Hindu have faith that [[w:Vishnu|Vishnu]] is real?"''' Usually, they will agree. If not, you might have to clarify further what constitutes faith. Then ask something along the lines of, '''"If the Hindu and the Christian both use faith to become confident about different gods being real, how can faith help me to determine which is real?"''' Pick an outsider religion that is as incompatible as possible with the interlocutor's religion: Hinduism is usually a good outsider to any Abrahamic religion, while Islam makes a good outsider to Hinduism because Islam is emphatically monotheistic. Avoid picking the dominant religion in your area as interlocutors may assume you are trying to convert them. You can also see the outsider test of faith in the form of a 5-step Socratic dialogue: #'''Wonder:''' How can we reliably know that this god is real? #'''Hypothesis:''' Having faith is a reliable way to know this god is real. #'''Elenchus:''' Do many religions use faith? (yes) Do religions all believe in the same god, with identical attributes? (no) How can one use faith to determine which religion has the correct god? (you can't — the elenchus, ideally provided by the interlocutor themselves) #'''Accept or revise:''' Faith does not reliably lead one to true belief regarding the reality of a specific god. #'''Act accordingly:''' Don't use faith to be confident about the reality of a god. You can then cycle back to wonder, and the interlocutor may provide other hypotheses. After many hypotheses are found insufficiently reliable, they may begin themselves to question whether there is a sufficiently reliable way to know. Be alert for the “[[Street Epistemology#The_"Spider_on_the_ceiling"|spider on the ceiling]]”! Outsider tests work best in the religious domain, where you find competing claims based on the same ways of knowing. They don't work well in pseudoscience and conspiracy theory domains for example, where you find many independent claims, and people who believe one such claim often also believe many others by using similar ways of knowing. With any kind of claim, you can help the interlocutor find more plausible explanations for the evidence they've provided, as well as real-life instances where people make errors when relying on the same way of knowing that they provided. ===Defeasibility tests === Sometimes the interlocutor is doxastically closed in the sense that they believe they cannot be mistaken. In a sense, their belief in the claim is stronger than their commitment to reason and evidence. You can use a [[w:defeasibility|defeasibility]] test to introduce a glimmer of uncertainty.<ref>Proving the Negative, [http://www.provingthenegative.com/2011/02/defeasibility-test.html The Defeasability Test], February 5, 2011</ref> To use the defeasibility test, start by asking "What evidence would change your confidence in the truth of this claim?" Another way to phrase it is literally as a test: “How might we test the belief, in a way that would be difficult to pass if the belief were false?" Avoid using scientific terms like "[[w:hypothesis|hypothesis]]" and “[[w:Falsifiability|falsifiable]]” with people unfamiliar with those terms - they may misinterpret you as asking them to demonstrate that their belief is false, when you are really asking them to devise a test that the belief should fail if it were false. Instead, use words like “test”, “study”, or “examine”. Your interlocutor may give evidence that would make them more confident. If they were already 100% confident, you might ask how this is possible. Your next step is to ask, '''"And what evidence would make you less confident?"''' They may say it would take extraordinary evidence to reduce their confidence — but do they have extraordinary evidence to support their confidence? They may also cite some specific evidence that would reduce their confidence - as in the "bones of Christ" dialogue in AMFCA [p 59-62]. If this happens, ask them to be more specific about the criteria. You might then discuss the ways in which one could easily discredit any claim of having acquired said evidence, since it would be extraordinary for such "evidence" to exist in the first place. They may suggest evidence that obviously does not or cannot exist, or would be highly unlikely to exist even if the belief were false. They may even claim that no evidence could alter their confidence — their belief is self-affirming. If they value evidence at all, try asking "If evidence has no power to alter your confidence, are you really believing based on evidence in the first place?" This may move them to bring up faith. You may also ask about human fallibility: "Do humans ever misconstrue reality?", and ask how they can distinguish a false self-affirming belief from a true one. They may say their belief is more fundamental than evidence and reason itself, which means you may have encountered a case of [[w:Presuppositional_apologetics|presuppositional apologetics]], which is beyond the scope of this course. ===What to avoid === Knowing what to do is half the story. You'll also need to know what not to do, particularly when "what not to do" is something that comes naturally or intuitively to people. The following are some easy-to-make mistakes: '''Arguing or debating:''' It is a common mistake to revert to the default mode of presenting facts and arguments when the interlocutor makes a claim that you believe to be wrong. It's very tempting to think that if you simply present relevant facts and reasoning, they will realize they are mistaken. Resist the temptation! Presenting good evidence to an unreliable epistemology that doesn't value evidence is not going to help. You will only turn a dialogue into a debate, and the interlocutor may double-down by finding ways to dispute or interpret contrary facts so as to defend their conclusion. This is known as the "[[w:Confirmation_bias#backfire_effect|backfire effect]]", net result being that one becomes more confident in a false belief over time due as one finds more ways to discount or reinterpret contrary data to one's own satisfaction. '''Targeting reasonable hypotheses:''' If the interlocutor makes some reasonable hypothesis or plausible claim, for example claiming the occurrence of a mundane event in a holy text, you may be tempted to target how they know it happened. Don't do it. Mundane evidence will do for mundane claims. Stay focused on the extraordinary claims, the ones that call for [[w:Sagan_standard|extraordinary evidence]]. '''Denigrating the interlocutor's personal experience:''' Similarly, don't challenge whether someone in fact experienced something that seemed to them to be transcendent or miraculous. Target instead the process by which they concluded that their experience had a supernatural cause. '''Constructing hypothetical scenarios:''' Avoid introducing hypothetical scenarios, because they are weaker than real scenarios. You can see when you are getting hypothetical when you use the [[w:Subjunctive_mood|subjunctive mood]]: "If you ''were'' to...". For example, when the interlocutor says they use faith, and you ask "What if someone ''were'' to use faith to know that there are unicorns living on a distant planet — how reliable is faith at guiding them to a true belief?" the interlocutor can point out that this is a made-up scenario, it doesn't actually happen. Instead, make the observation that many people right now are actually (not hypothetically!) using faith to believe in many competing religions. While thought experiments like [[w:Russell's_teapot|Russell's Teapot]] and [[w:Brain_in_a_vat|Brain-in-a-Vat]]<ref>Internet Encyclopedia of Philosophy, entry on [https://iep.utm.edu/brainvat/ The Brain in a Vat Argument].</ref> are common in philosophical arguments, prefer to keep the dialogue grounded in the here and now, rather than drifting into hypothetical worlds. Asking too many questions: The too-many-questions mistake shows up more in online comment threads: asking more than one question in your turn, interrupting with another question before the interlocutor has finished their answer, or asking a follow-up question without checking that you understood their previous answer. In comments and email it is tempting to deal with multiple topics and present multiple questions and skip the "Did I understand you?" step, because of the long turnaround time for replies. Don't do it. If you are impatient, upgrade to a call or face-to-face meeting. Think of your one best question and ask it. When the interlocutor replies, rephrase and summarize their point and ask whether you have understood them correctly. If you have, you can move on to the next question. Make the interlocutor's thought process crystal clear to them so that they can see the flaws, and that requires working one step at a time. === Assignment === Please describe each of the following concepts in your own words. How would you convey your understanding of each of these topics to your next-door neighbor, or your hair dresser? It is perfectly acceptable to simply reflect on each of these questions, choose a (real or imaginary) friend to discuss this with, or write your answers in a notebook or diary. *The Socratic method; *Effective dialogue questions; *Deepities; *Faith; *Relativism; *Outsider tests; *Defeasibility tests; *What to avoid. ==Ending the dialogue == Ending a dialogue in a polite, considerate, positive way can increase the interlocutor’s sense of having had a meaningful conversation with you. ===When to end the dialogue === Your best moment to end the dialogue is if the interlocutor experiences a moment of realization that they don't know what they thought they knew. Your best indication that this is happening is when you detect a "spider moment". See [[Street Epistemology#The "Spider_on_the_ceiling"|The "Spider on the ceiling"]]. Ending the dialogue there leaves the interlocutor pondering their own epistemology. They may continue reflecting on it long after the dialogue. If you end at another point, they are less likely to engage in such reflection. On the other hand if you see indications that the dialogue is heading downhill, it is better to end it on a positive note. Here are some signs that the dialogue is overdue for wrapping up: *The time you agreed on for the dialogue is up. *The dialogue wanes or becomes repetitive. *Either of you has somewhere else to go. *Either of you is becoming irritated or impatient. *Either of you is slowing, shutting down or losing interest. *Either of you is feeling overloaded - too many questions to think about. *Either of you seem determined to keep the other there until they "win". Do not hesitate to end the conversation if the interlocutor appears intoxicated, becomes verbally/physically aggressive, or seems otherwise incapable of understanding your questions. Productive dialogue is impossible under these circumstances. You may also uncover information which presents an ethical dilemma, such as someone who is developmentally disabled or critically ill and relying on their beliefs for emotional support. Refer to [[Street Epistemology#Developing_the_right_mindset|Developing the right mindset]] to ensure you are choosing to engage for the right reasons. If you determine that ethically responsible conditions no longer exist, it's time to wrap up the dialogue in a positive manner. === 7.2 Ending in a positive manner === These are some things you can do to help end the dialogue on a positive note: *Thank the person for their time. *Apologize for any conversational missteps. *Mention things they did that you thought well of, such as reconsidering beliefs, contemplating change, or engaging with difficult questions. *Describe a follow-up that have set for yourself as a result of the dialogue and invite them to do the same. *Invite them to contemplate a relevant question that remains unanswered. *If they have expressed interest in your sources, provide them with a link, book recommendation, or contact information. *Ask about a time or place where you might be likely to encounter them again, or exchange details and schedule a follow-up meeting. You can also ask one or more questions about the dialogue to get some feedback on how you are doing: *Did you enjoy the talk? *What questions did you find most intriguing? *What did you find frustrating? *What is the take-away for you? *What would you rather I did differently? If they experienced Socratic aporia, ask them to ponder the question that stimulated the aporia for the next time you meet. Also set yourself the task to learn more about any interesting ideas or insights that the interlocutor provided. Try inviting the interlocutor when reflecting on their belief to anticipate the questions that you might ask them the next time, so that they are well-prepared to answer them. Such reflection enables them to continue the dialogue in their imagination. The net effect should encourage them to think critically about their own epistemology if they were not doing so already. Should the interlocutor express an active willingness to reconsider their beliefs, don't leave them hanging. If your dialogue is with someone close to you, you are in a position to help them directly, but for strangers, try to ensure that they know what next steps are available and that they have sources of support. For example, you may refer the person to local community groups or other resources. ==After the talk == Reflect on your performance. Consider follow-up dialogues that can go much deeper than a one-time chat. Also, remember to take care of yourself. ===Reflecting on your performance === Try reflecting on the dialogue and noting the details soon afterwards. If you recorded the audio or video of your dialogue, consider waiting a day to watch or listen to the recording. The distance helps you reflect on the dialogue from a fresher perspective. Here are some important items to document after a dialogue: '''Basic data about the dialogue''' *Date, time and location *Name of the interlocutor *Statement of the belief or belief system investigated *Foundational belief and ways of knowing *Before-and-after confidence '''Summary of the dialogue''' *Important questions that you asked and the interlocutor's responses *Their method(s) for justifying the belief *How they first came to believe it *Their definition of faith (if discussed) *Turning points and important moments in the dialogue *How you ended the dialogue *The interlocutor's feedback regarding the dialogue '''Reflective evaluation''' *Whether rapport was good and what affected it *Specific strategies you used (defeasibility, outsider test of faith, etc) *Explanation of what worked well (or not) about each of your major questions. *Indicators of change in the interlocutor: Becoming aware of unreliable way of knowing, willingness to revise beliefs, contemplating change *Goals for follow-up dialogues with the interlocutor *Recommendations for dialogues with future interlocutors Consider using this [https://docs.google.com/document/d/16MNbTjNNV_1xqm_CjiqMefRWRUTzDvIhoGPiCjstHpY/edit Reflection template] for reflecting on your dialogues. [https://docs.google.com/document/d/17SnJXtGjWs_1f2SuyUaIHTd_OZrahEDguSNR3sxU-1U/edit Here is an example] of the sheet filled out after [https://www.youtube.com/watch?v=6QerIfN7SQs this interview]. Or, create your own template customized to your particular approach to SE. After completing your self-evaluation, consider sharing the experience with others on the Street Epistemology Facebook Group, Youtube, or other forums and social media platforms. Ask for advice and suggestions for improvement on your next encounter. Sharing your experience is very helpful for other, less-experienced Street Epistemologists. ===Continue the relationship === There's no requirement that you maintain contact after your talks, but if you wish to do that, consider exchanging contacts with the interlocutor and setting up a lunch, dinner, coffee or other occasion to follow up with them and continue the dialogue. If the interlocutor is contemplating their beliefs and ways of knowing, you can offer to stay in touch and be available to talk with them. Prepare well if you are planning to meet with an interlocutor again. Review your last meeting notes or recording. Think about appropriate follow-up questions and possible directions to explore next. If you posed a parting thought, ask if they have any new ideas based on that. ===Taking care of yourself === If Street Epistemology remains a hobby that takes a back-seat to family, work, and leisure time, you will find it more fulfilling, and reduce the likelihood of burnout. Some signs of burnout include loss of empathy for your interlocutor, increased frustration in the face of doxastic closure, and loss of humility during your talks. Here are some suggestions on how to take care of yourself. *Limit your dialogues to a maximum time, never more than an hour, but 10-20 minutes is better. End the dialogue politely if it extends past your personal limit. *When engaging by correspondence or over a comment thread, watch out for rumination: spending lots of time and effort thinking about what to say next, or replaying the dialogue over in your head. In that case, it may be time to bow out of the dialogue. *You might review [[Street Epistemology#Developing_the_right_mindset|Developing the right mindset]], watch some recorded interviews, and talk about what's happening. You may even find it helps to take a break: disconnect entirely, take a vacation where you do nothing related to Street Epistemology for a few weeks. Then dive back in when you’ve regained a positive frame of mind. ==You're ready! == Congratulations on completing this Street Epistemology Course! You should now have a good idea of how to do it: the mindset involved, when and where to engage, how to strike up dialogue, how to keep it on track and be respectful, how to finish up the dialogue, and of course how to help the interlocutor evaluate their own epistemology. If you have suggestions for this course, please leave feedback on the [[Talk:Street_Epistemology|course discussion page]]. Best of luck in helping humanity leave behind unreliable ways of knowing! === Assignment === *Practice Street Epistemology. *Improve your practice of Street Epistemology. ==Appendices == ===Terminology === Many terms are from Dr. Peter Boghossian's "A Manual for Creating Atheists". Here are some helpful definitions of terms as practitioners of Street Epistemology understand them. For crucial words like "faith" and "knowledge", remember to negotiate with your interlocutor to agree on what they mean in the context of your dialogue - or you will only be talking past each other. *'''[[w:Apologetics|Apologetics]]:''' Reasoned arguments or writings in justification of something, typically a theory or religious doctrine. See John W. Loftus' 2015 book ''How to Defend the Christian Faith: Advice from an Atheist''<ref>{{cite book |last=Loftus |first=John W. |author-link=w:John_W._Loftus |date=November 1, 2015 |title=How to Defend the Christian Faith: Advice from an Atheist |publisher=Pitchstone Publishing |pages=280 |isbn=978-1634310567}}</ref> for why apologetic arguments have failed to be persuasive. *[[w:Aporia|'''Aporia''']]: An expression of doubt. In Street Epistemology, a state of puzzlement caused by realizing that one does not in fact know what one thought one knew. In Socratic Dialogue, successful elenchus produces an aporia, which increases one's doxastic openness. *'''Deepity''': A statement that can be read in two different ways: one way that's true but trivial, and another that's much more intriguing but false.<ref>[https://www.theguardian.com/lifeandstyle/2013/may/25/change-your-life-life-deepities-oliver-burkeman This column will change your life:deepities], The Guardian, May 25, 2013, Oliver Burkeman</ref> *'''Doxastic Openness/Closure''': The term “[[w:Doxastic_logic|doxastic]]” refers to stances that one takes regarding one's beliefs. Doxastic openness as used in AMFCA [Ch. 3] is the willingness to revise beliefs in response to evidence, while doxastic closure is the unwillingness to revise beliefs. *[[w:Socratic_method#Method|'''Elenchus''']]: Refers to the question-and-answer part of Socratic dialogue, in which a hypothesis is refuted or cast doubt upon by developing counterexamples or deriving contradictions. *[[w:Epistemology|'''Epistemology''']]: The theory of knowledge, especially with regard to its methods, validity, and scope, and the distinction between justified belief and opinion. *[[w:Interlocutor_(linguistics)|'''Interlocutor''']]: A participant in a dialogue. From your perspective as the Street Epistemologist, the interlocutor is the person you are talking with, sometimes abbreviated as “IL”. *[[w:Knowledge|'''Knowledge''']]: As a working definition for examining beliefs, knowledge is justified true belief in a proposition. Differs from everyday usage of knowledge to refer to facts, information or skills acquired through experience or education. *[[w:Socratic_method|'''Socratic Method''']]: Named after the philosopher Socrates, this is also known as elenchus. It is the use of questions to probe a point of view, stimulate critical thinking, and evaluate the consistency of ideas. The Wikiversity course on [[Socratic methods]] provides additional instruction. *'''Street Epistemology''': Coined by Dr. Peter Boghossian in his book ''A Manual for Creating Atheists''. An activist approach to helping people reduce their reliance on faith as a way of knowing religious claims to be true. A method of rational dialogue that examines the way we form beliefs, to evaluate which ways of knowing are sufficiently reliable to justify belief. The word “street” indicates that the method is being used on the street with strangers, but in practice, it can be used in casual conversation with acquaintances or people one knows well. ===Resources === *{{cite book |last=Boghossian |first=Peter |author-link=w:Peter_Boghossian |date=November 1, 2013 |title=A Manual for Creating Atheists |publisher=Pitchstone Publishing |page=280 |isbn=978-1939578099}} — the book that started it all. AMFCA is a must read for anyone interested in conducting Street Epistemology *{{cite book |last=Ariely |first=Dan |author-link=w:Dan_Ariely |date=September 17, 2024 |title=Misbelief: What Makes Rational People Believe Irrational Things |publisher=Harper Perennial |pages=320 |isbn=978-0063280434}} *'''Atheos App'''<ref>http://www.atheos-app.com/</ref> — Teaches Street Epistemology by presenting multiple-choice responses to things that interlocutors might say. *'''www.streetepistemology.com'''<ref>http://www.streetepistemology.com</ref> — an aggregator site for news, debates, critiques, plugs, blogs, videos, and everything else related to Street Epistemology. *'''On-camera dialogues on YouTube, Periscope and Blab''' — By watching real people practicing Street Epistemology in the real world, you radically improve your practice of the techniques. *[https://docs.google.com/document/d/16MNbTjNNV_1xqm_CjiqMefRWRUTzDvIhoGPiCjstHpY/edit '''The Street Epistemology Reflection'''] — A template for reflecting on dialogues. Here is [https://docs.google.com/document/d/17SnJXtGjWs_1f2SuyUaIHTd_OZrahEDguSNR3sxU-1U/edit an example] of the sheet filled out after [https://www.youtube.com/watch?v=6QerIfN7SQs this interview]. == References == <references/> {{CourseCat}} [[Category:Life skills]] [[Category:Applied Wisdom]] [[Category:Clear Thinking]] [[Category:Philosophy]] [[Category:Epistemology]] [[Category:Courses]] 3b361n31s6opa1s4iba489301ld4m2j 0 285380 2819270 2819140 2026-07-24T13:51:55Z Young1lim 21186 /* Applications */ 2819270 wikitext text/x-wiki === Introduction === * Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]]) * Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]]) * Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]]) === Handling Repetition === * Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]]) * Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]]) === Handling a Big Work === * Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]]) * Functions & Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]]) * Functions & Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]]) * Functions & Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]]) === Handling Series of Data === ==== Background ==== * Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]]) ==== Basics ==== * Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]]) * Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]]) * Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]]) * Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]]) ==== Examples ==== * Spreadsheet Example Programs :: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]]) :: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]]) :: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]]) :: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]]) ==== Applications ==== * Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20260724.pdf |A.pdf]]) * Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]]) * Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]]) * Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]]) * Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]]) * Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]]) * Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]]) === Handling Various Kinds of Data === * Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]]) * Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]]) * Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]]) * Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]]) === Handling Low Level Operations === * Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]]) * Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]]) * Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]]) * Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]]) === Declarations === * Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]]) * Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]]) * Scope === Class Notes === * TOC ([[Media:TOC.20171007.pdf |TOC.pdf]]) * Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library * Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements * Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers * Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts * Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops * Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control * Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions * Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope * Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion * Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions * Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications * Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions * Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications * Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) * Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2) * Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO * Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions * Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications * Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum * Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List * Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing * Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing <!----------------------------------------------------------------------> </br> See also https://cprogramex.wordpress.com/ == '''Old Materials '''== until 201201 * Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]]) * Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]]) * Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]]) * Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]]) * Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]]) * Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]]) * Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]]) * Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]]) * Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]]) * Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]]) * Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]]) * Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]]) * Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]]) <br> until 201107 * Intro.1.A ([[Media:Intro.1.A.pdf |pdf]]) * Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]]) * Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]]) * Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]]) * Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]]) * Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]]) * Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]]) * Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]]) * Array.1.A ([[Media:Array.1.A.pdf |pdf]]) * Type.1.A ([[Media:Type.1.A.pdf |pdf]]) * Structure.1.A ([[Media:Structure.1.A.pdf |pdf]]) go to [ [[C programming in plain view]] ] [[Category:C programming language]] </br> modajr1w0lqvvg7qt6d0jazi30x20ky 0 305362 2819291 2819210 2026-07-24T16:35:52Z Dc.samizdat 2856930 /* Chiral symmetry operations */ 2819291 wikitext text/x-wiki {{Short description|Regular object in four dimensional geometry}} {{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}} {{Infobox 4-polytope | Name=24-cell | Image_File=Schlegel wireframe 24-cell.png | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<br>(vertices and edges) | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]] | Last=[[W:Omnitruncated tesseract|21]] | Index=22 | Next=[[W:Rectified 24-cell|23]] | Schläfli={3,4,3}<br>r{3,3,4} = <math>\left\{\begin{array}{l}3\\3,4\end{array}\right\}</math><br>{3^1,1,1} = <math>\left\{\begin{array}{l}3\\3\\3\end{array}\right\}</math> | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}} | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]] | Face_List=96 [[W:Triangle|{3}]] | Edge_Count=96 | Vertex_Count= 24 | Petrie_Polygon=[[W:Dodecagon|{12}]] | Coxeter_Group=[[W:F4 (mathematics)|F₄]], [3,4,3], order 1152<br>B₄, [4,3,3], order 384<br>D₄, [3^1,1,1], order 192 | Vertex_Figure=[[W:Cube|cube]] | Dual=[[W:Polytope#Self-dual polytopes|self-dual]] | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]] }} [[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]] In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C₂₄''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for "octahedral complex"), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}} The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}} ==Geometry== The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}} === Coordinates === The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. ==== Great squares ==== The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}} {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}} The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} ==== Great hexagons ==== The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell & Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} ==== Great triangles ==== The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} ==== Hypercubic chords ==== [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. ==== Geodesics ==== [[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]] The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} {| class="wikitable floatright" |+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell |- style="text-align:center;" ![[W:Coxeter plane|Coxeter plane]] !colspan=2|F₄ |- style="text-align:center;" !Graph |colspan=2|[[File:24-cell t0_F4.svg|100px]] |- style="text-align:center;" ![[W:Dihedral symmetry|Dihedral symmetry]] |colspan=2|[12] |- style="text-align:center;" !Coxeter plane !B₃ / A₂ (a) !B₃ / A₂ (b) |- style="text-align:center;" !Graph |[[File:24-cell t0_B3.svg|100px]] |[[File:24-cell t3_B3.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[6] |[6] |- style="text-align:center;" !Coxeter plane !B₄ !B₂ / A₃ |- style="text-align:center;" !Graph |[[File:24-cell t0_B4.svg|100px]] |[[File:24-cell t0_B2.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[8] |[4] |} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell & Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} === Constructions === [[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]] Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). ==== Relationships among interior polytopes ==== The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} ==== Boundary cells ==== Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element. {| class=wikitable |- align=center |\||style="background-color:#808080;"|v||style="background-color:#E6194B;"|e||style="background-color:#3CB44B;"|f||style="background-color:#FFE119;"|c |- align=right |align=left style="background-color:#808080;"|v||style="background-color:#E0F0FF"|'''24'''||8||12||6 |- align=right |align=left style="background-color:#E6194B;"|e||2||style="background-color:#f0FFE0"|'''96'''||3||3 |- align=right |align=left style="background-color:#3CB44B;"|f||3||3||style="background-color:#f0FFE0"|'''96'''||2 |- align=right |align=left style="background-color:#FFE119;"|c||6||12||8||style="background-color:#f0FFE0"|'''24''' |} Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation. In the [[W:uniform 4-polytope|uniform]] D₄ construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.<ref>[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]</ref> The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells. {| class=wikitable |\||style="background-color:#808080;"|v||style="background-color:#E6194B;"|e||style="background-color:#3CB44B;"|f1||style="background-color:#3CB44B;"|f2||style="background-color:#3CB44B;"|f3||style="background-color:#FFE119;"|c1||style="background-color:#FFE119;"|c2||style="background-color:#FFE119;"|c3 |- align=right |align=left style="background-color:#808080;"|v||style="background-color:#E0F0FF"|'''24'''||8||4||4||4||2||2||2 |- align=right |align=left style="background-color:#E6194B;"|e||2||style="background-color:#f0FFE0"|'''96'''||1||1||1||1||1||1 |- align=right |align=left style="background-color:#3CB44B;"|f1||3||3||style="background-color:#f0FFE0"|'''32'''||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|*||1||1||0 |- align=right |align=left style="background-color:#3CB44B;"|f2||3||3||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''32'''||style="background-color:#f0FFE0"|*||1||0||1 |- align=right |align=left style="background-color:#3CB44B;"|f3||3||3||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''32'''||0||1||1 |- align=right |align=left style="background-color:#FFE119;"|c1||6||12||4||4||0||style="background-color:#f0FFE0"|'''8'''||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|* |- align=right |align=left style="background-color:#FFE119;"|c2||6||12||4||0||4||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''8'''||style="background-color:#f0FFE0"|* |- align=right |align=left style="background-color:#FFE119;"|c3||6||12||0||4||4||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''8''' |} ==Symmetries, root systems, and tessellations== [[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F₄]] group, as shown in this F₄ Coxeter plane projection]] The 24 root vectors of the [[W:D4 (root system)|D₄ root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B₄ and C₄ simple Lie groups. The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F₄]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D₄; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F₄, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F₄ roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576. ===Quaternionic interpretation=== [[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection: * 1 order-1: 1 * 1 order-2: -1 * 6 order-4: ±i, ±j, ±k * 8 order-6: (+1±i±j±k)/2 * 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F₄ [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D₄ root lattice is the [[W:Dual lattice|dual]] of the F₄ and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}} Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}} ===Voronoi cells=== The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D₄]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D₄ lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F₄ lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''⁴. The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]]. ===Radially equilateral honeycomb=== The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}} A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia & Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia & Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia & Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia & Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram₅ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986<ref>{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman & Weinberg|1987}}}}</ref> to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference <math>8\pi</math>, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram₅ geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram₅ geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram₅. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell & Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the "left" is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell & Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon. ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical dodecagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]₅, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell & Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}} Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram₅ path. <s>Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew dodecagram₅ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram₅ has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.</s> At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram₅ geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|<blockquote>Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions).<br><br>Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br>where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br>where 2''q'' + ''r'' + 1 ≤ ''n''.<br><br>For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}</blockquote>|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column. {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']]{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}<br>[[File:Regular_star_figure_2(12,5).svg|100px]]<br><math>^{q7,q8}</math><br>[8] 10𝝅 {12/5} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(12,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]<br>[[File:Regular_star_polygon_24-11.svg|100px]]<br><math>^{q7,q7}</math><br>[24] 0𝝅 {1}? |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[12] 1𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12}? |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}<br>[[File:Regular_star_figure_8(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}<br>[[File:Regular_star_figure_8(3,1).svg|100px]]<br><math>^{q6,q6}</math><br>[24] 0𝝅 {1}? |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]<br>[[File:Regular_star_polygon_24-7.svg|100px]]<br><math>^{q6,-q6}</math><br>[12] 1𝝅 {2}? |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[36] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]<br>[[File:Regular_star_polygon_24-5.svg|100px]]<br><math>^{q4,q4}</math><br>[24] 0𝝅 {1}? |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]<br>[[File:Regular_star_figure_3(8,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(12,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/0}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], <s>passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each</s>.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, <s>each having one edge in each great hexagon plane, and</s> circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines. == Visualization == [[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]] === Cell rings === The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]). One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the "[[W:North Pole|North Pole]]". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "[[W:South Pole|South Pole]]" cell. This skeleton accounts for 18 of the 24 cells (2&nbsp;+&nbsp;{{gaps|8|×|2}}). See the table below. There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section. Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial "equatorial" cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices. This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="2" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 8 cells | First layer of meridian cells | style="text-align: center" | 60° |- | style="text-align: center" | 3 | style="text-align: center" | 6 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" |Equator |- | style="text-align: center" | 4 | style="text-align: center" | 8 cells | Second layer of meridian cells | style="text-align: center" | 120° | rowspan="2" | Southern Hemisphere |- | style="text-align: center" | 5 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 24 cells ! colspan="3" | |} [[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]] The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is "vertical", encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}} Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees. This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously. One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long. These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section. The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell. This great circle is self dual. This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells. The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract. Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long. Collectively these three subsets now produce another, six ring, discrete Hopf fibration. === Parallel projections === [[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]] The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron. The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed. The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope. === Perspective projections === The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection. The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell. {|class="wikitable" width=660 !colspan=3|Cell-first perspective projection |- valign=top |[[Image:24cell-perspective-cell-first-01.png|220px]]<BR>In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled. |[[Image:24cell-perspective-cell-first-02.png|220px]]<BR>In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent). |[[Image:24cell-perspective-cell-first-03.png|220px]]<BR>Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta. |- |colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the "equator" of the 24-cell, and bridge the two sets of cells. |} {| class="wikitable" width=440 |[[Image:24cell section anim.gif|220px]]<br>Animated cross-section of 24-cell |- |colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]<br>A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell). |- |colspan=3|[[File:Cell24Construction.ogv|450px]]<br>Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell |} == Related polytopes == === Three Coxeter group constructions === There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B₄ or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D₄ or [3^1,1,1] symmetry, and drawn tricolored with 8 octahedra each.<!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --> {| class="wikitable collapsible collapsed" !colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D₄, B₄, and F₄ symmetry |- ![[W:Rectified demitesseract|Rectified demitesseract]] ![[W:Rectified demitesseract|Rectified 16-cell]] !Regular 24-cell |- !D₄, [3^1,1,1], order 192 !B₄, [3,3,4], order 384 !F₄, [3,4,3], order 1152 |- |colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]] |- valign=top |width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells |width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells. |width=213|One set of 24 [[W:Octahedron|octahedral]] cells |- |colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''<br>(Each edge corresponds to one triangular face, colored by symmetry arrangement) |- align=center |[[Image:Rectified demitesseract verf.png|120px]] |[[Image:Rectified 16-cell verf.png|120px]] |[[Image:24 cell verf.svg|120px]] |} === Related complex polygons === The [[W:Regular complex polygon|regular complex polygon]] ₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is ₄[3]₄, order 96.{{Sfn|Coxeter|1991|p=}} The regular complex polytope ₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in <math>\mathbb{C}^2</math> has a real representation as a 24-cell in 4-dimensional space. ₃{4}₃ has 24 vertices, and 24 3-edges. Its symmetry is ₃[4]₃, order 72. {| class=wikitable width=600 |+ Related figures in orthogonal projections |- !Name !{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}} !₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} !₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} |- !Symmetry ![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152 !₄[3]₄, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96 !₃[4]₃, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72 |- align=center !Vertices |24||24||24 |- align=center !Edges |96 2-edges||24 4-edge||24 3-edges |- valign=top !valign=center|Image |[[File:24-cell t0 F4.svg|200px]]<BR>24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges. |[[File:Complex polygon 4-3-4.png|200px]]<BR>₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges. |[[File:Complex polygon 3-4-3-fill1.png|200px]]<BR>₃{4}₃ or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled. |} === Related 4-polytopes === Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]: * truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]]; * truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]]; * and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]]. The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or "[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]]." The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]]. === Related uniform polytopes === {{Demitesseract family}} {{24-cell_family}} The 24-cell can also be derived as a rectified 16-cell: {{Tesseract family}} {{Symmetric_tessellations}} ==See also== *[[W:Octacube (sculpture)|Octacube (sculpture)]] *[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]] == Notes == {{Regular convex 4-polytopes Notelist|wiki=W:}} == Citations == {{Regular convex 4-polytopes Reflist|wiki=W:}} == References == {{Refbegin}} {{Regular convex 4-polytopes Refs|wiki=W:}} <br> * {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}} * {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh & Nara|2021}}}} {{Refend}} ==External links== * [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes] * [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki] * [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space] * [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space] * [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations] * [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections] * [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }} * [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software] [[Category:Geometry]] [[Category:Polyscheme]] 56yq0r9u08h87ojlygog92fb499s5li 2819292 2819291 2026-07-24T16:37:14Z Dc.samizdat 2856930 /* Chiral symmetry operations */ 2819292 wikitext text/x-wiki {{Short description|Regular object in four dimensional geometry}} {{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}} {{Infobox 4-polytope | Name=24-cell | Image_File=Schlegel wireframe 24-cell.png | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<br>(vertices and edges) | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]] | Last=[[W:Omnitruncated tesseract|21]] | Index=22 | Next=[[W:Rectified 24-cell|23]] | Schläfli={3,4,3}<br>r{3,3,4} = <math>\left\{\begin{array}{l}3\\3,4\end{array}\right\}</math><br>{3^1,1,1} = <math>\left\{\begin{array}{l}3\\3\\3\end{array}\right\}</math> | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}} | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]] | Face_List=96 [[W:Triangle|{3}]] | Edge_Count=96 | Vertex_Count= 24 | Petrie_Polygon=[[W:Dodecagon|{12}]] | Coxeter_Group=[[W:F4 (mathematics)|F₄]], [3,4,3], order 1152<br>B₄, [4,3,3], order 384<br>D₄, [3^1,1,1], order 192 | Vertex_Figure=[[W:Cube|cube]] | Dual=[[W:Polytope#Self-dual polytopes|self-dual]] | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]] }} [[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]] In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C₂₄''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for "octahedral complex"), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}} The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}} ==Geometry== The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}} === Coordinates === The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. ==== Great squares ==== The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}} {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}} The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} ==== Great hexagons ==== The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell & Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} ==== Great triangles ==== The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} ==== Hypercubic chords ==== [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. ==== Geodesics ==== [[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]] The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} {| class="wikitable floatright" |+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell |- style="text-align:center;" ![[W:Coxeter plane|Coxeter plane]] !colspan=2|F₄ |- style="text-align:center;" !Graph |colspan=2|[[File:24-cell t0_F4.svg|100px]] |- style="text-align:center;" ![[W:Dihedral symmetry|Dihedral symmetry]] |colspan=2|[12] |- style="text-align:center;" !Coxeter plane !B₃ / A₂ (a) !B₃ / A₂ (b) |- style="text-align:center;" !Graph |[[File:24-cell t0_B3.svg|100px]] |[[File:24-cell t3_B3.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[6] |[6] |- style="text-align:center;" !Coxeter plane !B₄ !B₂ / A₃ |- style="text-align:center;" !Graph |[[File:24-cell t0_B4.svg|100px]] |[[File:24-cell t0_B2.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[8] |[4] |} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell & Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} === Constructions === [[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]] Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). ==== Relationships among interior polytopes ==== The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} ==== Boundary cells ==== Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element. {| class=wikitable |- align=center |\||style="background-color:#808080;"|v||style="background-color:#E6194B;"|e||style="background-color:#3CB44B;"|f||style="background-color:#FFE119;"|c |- align=right |align=left style="background-color:#808080;"|v||style="background-color:#E0F0FF"|'''24'''||8||12||6 |- align=right |align=left style="background-color:#E6194B;"|e||2||style="background-color:#f0FFE0"|'''96'''||3||3 |- align=right |align=left style="background-color:#3CB44B;"|f||3||3||style="background-color:#f0FFE0"|'''96'''||2 |- align=right |align=left style="background-color:#FFE119;"|c||6||12||8||style="background-color:#f0FFE0"|'''24''' |} Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation. In the [[W:uniform 4-polytope|uniform]] D₄ construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.<ref>[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]</ref> The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells. {| class=wikitable |\||style="background-color:#808080;"|v||style="background-color:#E6194B;"|e||style="background-color:#3CB44B;"|f1||style="background-color:#3CB44B;"|f2||style="background-color:#3CB44B;"|f3||style="background-color:#FFE119;"|c1||style="background-color:#FFE119;"|c2||style="background-color:#FFE119;"|c3 |- align=right |align=left style="background-color:#808080;"|v||style="background-color:#E0F0FF"|'''24'''||8||4||4||4||2||2||2 |- align=right |align=left style="background-color:#E6194B;"|e||2||style="background-color:#f0FFE0"|'''96'''||1||1||1||1||1||1 |- align=right |align=left style="background-color:#3CB44B;"|f1||3||3||style="background-color:#f0FFE0"|'''32'''||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|*||1||1||0 |- align=right |align=left style="background-color:#3CB44B;"|f2||3||3||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''32'''||style="background-color:#f0FFE0"|*||1||0||1 |- align=right |align=left style="background-color:#3CB44B;"|f3||3||3||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''32'''||0||1||1 |- align=right |align=left style="background-color:#FFE119;"|c1||6||12||4||4||0||style="background-color:#f0FFE0"|'''8'''||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|* |- align=right |align=left style="background-color:#FFE119;"|c2||6||12||4||0||4||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''8'''||style="background-color:#f0FFE0"|* |- align=right |align=left style="background-color:#FFE119;"|c3||6||12||0||4||4||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''8''' |} ==Symmetries, root systems, and tessellations== [[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F₄]] group, as shown in this F₄ Coxeter plane projection]] The 24 root vectors of the [[W:D4 (root system)|D₄ root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B₄ and C₄ simple Lie groups. The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F₄]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D₄; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F₄, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F₄ roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576. ===Quaternionic interpretation=== [[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection: * 1 order-1: 1 * 1 order-2: -1 * 6 order-4: ±i, ±j, ±k * 8 order-6: (+1±i±j±k)/2 * 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F₄ [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D₄ root lattice is the [[W:Dual lattice|dual]] of the F₄ and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}} Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}} ===Voronoi cells=== The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D₄]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D₄ lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F₄ lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''⁴. The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]]. ===Radially equilateral honeycomb=== The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}} A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia & Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia & Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia & Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia & Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram₅ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986<ref>{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman & Weinberg|1987}}}}</ref> to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference <math>8\pi</math>, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram₅ geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram₅ geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram₅. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell & Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the "left" is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell & Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon. ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical dodecagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]₅, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell & Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}} Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram₅ path. <s>Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew dodecagram₅ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram₅ has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.</s> At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram₅ geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|<blockquote>Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions).<br><br>Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br>where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br>where 2''q'' + ''r'' + 1 ≤ ''n''.<br><br>For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}</blockquote>|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column. {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']]{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}<br>[[File:Regular_star_figure_2(12,5).svg|100px]]<br><math>^{q7,q8}</math><br>[8] 10𝝅 {12/5} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(12,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]<br>[[File:Regular_star_polygon_24-11.svg|100px]]<br><math>^{q7,q7}</math><br>[24] 0𝝅 {1}? |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,-q7}</math><br>[12] 1𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12}? |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}<br>[[File:Regular_star_figure_8(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}<br>[[File:Regular_star_figure_8(3,1).svg|100px]]<br><math>^{q6,q6}</math><br>[24] 0𝝅 {1}? |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]<br>[[File:Regular_star_polygon_24-7.svg|100px]]<br><math>^{q6,-q6}</math><br>[12] 1𝝅 {2}? |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[36] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]<br>[[File:Regular_star_polygon_24-5.svg|100px]]<br><math>^{q4,q4}</math><br>[24] 0𝝅 {1}? |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]<br>[[File:Regular_star_figure_3(8,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(12,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/0}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], <s>passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each</s>.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, <s>each having one edge in each great hexagon plane, and</s> circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines. == Visualization == [[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]] === Cell rings === The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]). One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the "[[W:North Pole|North Pole]]". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "[[W:South Pole|South Pole]]" cell. This skeleton accounts for 18 of the 24 cells (2&nbsp;+&nbsp;{{gaps|8|×|2}}). See the table below. There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section. Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial "equatorial" cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices. This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="2" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 8 cells | First layer of meridian cells | style="text-align: center" | 60° |- | style="text-align: center" | 3 | style="text-align: center" | 6 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" |Equator |- | style="text-align: center" | 4 | style="text-align: center" | 8 cells | Second layer of meridian cells | style="text-align: center" | 120° | rowspan="2" | Southern Hemisphere |- | style="text-align: center" | 5 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 24 cells ! colspan="3" | |} [[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]] The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is "vertical", encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}} Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees. This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously. One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long. These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section. The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell. This great circle is self dual. This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells. The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract. Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long. Collectively these three subsets now produce another, six ring, discrete Hopf fibration. === Parallel projections === [[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]] The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron. The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed. The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope. === Perspective projections === The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection. The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell. {|class="wikitable" width=660 !colspan=3|Cell-first perspective projection |- valign=top |[[Image:24cell-perspective-cell-first-01.png|220px]]<BR>In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled. |[[Image:24cell-perspective-cell-first-02.png|220px]]<BR>In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent). |[[Image:24cell-perspective-cell-first-03.png|220px]]<BR>Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta. |- |colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the "equator" of the 24-cell, and bridge the two sets of cells. |} {| class="wikitable" width=440 |[[Image:24cell section anim.gif|220px]]<br>Animated cross-section of 24-cell |- |colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]<br>A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell). |- |colspan=3|[[File:Cell24Construction.ogv|450px]]<br>Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell |} == Related polytopes == === Three Coxeter group constructions === There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B₄ or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D₄ or [3^1,1,1] symmetry, and drawn tricolored with 8 octahedra each.<!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --> {| class="wikitable collapsible collapsed" !colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D₄, B₄, and F₄ symmetry |- ![[W:Rectified demitesseract|Rectified demitesseract]] ![[W:Rectified demitesseract|Rectified 16-cell]] !Regular 24-cell |- !D₄, [3^1,1,1], order 192 !B₄, [3,3,4], order 384 !F₄, [3,4,3], order 1152 |- |colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]] |- valign=top |width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells |width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells. |width=213|One set of 24 [[W:Octahedron|octahedral]] cells |- |colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''<br>(Each edge corresponds to one triangular face, colored by symmetry arrangement) |- align=center |[[Image:Rectified demitesseract verf.png|120px]] |[[Image:Rectified 16-cell verf.png|120px]] |[[Image:24 cell verf.svg|120px]] |} === Related complex polygons === The [[W:Regular complex polygon|regular complex polygon]] ₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is ₄[3]₄, order 96.{{Sfn|Coxeter|1991|p=}} The regular complex polytope ₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in <math>\mathbb{C}^2</math> has a real representation as a 24-cell in 4-dimensional space. ₃{4}₃ has 24 vertices, and 24 3-edges. Its symmetry is ₃[4]₃, order 72. {| class=wikitable width=600 |+ Related figures in orthogonal projections |- !Name !{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}} !₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} !₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} |- !Symmetry ![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152 !₄[3]₄, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96 !₃[4]₃, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72 |- align=center !Vertices |24||24||24 |- align=center !Edges |96 2-edges||24 4-edge||24 3-edges |- valign=top !valign=center|Image |[[File:24-cell t0 F4.svg|200px]]<BR>24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges. |[[File:Complex polygon 4-3-4.png|200px]]<BR>₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges. |[[File:Complex polygon 3-4-3-fill1.png|200px]]<BR>₃{4}₃ or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled. |} === Related 4-polytopes === Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]: * truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]]; * truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]]; * and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]]. The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or "[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]]." The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]]. === Related uniform polytopes === {{Demitesseract family}} {{24-cell_family}} The 24-cell can also be derived as a rectified 16-cell: {{Tesseract family}} {{Symmetric_tessellations}} ==See also== *[[W:Octacube (sculpture)|Octacube (sculpture)]] *[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]] == Notes == {{Regular convex 4-polytopes Notelist|wiki=W:}} == Citations == {{Regular convex 4-polytopes Reflist|wiki=W:}} == References == {{Refbegin}} {{Regular convex 4-polytopes Refs|wiki=W:}} <br> * {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}} * {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh & Nara|2021}}}} {{Refend}} ==External links== * [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes] * [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki] * [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space] * [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space] * [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations] * [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections] * [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }} * [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software] [[Category:Geometry]] [[Category:Polyscheme]] 3e4rva20rudknonf0n9zquajk4z6jra 2819295 2819292 2026-07-24T16:57:14Z Dc.samizdat 2856930 /* Chiral symmetry operations */ 2819295 wikitext text/x-wiki {{Short description|Regular object in four dimensional geometry}} {{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}} {{Infobox 4-polytope | Name=24-cell | Image_File=Schlegel wireframe 24-cell.png | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<br>(vertices and edges) | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]] | Last=[[W:Omnitruncated tesseract|21]] | Index=22 | Next=[[W:Rectified 24-cell|23]] | Schläfli={3,4,3}<br>r{3,3,4} = <math>\left\{\begin{array}{l}3\\3,4\end{array}\right\}</math><br>{3^1,1,1} = <math>\left\{\begin{array}{l}3\\3\\3\end{array}\right\}</math> | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}} | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]] | Face_List=96 [[W:Triangle|{3}]] | Edge_Count=96 | Vertex_Count= 24 | Petrie_Polygon=[[W:Dodecagon|{12}]] | Coxeter_Group=[[W:F4 (mathematics)|F₄]], [3,4,3], order 1152<br>B₄, [4,3,3], order 384<br>D₄, [3^1,1,1], order 192 | Vertex_Figure=[[W:Cube|cube]] | Dual=[[W:Polytope#Self-dual polytopes|self-dual]] | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]] }} [[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]] In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C₂₄''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for "octahedral complex"), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}} The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}} ==Geometry== The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}} === Coordinates === The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. ==== Great squares ==== The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}} {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}} The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} ==== Great hexagons ==== The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell & Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} ==== Great triangles ==== The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} ==== Hypercubic chords ==== [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. ==== Geodesics ==== [[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]] The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} {| class="wikitable floatright" |+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell |- style="text-align:center;" ![[W:Coxeter plane|Coxeter plane]] !colspan=2|F₄ |- style="text-align:center;" !Graph |colspan=2|[[File:24-cell t0_F4.svg|100px]] |- style="text-align:center;" ![[W:Dihedral symmetry|Dihedral symmetry]] |colspan=2|[12] |- style="text-align:center;" !Coxeter plane !B₃ / A₂ (a) !B₃ / A₂ (b) |- style="text-align:center;" !Graph |[[File:24-cell t0_B3.svg|100px]] |[[File:24-cell t3_B3.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[6] |[6] |- style="text-align:center;" !Coxeter plane !B₄ !B₂ / A₃ |- style="text-align:center;" !Graph |[[File:24-cell t0_B4.svg|100px]] |[[File:24-cell t0_B2.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[8] |[4] |} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell & Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} === Constructions === [[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]] Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). ==== Relationships among interior polytopes ==== The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} ==== Boundary cells ==== Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element. {| class=wikitable |- align=center |\||style="background-color:#808080;"|v||style="background-color:#E6194B;"|e||style="background-color:#3CB44B;"|f||style="background-color:#FFE119;"|c |- align=right |align=left style="background-color:#808080;"|v||style="background-color:#E0F0FF"|'''24'''||8||12||6 |- align=right |align=left style="background-color:#E6194B;"|e||2||style="background-color:#f0FFE0"|'''96'''||3||3 |- align=right |align=left style="background-color:#3CB44B;"|f||3||3||style="background-color:#f0FFE0"|'''96'''||2 |- align=right |align=left style="background-color:#FFE119;"|c||6||12||8||style="background-color:#f0FFE0"|'''24''' |} Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation. In the [[W:uniform 4-polytope|uniform]] D₄ construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.<ref>[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]</ref> The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells. {| class=wikitable |\||style="background-color:#808080;"|v||style="background-color:#E6194B;"|e||style="background-color:#3CB44B;"|f1||style="background-color:#3CB44B;"|f2||style="background-color:#3CB44B;"|f3||style="background-color:#FFE119;"|c1||style="background-color:#FFE119;"|c2||style="background-color:#FFE119;"|c3 |- align=right |align=left style="background-color:#808080;"|v||style="background-color:#E0F0FF"|'''24'''||8||4||4||4||2||2||2 |- align=right |align=left style="background-color:#E6194B;"|e||2||style="background-color:#f0FFE0"|'''96'''||1||1||1||1||1||1 |- align=right |align=left style="background-color:#3CB44B;"|f1||3||3||style="background-color:#f0FFE0"|'''32'''||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|*||1||1||0 |- align=right |align=left style="background-color:#3CB44B;"|f2||3||3||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''32'''||style="background-color:#f0FFE0"|*||1||0||1 |- align=right |align=left style="background-color:#3CB44B;"|f3||3||3||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''32'''||0||1||1 |- align=right |align=left style="background-color:#FFE119;"|c1||6||12||4||4||0||style="background-color:#f0FFE0"|'''8'''||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|* |- align=right |align=left style="background-color:#FFE119;"|c2||6||12||4||0||4||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''8'''||style="background-color:#f0FFE0"|* |- align=right |align=left style="background-color:#FFE119;"|c3||6||12||0||4||4||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''8''' |} ==Symmetries, root systems, and tessellations== [[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F₄]] group, as shown in this F₄ Coxeter plane projection]] The 24 root vectors of the [[W:D4 (root system)|D₄ root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B₄ and C₄ simple Lie groups. The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F₄]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D₄; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F₄, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F₄ roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576. ===Quaternionic interpretation=== [[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection: * 1 order-1: 1 * 1 order-2: -1 * 6 order-4: ±i, ±j, ±k * 8 order-6: (+1±i±j±k)/2 * 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F₄ [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D₄ root lattice is the [[W:Dual lattice|dual]] of the F₄ and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}} Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}} ===Voronoi cells=== The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D₄]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D₄ lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F₄ lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''⁴. The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]]. ===Radially equilateral honeycomb=== The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}} A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia & Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia & Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia & Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia & Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram₅ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986<ref>{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman & Weinberg|1987}}}}</ref> to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference <math>8\pi</math>, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram₅ geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram₅ geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram₅. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell & Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the "left" is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell & Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon. ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical dodecagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]₅, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell & Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}} Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram₅ path. <s>Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew dodecagram₅ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram₅ has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.</s> At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram₅ geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|<blockquote>Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions).<br><br>Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br>where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br>where 2''q'' + ''r'' + 1 ≤ ''n''.<br><br>For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}</blockquote>|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column. {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']]{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}<br>[[File:Regular_star_figure_2(12,5).svg|100px]]<br><math>^{q7,q8}</math><br>[8] 10𝝅 {12/5} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(12,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]<br>[[File:Regular_star_polygon_24-11.svg|100px]]<br><math>^{q7,q7}</math><br>[24] 0𝝅 {1}? |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,-q7}</math><br>[12] 1𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12}? |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=8{3}]]{{Efn|name=dodecagram}}<br>[[File:Regular_star_figure_8(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]<br>[[File:Regular_star_figure_3(8,3).svg|100px]]<br><math>^{q6,q6}</math><br>[24] 0𝝅 {1}? |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]<br>[[File:Regular_star_figure_3(8,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[12] 1𝝅 {2}? |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[36] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4,q4}</math><br>[24] 0𝝅 {1}? |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]<br>[[File:Regular_star_figure_3(8,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(12,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/0}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], <s>passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each</s>.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, <s>each having one edge in each great hexagon plane, and</s> circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines. == Visualization == [[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]] === Cell rings === The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]). One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the "[[W:North Pole|North Pole]]". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "[[W:South Pole|South Pole]]" cell. This skeleton accounts for 18 of the 24 cells (2&nbsp;+&nbsp;{{gaps|8|×|2}}). See the table below. There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section. Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial "equatorial" cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices. This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="2" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 8 cells | First layer of meridian cells | style="text-align: center" | 60° |- | style="text-align: center" | 3 | style="text-align: center" | 6 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" |Equator |- | style="text-align: center" | 4 | style="text-align: center" | 8 cells | Second layer of meridian cells | style="text-align: center" | 120° | rowspan="2" | Southern Hemisphere |- | style="text-align: center" | 5 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 24 cells ! colspan="3" | |} [[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]] The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is "vertical", encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}} Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees. This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously. One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long. These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section. The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell. This great circle is self dual. This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells. The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract. Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long. Collectively these three subsets now produce another, six ring, discrete Hopf fibration. === Parallel projections === [[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]] The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron. The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed. The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope. === Perspective projections === The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection. The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell. {|class="wikitable" width=660 !colspan=3|Cell-first perspective projection |- valign=top |[[Image:24cell-perspective-cell-first-01.png|220px]]<BR>In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled. |[[Image:24cell-perspective-cell-first-02.png|220px]]<BR>In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent). |[[Image:24cell-perspective-cell-first-03.png|220px]]<BR>Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta. |- |colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the "equator" of the 24-cell, and bridge the two sets of cells. |} {| class="wikitable" width=440 |[[Image:24cell section anim.gif|220px]]<br>Animated cross-section of 24-cell |- |colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]<br>A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell). |- |colspan=3|[[File:Cell24Construction.ogv|450px]]<br>Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell |} == Related polytopes == === Three Coxeter group constructions === There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B₄ or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D₄ or [3^1,1,1] symmetry, and drawn tricolored with 8 octahedra each.<!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --> {| class="wikitable collapsible collapsed" !colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D₄, B₄, and F₄ symmetry |- ![[W:Rectified demitesseract|Rectified demitesseract]] ![[W:Rectified demitesseract|Rectified 16-cell]] !Regular 24-cell |- !D₄, [3^1,1,1], order 192 !B₄, [3,3,4], order 384 !F₄, [3,4,3], order 1152 |- |colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]] |- valign=top |width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells |width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells. |width=213|One set of 24 [[W:Octahedron|octahedral]] cells |- |colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''<br>(Each edge corresponds to one triangular face, colored by symmetry arrangement) |- align=center |[[Image:Rectified demitesseract verf.png|120px]] |[[Image:Rectified 16-cell verf.png|120px]] |[[Image:24 cell verf.svg|120px]] |} === Related complex polygons === The [[W:Regular complex polygon|regular complex polygon]] ₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is ₄[3]₄, order 96.{{Sfn|Coxeter|1991|p=}} The regular complex polytope ₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in <math>\mathbb{C}^2</math> has a real representation as a 24-cell in 4-dimensional space. ₃{4}₃ has 24 vertices, and 24 3-edges. Its symmetry is ₃[4]₃, order 72. {| class=wikitable width=600 |+ Related figures in orthogonal projections |- !Name !{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}} !₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} !₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} |- !Symmetry ![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152 !₄[3]₄, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96 !₃[4]₃, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72 |- align=center !Vertices |24||24||24 |- align=center !Edges |96 2-edges||24 4-edge||24 3-edges |- valign=top !valign=center|Image |[[File:24-cell t0 F4.svg|200px]]<BR>24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges. |[[File:Complex polygon 4-3-4.png|200px]]<BR>₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges. |[[File:Complex polygon 3-4-3-fill1.png|200px]]<BR>₃{4}₃ or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled. |} === Related 4-polytopes === Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]: * truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]]; * truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]]; * and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]]. The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or "[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]]." The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]]. === Related uniform polytopes === {{Demitesseract family}} {{24-cell_family}} The 24-cell can also be derived as a rectified 16-cell: {{Tesseract family}} {{Symmetric_tessellations}} ==See also== *[[W:Octacube (sculpture)|Octacube (sculpture)]] *[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]] == Notes == {{Regular convex 4-polytopes Notelist|wiki=W:}} == Citations == {{Regular convex 4-polytopes Reflist|wiki=W:}} == References == {{Refbegin}} {{Regular convex 4-polytopes Refs|wiki=W:}} <br> * {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}} * {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh & Nara|2021}}}} {{Refend}} ==External links== * [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes] * [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki] * [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space] * [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space] * [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations] * [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections] * [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }} * [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software] [[Category:Geometry]] [[Category:Polyscheme]] e9x9xdcw9tyfpxrump561x7467oht5x 2819296 2819295 2026-07-24T17:00:59Z Dc.samizdat 2856930 /* Chiral symmetry operations */ 2819296 wikitext text/x-wiki {{Short description|Regular object in four dimensional geometry}} {{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}} {{Infobox 4-polytope | Name=24-cell | Image_File=Schlegel wireframe 24-cell.png | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<br>(vertices and edges) | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]] | Last=[[W:Omnitruncated tesseract|21]] | Index=22 | Next=[[W:Rectified 24-cell|23]] | Schläfli={3,4,3}<br>r{3,3,4} = <math>\left\{\begin{array}{l}3\\3,4\end{array}\right\}</math><br>{3^1,1,1} = <math>\left\{\begin{array}{l}3\\3\\3\end{array}\right\}</math> | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}} | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]] | Face_List=96 [[W:Triangle|{3}]] | Edge_Count=96 | Vertex_Count= 24 | Petrie_Polygon=[[W:Dodecagon|{12}]] | Coxeter_Group=[[W:F4 (mathematics)|F₄]], [3,4,3], order 1152<br>B₄, [4,3,3], order 384<br>D₄, [3^1,1,1], order 192 | Vertex_Figure=[[W:Cube|cube]] | Dual=[[W:Polytope#Self-dual polytopes|self-dual]] | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]] }} [[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]] In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C₂₄''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for "octahedral complex"), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}} The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}} Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}} ==Geometry== The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]]. The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}} === Coordinates === The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure. ==== Great squares ==== The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with the 8 vertices that are permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells. In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|name=radially equilateral|group=}} {{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|instance=1}} The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example: {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}} ==== Great hexagons ==== The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces. If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows: 8 vertices obtained by permuting the ''integer'' coordinates: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> all 24 of which lie at distance 1 from the origin. [[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}} The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell & Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}} ==== Great triangles ==== The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon. For example, in unit radius coordinates: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} ==== Hypercubic chords ==== [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing its 3 great circle polygons and its 4 chord lengths.|alt=]] The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares. Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = <small>{{sfrac|{{pi}}|3}}</small> of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = <small>{{pi}}</small> away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices. To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract. ==== Geodesics ==== [[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]] The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}} {| class="wikitable floatright" |+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell |- style="text-align:center;" ![[W:Coxeter plane|Coxeter plane]] !colspan=2|F₄ |- style="text-align:center;" !Graph |colspan=2|[[File:24-cell t0_F4.svg|100px]] |- style="text-align:center;" ![[W:Dihedral symmetry|Dihedral symmetry]] |colspan=2|[12] |- style="text-align:center;" !Coxeter plane !B₃ / A₂ (a) !B₃ / A₂ (b) |- style="text-align:center;" !Graph |[[File:24-cell t0_B3.svg|100px]] |[[File:24-cell t3_B3.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[6] |[6] |- style="text-align:center;" !Coxeter plane !B₄ !B₂ / A₃ |- style="text-align:center;" !Graph |[[File:24-cell t0_B4.svg|100px]] |[[File:24-cell t0_B2.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[8] |[4] |} The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell & Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}} The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}} The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex. The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24².{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}} The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}} The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once. Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}} === Constructions === [[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Efn|name=three isoclinic 16-cells}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]] Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼₄]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell. ==== Reciprocal constructions from 8-cell and 16-cell ==== The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}} We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2<small>A</small> and Fig 8.2<small>B</small>|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}} ==== Diminishings ==== We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}} ===== 8-cell ===== Starting with a complete 24-cell, remove the 8 orthogonal vertices of a 16-cell (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}} ===== 16-cell ===== Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set of 6) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}} ==== Tetrahedral constructions ==== The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center. The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}} The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F₄]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}} ==== Cubic constructions ==== The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint. The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells). ==== Relationships among interior polytopes ==== The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|A simple way of stating this relationship is that the common core of the {{radic|2}}-radius 4-polytopes is the unit-radius 24-cell. The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius {{radic|1/2}}.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]".}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/2 (1/4 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges. {{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube long diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}} The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]] The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{Efn|name=root 2 chords}} The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of the same radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}} ==== Boundary cells ==== Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other). Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}} As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}} === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element. {| class=wikitable |- align=center |\||style="background-color:#808080;"|v||style="background-color:#E6194B;"|e||style="background-color:#3CB44B;"|f||style="background-color:#FFE119;"|c |- align=right |align=left style="background-color:#808080;"|v||style="background-color:#E0F0FF"|'''24'''||8||12||6 |- align=right |align=left style="background-color:#E6194B;"|e||2||style="background-color:#f0FFE0"|'''96'''||3||3 |- align=right |align=left style="background-color:#3CB44B;"|f||3||3||style="background-color:#f0FFE0"|'''96'''||2 |- align=right |align=left style="background-color:#FFE119;"|c||6||12||8||style="background-color:#f0FFE0"|'''24''' |} Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation. In the [[W:uniform 4-polytope|uniform]] D₄ construction, {{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}}, the face and cell rows and columns split into 3 partitions.<ref>[https://bendwavy.org/klitzing/incmats/ico.htm 24-cell: o3x3o *b3o]</ref> The dual of this construction will have 3 partitions of vertices and edges, and 1 class each of faces and cells. {| class=wikitable |\||style="background-color:#808080;"|v||style="background-color:#E6194B;"|e||style="background-color:#3CB44B;"|f1||style="background-color:#3CB44B;"|f2||style="background-color:#3CB44B;"|f3||style="background-color:#FFE119;"|c1||style="background-color:#FFE119;"|c2||style="background-color:#FFE119;"|c3 |- align=right |align=left style="background-color:#808080;"|v||style="background-color:#E0F0FF"|'''24'''||8||4||4||4||2||2||2 |- align=right |align=left style="background-color:#E6194B;"|e||2||style="background-color:#f0FFE0"|'''96'''||1||1||1||1||1||1 |- align=right |align=left style="background-color:#3CB44B;"|f1||3||3||style="background-color:#f0FFE0"|'''32'''||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|*||1||1||0 |- align=right |align=left style="background-color:#3CB44B;"|f2||3||3||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''32'''||style="background-color:#f0FFE0"|*||1||0||1 |- align=right |align=left style="background-color:#3CB44B;"|f3||3||3||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''32'''||0||1||1 |- align=right |align=left style="background-color:#FFE119;"|c1||6||12||4||4||0||style="background-color:#f0FFE0"|'''8'''||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|* |- align=right |align=left style="background-color:#FFE119;"|c2||6||12||4||0||4||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''8'''||style="background-color:#f0FFE0"|* |- align=right |align=left style="background-color:#FFE119;"|c3||6||12||0||4||4||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|'''8''' |} ==Symmetries, root systems, and tessellations== [[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F₄]] group, as shown in this F₄ Coxeter plane projection]] The 24 root vectors of the [[W:D4 (root system)|D₄ root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: "... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces."}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B₄ and C₄ simple Lie groups. The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F₄]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D₄; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F₄, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F₄ roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576. ===Quaternionic interpretation=== [[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection: * 1 order-1: 1 * 1 order-2: -1 * 6 order-4: ±i, ±j, ±k * 8 order-6: (+1±i±j±k)/2 * 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F₄ [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D₄ root lattice is the [[W:Dual lattice|dual]] of the F₄ and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}} Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}} Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca et. al.|2007}} ===Voronoi cells=== The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D₄]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D₄ lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F₄ lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''⁴. The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]]. ===Radially equilateral honeycomb=== The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}} A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. <!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --> The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}} Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}} == Rotations == The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' ²''r'' ³."}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}} === The 3 Cartesian bases of the 24-cell === There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of completely orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}} === Planes of rotation === [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia & Thomas|2017|loc=§7. Conclusions|ps=; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia & Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}} ==== Simple rotations ==== [[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]]. When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively.{{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}} ==== Double rotations ==== [[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0. Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions "right" and "left", since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating "to the left", the right rotation is not rotating "to the right", and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia & Thomas|2017|loc=§5. A useful mapping|pp=12−13}} In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie and Clifford dodecagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}} ==== Isoclinic rotations ==== When the angles of rotation in the two completely orthogonal invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia & Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the pair of invariant planes become pairs of invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. (In the 4-dimensional case, the orthogonal distance equals half the total Pythagorean distance.) All vertices are displaced to a vertex more than one edge length away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon 120 degrees away. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 120 degrees away.|name=Clifford displacement}} The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; "[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''."}} In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a non-adjacent vertex {{radic|3}} and 120° distant. The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ are adjacent vertices, {{radic|1}} apart. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 180° isoclinic rotation, and one quarter of the 24-cell's double-loop decagram₅ Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. The helix of {{radic|3}} chords closes into a loop only after twelve {{radic|3}} chords: a 720° isoclinic rotation{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly or enantiomorphously congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it has two chiral forms (left and right).{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} over a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] {12/5} dodecagram with {{radic|3}} edges. All 24 vertices rotate at once, on two Clifford parallel dodecagon isoclines. Each vertex visits half the 24 vertex positions. Although each isocline is a circular spiral through all 4 dimensions, not a 2-dimensional circle in the plane, like an ordinary great circle it is a geodesic, because it is the shortest circle through those 12 vertices.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}} A 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After six 60° rotational displacements each vertex has departed from six vertex positions and reached a seventh vertex position adjacent to its antipodal vertex. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but its [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986<ref>{{Cite book|title=Elementary particles and the laws of physics|chapter=The reason for antiparticles|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987|ref={{SfnRef|Feynman & Weinberg|1987}}}}</ref> to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same rotational direction through six more 60° isoclinic displacements, the 24 moving vertices will pass through the other half of the vertices, and each vertex will arrive back at the vertex position it departed from, after tracing a closed helical loop over twelve {{radic|3}} chords. It takes a 720 degree isoclinic rotation for each vertex to traverse a geodesic circle of circumference <math>8\pi</math>, [[W:Winding number|winding]] around the 24-cell 5 times and returning the 24-cell to its original orientation.{{Efn|In a 720° isoclinic rotation of a rigid 24-cell the 24 vertices rotate along two Clifford parallel dodecagram₅ geodesic loops (12 vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}} The twin dodecagram winding paths that the vertices take as they loop five times around the 24-cell form a double helix bent into a ring.{{Efn|The 24-cell's helical dodecagram₅ geodesic is bent into a twisted ring in the fourth dimension. Its [[W:Screw thread|screw thread]] maintains the same chirality{{Efn|name=Clifford polygon}} and even/odd parity of rotation (black or white) throughout.{{Efn|name=black and white}} Two Clifford parallel 12-vertex circular helixes form a Möbius strip one edge wide, a 4-dimensional circular double helix.{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>.}}}} This 60° isocline is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {12/5} or dodecagram₅. Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} === Clifford parallel polytopes === Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 2 disjoint subsets of 12 vertices (dodecagrams), each skew [[#Helical hdodecagrams and their isoclines|dodecagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 12 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of two Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles as either its left or right rotation.{{Efn|Each set of four [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of two Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the same discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram₃ forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}} Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell & Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or linage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. If so, that was not an error in our visualization; there are two chiral images we can ascribe to the 24-cell, from mirror-image viewpoints which turn the 24-cell inside-out. But from either viewpoint, the 16-cell to the "left" is the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.{{Efn|name=clasped hands}}|name=three isoclinic 16-cells}} All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}} Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel subspaces that comprise the 3-sphere.{{Sfn|Tyrrell & Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces. === Rings === In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]]. The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix dodecagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon. ==== 4-cell rings ==== Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in a great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring. ==== 6-cell rings ==== [[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}} Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]]. Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] <math>F_a, F_b, F_c, F_d</math>. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}} Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}} ==== Helical dodecagrams and their isoclines ==== Another kind of geodesic fiber, the [[#Isoclinic rotations|helical dodecagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''fifth'' vertex of a skew [[W:Dodecagon#Related figures|dodecagram]]₅, which in the unit-radius, unit-edge-length 24-cell has twelve {{radic|3}} edges. The dodagram does not lie in a single central plane, but is composed of twelve linked {{radic|3}} chords from different hexagon great circles. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell linking non-adjacent vertices, that winds five times around the 24-cell before completing its twelve-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell & Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] {12/5} dodecagram.{{Efn|name=double threaded}} Each fibration of four 6-cell rings contains four such dodecagram isoclines, two black and two white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. Two chiral sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]] run through each [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of the skew dodecagrams lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white dodecagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} The fibration's right (or left) rotation traverses a black isocline and a white isocline in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}} Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew dodecagram contains one {{radic|3}} chord of each color, and visits all 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew dodecagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=dodecagrams hitting vertex of 6-cell ring}}}} The path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the dodecagram₅ path. <s>Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew dodecagram₅ isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic dodecagram₅ has {{radic|3}} edges which all bend either left or right at every fifth vertex along a geodesic spiral of potentially either chirality (left or right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting two verticies of each of those same 6 octahedra in a 720° rotation.|name=Petrie and Clifford dodecagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew dodecagram and begins to repeat itself, circling again through the black (or white) vertices and cells.</s> At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it, missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The black and white isoclines belong to the same fibration.|name=isoclines at hexagons}} and four dodecagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each dodecagram isocline hits only one end of an axis, unlike a great circle in the plane which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of one of the 24-cell's 12 axes.|name=dodecagram isoclines at an axis}} Two dodecagram isoclines (one black and one white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 8 distinct dedecagram isoclines in the 24-cell (4 black and 4 white). Each dodecagram is a skew ''Clifford polygon'' of no inherent chirality, that acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}} ==== Helical octagrams and their isoclines ==== The 24-cell contains 18 helical {8/3} [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a zig-zag Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in the context of a particular rotation. Adjacent vertices on the {8/3} octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each great square to its completely orthogonal great square in a twisting displacement, and each vertex to a vertex 90° away over a rotational curve. The rotational curve over each {{radic|2}} chord of the {8/3} octagram makes three 90° left (or right) turns. Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}} In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram_8{4}=4{2}]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes. The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}} This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical dodecagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}} {| class="wikitable" width=610 !colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]] |- ![[16-cell#Rotations|Edge path]] ![[W:Petrie polygon|Petrie polygon]]s ![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]] ![[#Great squares|Discrete fibration]] ![[16-cell#Helical construction|Diameter chords]] |- ![[16-cell#Helical construction|16-cells]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |align=center|[[File:Regular_star_figure_3(8,3).svg|120px]] |align=center|[[File:Regular_star_figure_2(12,1).svg|120px]] |align=center|[[File:Regular_star_polygon_24-5.svg|120px]] |align=center|[[File:Regular_star_figure_6(4,1).svg|120px]] |align=center|[[File:Regular_star_figure_12(2,1).svg|120px]] |- |The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}} |2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h₂'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}} |Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other. |Two vertices four {{radic|2}} chords apart on a Petrie polygon are antipodal vertices joined by a {{radic|4}} axis. |} ===Characteristic orthoscheme=== {| class="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center| |align=center| |align=center| |align=center| |} Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center. === Reflections === The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a great hexagon invariant plane takes each of the 24 vertices to and through eleven other vertices and back to itself, on a skew [[#Helical dodecagrams and their isoclines|dodecagram₅ geodesic isocline]] that winds five times around the 3-sphere on every fifth vertex of the dodecagram. Any pair of antipodal vertices performing such an orbit visits 2 * 12 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in the twelve steps of a single 720° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection. Tracing the orbit of one vertex during the 720° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|<blockquote>Let Q denote a rotation, R a reflection, T a translation, and let Q^''q'' R^''r'' T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q² is a double rotation (in four dimensions).<br><br>Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br>where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br>where 2''q'' + ''r'' + 1 ≤ ''n''.<br><br>For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}</blockquote>|name=transformations}} The vertex follows an [[#Helical dodecagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects non-adjacent vertices , but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow a great circle in the plane, it is a great circle of another kind that curves in two completely orthogonal directions at once, and winds through all four dimensions. === Chiral symmetry operations === A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror facets.{{Efn|name=transformations}} Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once. Each row of the table describes a class of rotational displacements which comprise a distinct isoclinic rotation of the rigid 24-cell. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The 24 vertices of the moving planes move in parallel between the left and right planes over the '''isocline''' chord paths pictured. For example, the <math>[32]R_{q7,q8}</math> rotation class consists of [32] plane displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} There are [32] distinct rotational plane displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations.'' One of the [32] plane displacements in this class moves the representative [[#Great hexagons|vertex coordinate]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}} Corresponding vertices in the left and right hexagon planes are 5 vertices apart on a Petrie polygon of the 24-cell, so the {{radic|3}} displacement chords of the 24 moving vertices form 2 disjoint skew {12/5} dodecagram helixes, pictured in the isocline column. {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']]{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}} |- !Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{Efn|name=isoclinic geodesic}}|name=isocline}} !colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}} !colspan=5|Left planes <math>ql</math>{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}} !colspan=5|Right planes <math>qr</math> |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/10}=2{12/5}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {12/5} [[W:Dodecagon|Related figures]] with {{radic|3}} edges and a circumference of 8𝝅. The 4 disjoint skew [[#Helical hdodecagrams and their isoclines|dodecagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew dodecagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford dodecagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} |name=dodecagram}}<br>[[File:Regular_star_figure_2(12,5).svg|100px]]<br><math>^{q7,q8}</math><br>[8] 10𝝅 {12/5} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{Efn|name=north pole relative coordinate}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(12,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/11}]]<br>[[File:Regular_star_polygon_24-11.svg|100px]]<br><math>^{q7,q7}</math><br>[24] 0𝝅 {1}? |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,-q7}</math><br>[12] 1𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/7}]]<br>[[File:Regular_star_polygon_24-7.svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12}? |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|<math>B_4</math> symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math> symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/5}]]<br>[[File:Regular_star_polygon_24-5.svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]<br>[[File:Regular_star_figure_3(8,3).svg|100px]]<br><math>^{q6,q6}</math><br>[24] 0𝝅 {1}? |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]<br>[[File:Regular_star_figure_3(8,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[12] 1𝝅 {2}? |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[36] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4,q4}</math><br>[24] 0𝝅 {1}? |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/3}=3{8}]]<br>[[File:Regular_star_figure_3(8,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}<br>[[File:Regular_star_figure_2(12,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/0}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |} In a rotation class <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> which has [32] distinct rotational displacements, [16] left rotations and [16] right rotations,. which are not congruent but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are [[W:chiral|chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes. Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic twisting rotational displacement.{{Efn|name=two angles between central planes}} For example, the <math>[32]R_{q7,q8}</math> rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 12 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], <s>passing through all 4 planes of the <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each</s>.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents the helical paths of the vertices as they move between planes in the left and right plane sets. In the <math>[32]R_{q7,q8}</math> example it can be seen as a set of 2 Clifford parallel skew {12/5} dodecagrams, <s>each having one edge in each great hexagon plane, and</s> circular helixes which skew to the left (or right) at each vertex throughout the left (or right) double rotation.{{Efn|name=clasped hands}} The 24 vertices circulate on the two parallel {12/5} isoclines. == Visualization == [[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]] === Cell rings === The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]]. For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]). One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie and Clifford dodecagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[W:chiral|chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the "[[W:North Pole|North Pole]]". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "[[W:South Pole|South Pole]]" cell. This skeleton accounts for 18 of the 24 cells (2&nbsp;+&nbsp;{{gaps|8|×|2}}). See the table below. There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section. Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial "equatorial" cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices. This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="2" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 8 cells | First layer of meridian cells | style="text-align: center" | 60° |- | style="text-align: center" | 3 | style="text-align: center" | 6 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" |Equator |- | style="text-align: center" | 4 | style="text-align: center" | 8 cells | Second layer of meridian cells | style="text-align: center" | 120° | rowspan="2" | Southern Hemisphere |- | style="text-align: center" | 5 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 24 cells ! colspan="3" | |} [[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]] The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is "vertical", encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}} Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees. This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously. One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long. These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section. The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell. This great circle is self dual. This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells. The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract. Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long. Collectively these three subsets now produce another, six ring, discrete Hopf fibration. === Parallel projections === [[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]] The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron. The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed. The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope. === Perspective projections === The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection. The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell. {|class="wikitable" width=660 !colspan=3|Cell-first perspective projection |- valign=top |[[Image:24cell-perspective-cell-first-01.png|220px]]<BR>In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled. |[[Image:24cell-perspective-cell-first-02.png|220px]]<BR>In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent). |[[Image:24cell-perspective-cell-first-03.png|220px]]<BR>Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta. |- |colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the "equator" of the 24-cell, and bridge the two sets of cells. |} {| class="wikitable" width=440 |[[Image:24cell section anim.gif|220px]]<br>Animated cross-section of 24-cell |- |colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]<br>A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell). |- |colspan=3|[[File:Cell24Construction.ogv|450px]]<br>Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell |} == Related polytopes == === Three Coxeter group constructions === There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B₄ or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D₄ or [3^1,1,1] symmetry, and drawn tricolored with 8 octahedra each.<!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --> {| class="wikitable collapsible collapsed" !colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D₄, B₄, and F₄ symmetry |- ![[W:Rectified demitesseract|Rectified demitesseract]] ![[W:Rectified demitesseract|Rectified 16-cell]] !Regular 24-cell |- !D₄, [3^1,1,1], order 192 !B₄, [3,3,4], order 384 !F₄, [3,4,3], order 1152 |- |colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]] |- valign=top |width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells |width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells. |width=213|One set of 24 [[W:Octahedron|octahedral]] cells |- |colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''<br>(Each edge corresponds to one triangular face, colored by symmetry arrangement) |- align=center |[[Image:Rectified demitesseract verf.png|120px]] |[[Image:Rectified 16-cell verf.png|120px]] |[[Image:24 cell verf.svg|120px]] |} === Related complex polygons === The [[W:Regular complex polygon|regular complex polygon]] ₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is ₄[3]₄, order 96.{{Sfn|Coxeter|1991|p=}} The regular complex polytope ₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in <math>\mathbb{C}^2</math> has a real representation as a 24-cell in 4-dimensional space. ₃{4}₃ has 24 vertices, and 24 3-edges. Its symmetry is ₃[4]₃, order 72. {| class=wikitable width=600 |+ Related figures in orthogonal projections |- !Name !{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}} !₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} !₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} |- !Symmetry ![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152 !₄[3]₄, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96 !₃[4]₃, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72 |- align=center !Vertices |24||24||24 |- align=center !Edges |96 2-edges||24 4-edge||24 3-edges |- valign=top !valign=center|Image |[[File:24-cell t0 F4.svg|200px]]<BR>24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges. |[[File:Complex polygon 4-3-4.png|200px]]<BR>₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges. |[[File:Complex polygon 3-4-3-fill1.png|200px]]<BR>₃{4}₃ or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled. |} === Related 4-polytopes === Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]: * truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]]; * truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]]; * and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]]. The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or "[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]]." The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]]. === Related uniform polytopes === {{Demitesseract family}} {{24-cell_family}} The 24-cell can also be derived as a rectified 16-cell: {{Tesseract family}} {{Symmetric_tessellations}} ==See also== *[[W:Octacube (sculpture)|Octacube (sculpture)]] *[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]] == Notes == {{Regular convex 4-polytopes Notelist|wiki=W:}} == Citations == {{Regular convex 4-polytopes Reflist|wiki=W:}} == References == {{Refbegin}} {{Regular convex 4-polytopes Refs|wiki=W:}} <br> * {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4|ref={{SfnRef|Ghyka|1977}}}} * {{cite journal|last1=Itoh|first1=Jin-ichi|last2=Nara|first2=Chie|doi=10.1007/s00022-021-00575-6|doi-access=free|issue=13|journal=[[W:Journal of Geometry|Journal of Geometry]]|title=Continuous flattening of the 2-dimensional skeleton of a regular 24-cell|volume=112|year=2021|ref=SfnRef|Itoh & Nara|2021}}}} {{Refend}} ==External links== * [https://bendwavy.org/klitzing/incmats/ico.htm ico], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes] * [https://polytope.miraheze.org/wiki/Icositetrachoron Icositetrachoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki] * [http://hi.gher.space/wiki/Xylochoron Xylochoron], at [http://hi.gher.space/wiki/Main_Page Higher space] * [https://www.qfbox.info/4d/24-cell The 24-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space] * [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations] * [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections] * [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }} * [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software] [[Category:Geometry]] [[Category:Polyscheme]] 4kgkz9t4805fn3jkraso3r0kz1kq6tt 4 305980 2819288 2811607 2026-07-24T16:09:54Z Atcovi 276019 /* See also */ +[[w:Wikipedia:Signs of AI writing|Wikipedia:Signs of AI writing]] 2819288 wikitext text/x-wiki {{policy|WV:AI}} This policy specifies the requirements for contributing [[w:Generative artificial intelligence|AI-generated content]] (text and media) to [[Main page|Wikiversity]]. AI-generated content is permitted where it follows good [[w:scholarly method|scholarly practice]]s, including: * '''Edit summary''': The origin of contributed AI text is clearly indicated in the [[Wikiversity:FAQ/Editing/Edit summary|edit summary]] along with a publicly accessible link to the chatbot conversation (or a copy of the transcript) to maximise transparency. * '''Verifiability''' / '''Citations''': AI text is [[Wikiversity:Verifiability|verified]] by the contributor. [[Wikiversity:Cite sources|Citations]] are [[w:Fact-checked|fact-checked]] for appropriateness and relevance by the contributor. * '''Human revision''': AI text is revised and rewritten by the contributor (or originally human-written and improved using genAI) * '''Copyright''': AI content must be compatible with Wikiversity's [[Wikiversity:Copyrights|licensing requirements]] (CC BY-SA). * '''Template''': Display the {{tl|AI-generated}} template at the top of pages, or below the description template for files, when the content contains substantial AI-generated material. Contributors wanting to use AI-generated content in ways not covered by this policy should seek community input by discussing at the [[Wikiversity:Colloquium|Colloquium]]. ==See also== ;Meta * [[meta:Artificial intelligence/Policies by project|Artificial intelligence/Policies by project]] (List) ;Wikimedia projects * [[b:Wikibooks:Artificial intelligence|Wikibooks:Artificial intelligence]] (Policy) * [[c:Commons:AI-generated media|Wikimedia Commons:AI-generated media]] (Policy) * [[w:Wikipedia:Large language models|Wikipedia:Large language models]] (Information page) * [[w:Wikipedia:Writing articles with large language models|Writing articles with large language models]] (Guideline) * [[w:Wikipedia:Signs of AI writing|Wikipedia:Signs of AI writing]] (Detection) ;Wikiversity project guidelines * [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]] (Motivation and emotion) ==External links== ;Wiki Education Foundation * [https://dashboard.wikiedu.org/training/students/generative-ai Using generative AI tools with Wikipedia] (Training module) [[Category:Artificial intelligence]] 3k1zab6ism55cxhpntg8rq2jq78r3b7 0 319133 2819350 2811461 2026-07-25T10:01:01Z ~2026-41522-22 3103197 /* Entertainment */ Fix typo. Change "Heroes Journey" to "Hero's Journey." 2819350 wikitext text/x-wiki {{AI-generated}} {{note|This is a user essay. It contains the advice or opinions of one or more Wikiversity contributors. It has not been reviewed by the Wikiversity community and does not necessarily reflect the views of the community.}} —Avoiding nonsense [[File:John Tenniel- Alice's mad tea party, colour.jpg|thumb|[[w:Alice's Adventures in Wonderland|Alice]] challenged the queen who believed six impossible things before breakfast.]] {{TOC right | limit|limit=2}} == Introduction == Perhaps Mark Twain<ref>{{Cite web|url=https://quoteinvestigator.com/2018/11/18/know-trouble/|title=It Ain’t What You Don’t Know That Gets You Into Trouble. It’s What You Know for Sure That Just Ain’t So – Quote Investigator®|date=2018-11-18|language=en-US|access-date=2025-03-02}}</ref> said it best: “It ain’t so much the things that people don’t know that makes trouble in this world, as it is the things that people know that ain’t so.” Why do people believe things that aren’t true? From [[w:Conspiracy_theory|conspiracy theories]]<ref>Speculations from minority viewpoints, often dismissed as conspiracy theories, [[/Conspiracy Theories That Turned Out to Be True/|sometimes turn out to be true]]. </ref> to [[w:Urban_legend|urban legends]], from [[w:Pseudoscience|pseudoscience]] to [[w:Pseudohistory|historical distortions]], misbeliefs shape the way individuals and societies interpret the world. This course is designed to explore the psychological, social, and cultural forces that drive false beliefs and their persistence. Through a blend of philosophy, cognitive science, and real-world case studies, we will examine how misbeliefs form, why they spread, and what strategies can help foster critical thinking and truth-seeking. Whether you’re interested in debunking myths, understanding human cognition, or navigating today’s complex information landscape, this course will provide valuable insights into the nature of belief—and how to distinguish fact from fiction.<ref>[[w:ChatGPT|ChatGPT]] generated this text responding to the prompt: “Write a short introduction to a course named ‘understanding misbelief’”.</ref> Misbeliefs are common and troublesome. We can learn to reduce their origination, spread, and harm. Approach misbeliefs as you would a burning candle; [[/Look but Don’t Touch/|look but don’t touch]]. == Objectives == The objectives of this course are to help students: * Avoid the temptations of misbeliefs. * Understand various motivations and benefits for holding misbeliefs and true beliefs; * Foster [[w:Intellectual_humility|intellectual humility]]—an awareness of the limits of our knowledge; * Find the [[Finding Courage|courage]] and [[wisdom]] to [[Embracing Ambiguity|embrace ambiguity]]. * Transition from [[Seeking True Beliefs#Rigidity|rigidly held beliefs]] to more carefully considered [[Seeking True Beliefs#Firmness|firmly held beliefs]]. * Understand the many mechanisms that foster misbelief including: motivations, benefits, vulnerabilities, and skills; * Reduce the prevalence, allure, and spread of misbeliefs; * [[Seeking True Beliefs|Choose true beliefs]]; and * Choose to abandon unhelpful misbeliefs. This course has no prerequisites, and all students are welcome. Several companion courses are available that can help students gain additional background or bolster their understanding of various concepts and techniques useful in evaluating information. These include: * [[Facing Facts]] — Embracing Reality * [[Evaluating Journalism Standards]] — Seeking reliable information sources * [[Evaluating Information]] — Fact or Fiction? * [[Embracing Ambiguity]] —Keep thinking * [[Intellectual honesty|Intellectual Honesty]] — Accurately communicating true beliefs * [[Virtues/Good Faith]] – The virtue of truthfulness * [[Forming beliefs|Forming Beliefs]] — Evaluating what you accept as true * [[Knowing How You Know|Knowing how you know]] * [[Seeking True Beliefs]] —Excellence in the Quest for Knowledge * [[Evaluating Evidence]] — Seeking Reality * [[Navigating Social Proof]] —Going along to get along * [[Fostering Curiosity]] — Wondering why Study these companion courses any time they may be helpful. Specific companion course suggestions appear in relevant sections throughout this course. == How Sure Are You? == Most of us believe we are reliable judges of what is true and what is false, distinguishing between fact and fiction, myth and reality. We hold numerous [[Forming beliefs|beliefs]] that we consider either true or false. Some of these beliefs are essential to us because they shape our identities, friendships, and communities. One objective of this course is to challenge that assumption and to foster [[w:Intellectual_humility|intellectual humility]]—an awareness of the limits of our knowledge === Assignment: === # Consider the statements in [[/How sure are you?/|this list of popular beliefs]]. # Choose several to rate according to your degree of certainty. # Are you mostly certain or uncertain of the accuracy of these statements? # Study the [[Seeking True Beliefs#Firmness|section on Firmness]] in the Wikiversity course on [[Seeking True Beliefs]]. # Work to hold your well-considered beliefs [[Seeking True Beliefs#Firmness|firmly]] rather than [[Seeking True Beliefs#Flaccidity|flaccidly]] or [[Seeking True Beliefs#Rigidity|rigidly]]. # Optionally study the Wikiversity courses on [[Knowing How You Know]], [[Seeking True Beliefs]], and [[Embracing Ambiguity]]. # For each statement that you marked as definitely true or definitely false, research the correct statement using [[Evaluating Information|reliable sources]] and by carefully [[Evaluating Evidence|evaluating evidence]]. == Characterizing Misbelief == For the purposes of this course, a ''misbelief'' is a [[Seeking True Beliefs#Rigidity|rigidly held]] belief that individuals prefer over a [[Seeking True Beliefs|true belief]]. ''Misbelievers'' are people who hold such misbeliefs. Misbelievers who share similar misbeliefs may engage in discussions, debates, and the exchange of those beliefs within [[w:Cult|cults]], develop [[w:Conspiracy_theory|conspiracy theories]], or firmly adhere to various [[w:Pseudoscience|pseudoscientific]] claims, [[w:Hoax|hoaxes]] or [[w:Ideology|ideologies]]. A ''misbelief'' differs from a ''misunderstanding'' because it is [[Seeking True Beliefs#Rigidity|rigidly held]], and resists correction despite [[Evaluating Evidence|evidence]] that falsifies the misbelief. [[w:Certainty|Certainty]] displaces [[Fostering Curiosity|curiosity]] and [[w:Doubt|doubt]] in sustaining misbelief. A misbelief is based on a [[Exploring Worldviews|worldview]] that is not aligned with the best available [[Exploring Worldviews/Aligning worldviews|understanding of reality]]. == Prevalence == How many people are misbelievers? In August 2004, a poll by Zogby International showed that 49 percent of New York City residents, with a margin of error of 3.5 percent, believed that officials of the U.S. government “knew in advance that attacks were planned on or around [[w:September_11_attacks|September 11, 2001]], and that they consciously failed to act".<ref>[http://www.ask-force.org/web/Discourse/Sunstein-Conspiracy-Theories-2009.pdf Symposium on Conspiracy Theories, Conspiracy Theories: Causes and Cures], Cass R. Sunstein, Law, Harvard University and Adrian Vermeule Law, Harvard University</ref> In any case, roughly half the people have it wrong. The [[w:Barack_Obama_citizenship_conspiracy_theories|Barack Obama citizenship conspiracy theories]] had a significant number of believers and had significant [[w:Barack_Obama_citizenship_conspiracy_theories#cite_note-HarrisPoll201003-11|political impact]]. In a [[w:Harris_Poll|Harris Poll]] online survey of 2,320 adults conducted in March 2010, 25% of the respondents said they believed that Obama was "[[w:Barack_Obama_citizenship_conspiracy_theories|not born in the United States]] and so is not eligible to be president". In a July 2010 CNN poll of adult Americans, 16% said they had doubts that Obama was born in the United States, and a further 11% were certain that he was not. A 2012 study found that 63 per cent of registered voters in the United States buy into at least one conspiracy theory<ref>[https://portal.fdu.edu/fdupoll-archive/outthere/final.pdf Conspiracy theories prosper: 25% of Americans are “truthers”], Fairleigh Dickenson University Public Mind, January 17, 2013.</ref> In May 2021 34% of the 1,230 respondents polled agreed with the statement “Elites, from government and Hollywood, are engaged in a massive child sex trafficking racket”.<ref>{{Cite journal|last=Uscinski|first=Joseph|last2=Enders|first2=Adam|last3=Klofstad|first3=Casey|last4=Seelig|first4=Michelle|last5=Drochon|first5=Hugo|last6=Premaratne|first6=Kamal|last7=Murthi|first7=Manohar|date=2022-07-20|title=Have beliefs in conspiracy theories increased over time?|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC9299316/|journal=PLOS ONE|language=en|volume=17|issue=7|pages=e0270429|doi=10.1371/journal.pone.0270429|issn=1932-6203}}</ref> Most Americans (71%) have heard of a conspiracy theory circulating widely online that alleges that powerful people intentionally planned the [[w:COVID-19_misinformation|coronavirus outbreak]]. And a quarter of U.S. adults see at least some truth in it – including 5% who say it is definitely true and 20% who say it is probably true.<ref>{{Cite web|url=https://www.pewresearch.org/short-reads/2020/07/24/a-look-at-the-americans-who-believe-there-is-some-truth-to-the-conspiracy-theory-that-covid-19-was-planned/|title=A look at the Americans who believe there is some truth to the conspiracy theory that COVID-19 was planned|last=Schaeffer|first=Katherine|date=2020-07-24|website=Pew Research Center|language=en-US|access-date=2025-03-02}} Katherine Schaeffer, Pew Research Center, July 24, 2020</ref> While boosting his career as a professional conspiracy theorist, [[w:Alex_Jones|Alex Jones]] was [[w:Sandy_Hook_Elementary_School_shooting#Alex_Jones|ordered to pay]] $965 million in damages to the families of Sandy Hook victims for repeatedly spreading disproven conspiracy theories about the 2012 [[w:Sandy_Hook_Elementary_School_shooting#Alex_Jones|Sandy Hook Elementary School shooting]], including claiming that it was a "false flag" operation perpetrated by gun control advocates, that "no one died" in Sandy Hook, and that the incident was "staged", "synthetic", "manufactured", "a giant hoax" and "completely fake with actors". Several Fox programs had broadcast false statements that Dominion's voting machines had been rigged to steal the 2020 United States presidential election from then-president Donald Trump. [[w:Dominion_Voting_Systems_v._Fox_News_Network|Fox News agreed to pay Dominion $787.5 million]] and acknowledged the court's earlier ruling that Fox had broadcast false statements about Dominion. == The Allure of Misbeliefs == Misbeliefs can be incredibly appealing despite being inaccurate or even harmful.  The [[/The Allure of Misbelief: Why People Embrace Falsehoods Over Truths/|allure of misbelief]] stems from deep psychological, social, and existential needs that make certain falsehoods more attractive than the truth. === Assignment === # Read this essay on the [[Understanding Misbelief/The Allure of Misbelief: Why People Embrace Falsehoods Over Truths|Allure of misbelief]]. # What, if any, of the factors described in [[Understanding Misbelief/The Allure of Misbelief: Why People Embrace Falsehoods Over Truths|the essay]] attract you toward your rigidly held beliefs? # Read this essay on the [[/The Utility of Misbelief/|Utility of Misbelief]]. # What, if any, of the factors described in the essay attract you toward your rigidly held beliefs? # Read this essay on [[/The Utility of Misbelief: A Critical Examination/|a critical examination of the utility of misbelief]]. # Read this essay on the [[/Benefits of True Beliefs/|Benefits of True Beliefs]]. # Read the essay [[/Facts Are Stubborn/|Facts Are Stubborn]]. # What factors motivate you toward misbeliefs or true beliefs? == Motivations == [[Forming beliefs|Beliefs]] shape how individuals interpret the world, influencing their actions, relationships, and overall worldview.  While truth is often held as the ultimate ideal, human psychology does not always prioritize factual accuracy. Instead, [[Forming beliefs|belief formation]] is deeply intertwined with emotional, social, and cognitive motivations. Misbeliefs—those that deviate from reality—persist not merely due to ignorance but because they fulfill fundamental human needs, such as connection, [[w:Belongingness|belonging]], and a sense of special knowledge. In contrast, true beliefs often align with motivations related to competence, safety, and integrity. Understanding the competing motivations that drive misbeliefs and true beliefs can help explain why falsehoods are sometimes more appealing than reality and why truth remains an essential yet challenging pursuit. === Assignment === # Read the essay [[/Exploring Motivations for Misbelief and True beliefs/|Exploring Motivations for Misbelief and True beliefs]]. # Read the Socratic Dialogue [[Wikidialogue/Are true beliefs more useful than misbeliefs?|Are true beliefs more useful than misbeliefs]]? # What motivates you to choose misbeliefs or true beliefs? == The Dangers of Misbelief == Beliefs shape how we interpret the world, make decisions, and interact with others.<ref>[[w:ChatGPT|ChatGPT]] generated this text responding to the prompt: “Write an essay describing dangers of misbelief”.</ref> While well-founded beliefs based on [[Evaluating Evidence|evidence]] and rational analysis guide us toward truth and progress, misbeliefs—false, unfounded, or irrational convictions—pose significant dangers. Misbeliefs can distort reality, influence poor decision-making, fuel conflicts, and enable exploitation. In an era where misinformation spreads rapidly through digital networks, understanding the dangers of misbelief is crucial for both individuals and societies. These dangers include poor decision-making, political and social polarization, exploitation and manipulation, societal and environmental harm, psychological and existential costs, among others. === Assignment === Read this essay describing the [[/The Dangers of Misbelief/|dangers of misbelief]]. == The Genesis of Misbeliefs == Several environmental conditions, cognitive features, personality factors, and social elements, make us vulnerable to rigidly holding onto misbeliefs. These are described in detail in the sections below. == Stress Sets the Stage == [[File:The Scream.jpg|thumb|The Scream by Edvard Munch (1893)]] [[w:Psychological_stress|Stress]] with its numerous causes and consequences, has a significant impact on our well-being. Stress is very common. A recent (2022) National Center for Health Statistics report, revealed that 21.4% of Americans report symptoms of anxiety disorder or depression disorder.<ref>National Center for Health Statistics. U.S. Census Bureau, Household Pulse Survey, 2020–2024. Anxiety and Depression. Generated interactively: November 20, 2024from <nowiki>https://www.cdc.gov/nchs/covid19/pulse/mental-health.htm</nowiki></ref> [[w:Stressor|Stressors]]—events or conditions that cause stress—are more likely to affect the health of an individual when they are "chronic, highly disruptive, or perceived as uncontrollable". While there are various types of stressors, [[w:Psychological_stress#Daily_hassles/microstressors|daily hassles]] and [[w:Psychological_stress#Ambient_stressors|ambient stressors]] are the most common and prevalent in most adults. While [[/Possible Causes of Everyday Stress/|everyday stressors]] are a somewhat expected part of life, and most people manage to cope with them, [[Understanding Misbelief/Possible Causes of Unpredictable Stress|''unpredictable'' stress]], arising from sudden or unforeseen events, can be particularly detrimental. These events are often perceived as uncontrollable and can impair our ability to think rationally and reflectively, making us susceptible to false beliefs. The [[w:COVID-19_pandemic|COVID-19 pandemic]] was an especially intense, extensive, and long-lasting source of unpredictable stress. In general, humans don’t cope well with unpredictable stress.<ref>{{cite book|title=Misbelief: What Makes Rational People Believe Irrational Things|last=Ariely|first=Dan|date=September 17, 2024|publisher=Harper Perennial|isbn=978-0063280434|pages=320|author-link=w:Dan_Ariely}} @77 of 488</ref> People may choose misbeliefs to help them cope with stress. Stress, and especially [[/Possible Causes of Unpredictable Stress/|unpredictable stress]], may lead to the development of ''[[Understanding Misbelief#Learned helplessness|learned helplessness]]'', discussed in the next section. === Assignment: === # Browse this list of [[Understanding Misbelief/Possible Causes of Everyday Stress|everyday stressors]]. # Identify sources of everyday stress that you encounter. # Browse this list of [[Understanding Misbelief/Possible Causes of Unpredictable Stress|unpredictable stress]]. # Identify sources of unpredictable stress that you encounter. # Which, if any, of these stressors cause you to feel discouraged or even helpless? # Seek constructive approaches to reducing stress. The Wikiversity course on [[Finding Equanimity]] may be helpful. === Learned helplessness === Uncontrollable events, such as unpredictable stress, can lead to a condition called [[w:Learned_helplessness|learned helplessness]]. This condition undermines a person’s [[What you can change and what you cannot#Agency|agency]] and willingness to [[What you can change and what you cannot|control events]], even when there are available means of control. Quoting [[w:Martin_Seligman|Martin Seligman]]: <blockquote>Not only do we face events that we can control by our actions, but we also face many events about which we can do nothing at all. Such uncontrollable events can significantly debilitate organisms: they produce passivity in the face of trauma, inability to learn that responding is effective, and emotional stress in animals, and possibly depression in man.<ref>[https://axelkra.us/wp-content/uploads/2021/04/document.pdf Learned Helplessness]; Martin E. P. Seligman, 1972, Departments of Psychiatry and Psychology, University of Pennsylvania, Philadelphia, Pennsylvania. </ref></blockquote>If enough unfortunate events beyond your control occur, you may eventually become overwhelmed and stop trying to help yourself. Your vitality and zest are gone, you are listless and discouraged, and you believe that nothing you do even matters. You have lost the struggle and learned to become helpless, and you are now passive and complacent even though you [[What you can change and what you cannot#Agency|could take action]] to help yourself. Perhaps rethinking how you explain these events to yourself can help you cope better.<ref>Adapted from the EmotionalCompetency.com entry on [https://www.emotionalcompetency.com/helpless.htm Learned Helplessness], with permission of the author. </ref> Uncontrollable events disrupt peoples' subsequent [[Solving Problems|problem]] solving skills. How people choose to [[Attributing Blame|explain the causes]] of these bad events affect their response in a variety of ways, including motivation, emotion, cognition, and behavior. People tend to define the extent of their helplessness—their lack of control or incompetency—as being either pervasive or narrow, or short term or long term. Various misbeliefs may seem to explain what is happening, identifying who or what is to [[Attributing Blame|blame]], may suggest specific actions you can take, and provide you some control over adverse events in your life. It is now clear what is ''really''happening, who is to blame and what you can do about it. Misbeliefs help you regain control. ==== Assignment ==== # Complete the Wikiversity course [[Sustaining Agency]]. # Avoid succumbing to [[Transcending Conflict|learned helplessness]] by sustaining your agency. === Intergroup Conflict === Recent research suggests a possible reciprocal relationship between [[w:Conspiracy_theory|conspiracy theories]] and violent [[w:Group_conflict|intergroup conflicts]]. Intergroup conflicts can strengthen belief in conspiracy theories, which, in turn, can radicalize the societies involved. This radicalization can hinder peaceful [[Transcending Conflict|conflict resolution]].<ref>{{Cite journal|last=Hebel-Sela|first=Shira|last2=Hameiri|first2=Boaz|last3=Halperin|first3=Eran|date=2022-10-01|title=The vicious cycle of violent intergroup conflicts and conspiracy theories|url=https://www.sciencedirect.com/science/article/abs/pii/S2352250X22001439|journal=Current Opinion in Psychology|volume=47|pages=101422|doi=10.1016/j.copsyc.2022.101422|issn=2352-250X}}</ref> ==== Assignment ==== # Study the Wikiversity course [[Transcending Conflict]]. # Transcend conflict. === Misattributions === We are not good at identifying the causes of our stress. As people seek to maintain control over their lives and understand what is happening in the word around them, they naturally seek to attribute various events to specific causes. Unfortunately, people often make mistakes and incorrectly attribute causes to various effects. Attribution errors occur at various [[Layers of Human Interaction|layers of our human interaction]], including the emotional, personality, and cognitive layers of ourselves. This section will address misattributing emotions. Experiments have shown that people often make a [[w:Misattribution_of_arousal|mistake in identifying the true cause of their arousal]]. For instance, when experiencing physiological responses associated with fear, individuals may mistakenly label those responses as romantic arousal. This error occurs because many stimuli share similar physiological symptoms, such as elevated blood pressure or shortness of breath. More generally, many people lack skill in recognizing, interpreting, and responding constructively to emotions in themselves and others. ==== Assignment ==== # Complete the Wikiversity course on [[Recognizing Emotions|recognizing emotions]]. # Practice accurately recognizing emotions as they arise in you and in others. # Optionally complete the Wikiversity curriculum on [[Emotional Competency|Emotional competency]]. # The guide to [[Studying Emotional Competency]] suggests a path for studying the emotional competency material and presents a thorough and orderly tour of the entire curriculum. # Improve your emotional competency. # Study the Wikiversity course on [[Attributing Blame]]. # Become accurate in attributing blame. === Scarcity Mindset === When you have too much on your mind, it is more difficult to think clearly, and you are more likely to make mental errors. Research indicates that people experiencing insufficient resources can foster a “scarcity” mindset, where individuals focus excessively on the scarce resource, potentially neglecting other important aspects of their lives.<ref>{{Cite journal|last=Huijsmans|first=Inge|last2=Ma|first2=Ili|last3=Micheli|first3=Leticia|last4=Civai|first4=Claudia|last5=Stallen|first5=Mirre|last6=Sanfey|first6=Alan G.|date=2019-06-11|title=A scarcity mindset alters neural processing underlying consumer decision making|url=https://www.pnas.org/doi/full/10.1073/pnas.1818572116|journal=Proceedings of the National Academy of Sciences|volume=116|issue=24|pages=11699–11704|doi=10.1073/pnas.1818572116|pmc=PMC6575633|pmid=31123150}}</ref> This scarcity mindset can result in a reduction in freely available mental resources—referred to as ''[[w:Cognitive_load|cognitive bandwidth]]''—because they are being diverted to another task.<ref>{{cite book|title=Misbelief: What Makes Rational People Believe Irrational Things|last=Ariely|first=Dan|date=September 17, 2024|publisher=Harper Perennial|isbn=978-0063280434|pages=320|author-link=w:Dan_Ariely}} @91 of 488</ref> In one study it appears that [[w:Poverty|poverty]] itself reduces cognitive capacity. The researchers suggest this is because poverty-related concerns [[w:Cognitive_load|consume mental resources]], leaving less for other tasks.<ref>Poverty Impedes Cognitive Function, by Anandi Mani, Sendhil Mullainathan , Eldar Shafir , and Jiaying Zhao. Science. 30 Aug 2013 Vol 341, Issue 6149 pp. 976-980.</ref> Several studies show that [[w:Scarcity|scarcity]], which ranges from external experiences of financial insecurity, a lack of time, or [[w:Social_isolation|social isolation]], can become an internalized [[w:Mindset|mindset]] that inhibits cognitive functioning and effective [[w:Coping|coping]].<ref>Mitsui, Kristi (2022) "The Relationship Between Coping Mechanisms and the Scarcity Mindset," Undergraduate Research: Vol. 2: Iss. 2, Article 21. Available at: <nowiki>https://kb.gcsu.edu/undergraduateresearch/vol2/iss2/21</nowiki></ref> Analyses showed that experiencing scarcity was associated with pessimism and maladaptive forms of coping. Additionally, there was evidence for [[w:Social_support|social support]] as a potential moderating factor for the consequences of scarcity. In one study behavioral and neural evidence suggests that inducing a scarcity mindset significantly dampens the ability to empathize with others’ pain during both the early and late stages of empathic processing. These findings shed light on our understanding of how a scarcity mindset may influence social emotions and behaviors.<ref>Scarcity mindset reduces empathic responses to others’ pain: the behavioral and neural evidence, by Wanchen Li, Jing Meng, and Fang Cui. Social Cognitive and Affective Neuroscience, Volume 18, Issue 1, 2023, nsad012, <nowiki>https://doi.org/10.1093/scan/nsad012</nowiki></ref> ==== Assignment ==== # [[Living Wisely/Take Care|Take care]] of yourself and others. # Maintain resilience, [[Clear Thinking/Curriculum|think clearly]], [[Finding Equanimity|find equanimity]], and [[Sustaining Agency|sustain your agency]] despite the turmoil in your life. === Economic Inequality === [[w:Economic_inequality|Economic inequality]]—a large difference in income and wealth between the richest and poorest people—has many effects that increase stress, including the direct negative effect of reducing [[w:Group_cohesiveness|social belonginess]] in a community while weakening people’s resilience.<ref>{{cite book|title=Misbelief: What Makes Rational People Believe Irrational Things|last=Ariely|first=Dan|date=September 17, 2024|publisher=Harper Perennial|isbn=978-0063280434|pages=320|author-link=w:Dan_Ariely}} @104 of 488</ref> Effects of income inequality, researchers have found, include higher rates of health and social problems, and lower rates of social goods, a lower population-wide satisfaction and happiness and even a lower level of economic growth when human capital is neglected for high-end consumption. 2013 Economics Nobel prize winner [[w:Robert_J._Shiller|Robert J. Shiller]] said that rising inequality in the United States and elsewhere is the most important problem. ==== Assignment ==== # [[Living Wisely/Take Care|Take care]] of yourself and others. # Maintain resilience, [[Clear Thinking/Curriculum|think clearly]], [[Finding Equanimity|find equanimity]], and [[Sustaining Agency|sustain your agency]] despite the turmoil in your life. === Unfair Treatment === People have no trouble complaining. Many can passionately recite their [[/A Litany of Grievances/|list of grievances]] whenever they find an audience, willing or not. Author and journalist [[w:Frank_Bruni|Frank Bruni]] claims we are living in the age of grievance.<ref>{{cite book|title=The Age of Grievance|last=Bruni|first=Frank|date=April 30, 2024|publisher=Avid Reader Press|isbn=978-1668016435|pages=288|author-link=w:Frank_Bruni}}</ref> To gain some sense of control over their [[w:Psychological_stress#Daily_hassles/microstressors|daily hassles]], unpredictable stress, and specific grievances, these aggrieved people find someone or something to [[Attributing Blame|blame]]. Each of these grievances is regarded as evidence of an injustice<ref>Dan Ariely uses the term “hard done by” in his book on Misbelief. </ref> directed toward ''me''. This is unfair I tell you! Because ''something'' has to change the injustice leads to [[Resolving Anger|anger]], and [[w:Punishment|retribution]] becomes imperative. [[w:Scapegoating|Scapegoating]] becomes common, although the blame is typically unwarranted. ==== Assignment ==== # [[Understanding Misbelief/A Litany of Grievances|Choose a grievance from this]] list of grievances or some other source to focus on for this assignment. # Write down the causes contributing to that grievance. Decide who to blame for the injustice. #* How do you know? Explain the connection in detail. #* Are you bristling from oppression because of this grievance? Why? # Complete the Wikiversity course on [[Attributing Blame]]. # Perform a careful [[Attributing Blame#Cause-Effect Analysis|cause-effect analysis]] to identify the many contributing causes of the chosen grievance. #* Does this more careful analysis identify additional causes contributing to the chosen grievance? # If this grievance still makes you angry, complete the Wikiversity course on [[Resolving Anger]]. # [[Living Wisely/Take Care|Take care]] of yourself and others. # Maintain [[Understanding Misbelief#Resilience|resilience]], [[Clear Thinking/Curriculum|think clearly]], [[Finding Equanimity|find equanimity]], and [[Sustaining Agency|sustain your agency]] despite the turmoil in your life. === Connecting the Dots from Stress to Misbelief === Stress is an integral part of life, with [[Understanding Misbelief/Possible Causes of Everyday Stress|everyday stressors]] often manageable. However, [[Understanding Misbelief/Possible Causes of Unpredictable Stress|unpredictable stress]], perceived as uncontrollable, can be particularly harmful. It impairs [[Clear Thinking/Curriculum|rational thinking]] and increases susceptibility to false beliefs. Uncontrollable stress can lead to [[w:Learned_helplessness|learned helplessness]], where individuals lose the will to exert control even when solutions exist. Seeking to make sense of events, people often misattribute causes, struggling to identify the true sources of stress or recognize their emotions. A scarcity mindset adds to this burden, diminishing [[Solving Problems|problem-solving]] and resilience. [[w:Economic_inequality|Economic inequality]] intensifies stress by eroding [[w:Group_cohesiveness|social belonging]] and weakening [[w:Psychological_resilience|resilience]]. In response to stress and uncertainty, people often seek control by [[Attributing Blame|assigning blame]], which can escalate to unwarranted [[w:Scapegoating|scapegoating]] fueled by [[Resolving Anger|anger]] and the desire for [[w:Punishment|retribution]]. === Resilience === Increasing [[w:Psychological_resilience|resilience]]—the ability to [[w:Coping_(psychology)|cope]] mentally and emotionally with a crisis—can help to manage stress. Many factors influence a person's level of resilience. Internal factors include personal characteristics such as [[w:Self-esteem|self-esteem]], [[w:Emotional_self-regulation|self-regulation]], and a [[w:Optimism|positive outlook]] on life. External factors include [[w:Social_support|social support]] systems, including relationships with family, [[Being Friends|friends]], and [[Creating Communities|community]], as well as access to resources and opportunities. Resilience is a "positive adaptation" to a stressful or adverse situation. When a person is "bombarded by daily stress, it disrupts their internal and external sense of balance, presenting challenges as well as opportunities." The [[w:Psychological_stress|routine stressors]] of daily life can have positive impacts, which promote resilience. Some people can handle stress better than others. Although different levels of stress vary among different individuals, stress allows people to practice resilience over time. People who have developed [[w:Secure_attachment|secure attachment]] during childhood cope better because they are able to go through life knowing that if something bad happens, someone will be there to help out.<ref>{{cite book|title=Misbelief: What Makes Rational People Believe Irrational Things|last=Ariely|first=Dan|date=September 17, 2024|publisher=Harper Perennial|isbn=978-0063280434|pages=320|author-link=w:Dan_Ariely}} @97 of 488</ref> [[w:Psychological_resilience#Social_support|Social support]] is an important factor in the development of resilience. Social support requires solidarity and trust, intimate communication, and mutual obligation both within and outside the family and community. ==== Assignment ==== # Read this list of [[w:Psychological_resilience#Developing_and_sustaining_resilience|suggestions for developing and sustaining resilience]]. # Employ tactics that you find effective in increasing your resilience. # Do your best to [[Sustaining Agency|sustain your agency]] despite the turmoil in your life. === Assignment === Cope with stress, increase your resilience, and reduce misbeliefs by taking actions selected from the following list. # [[Living Wisely/Take Care|Take care]]. Give care. # [[Creating Communities|Join communities]] (In real life). # [[Alleviating Loneliness|Alleviate loneliness]]. # Improve your [[Studying Emotional Competency|emotional competency]]. # [[Earning Trust|Earn trust]], grant trust, [[w:Wikipedia:Assume_good_faith|assume good faith]]. # Practice [[Knowing Someone/Deep Listening|deep listening]] and [[Practicing Dialogue|dialogue]]. # Use [[Socratic Methods|Socratic methods]] to explore beliefs. # [[Clear Thinking/Curriculum|Think more clearly]] #* [[Facing Facts|Face Facts]]. #* [[Evaluating Evidence|Evaluate evidence]] skillfully. #* [[Evaluating Journalism Standards|Evaluate journalism standards]]. #* [[Evaluating Information|Evaluate information]] carefully. #* [[Seeking True Beliefs|Seek true beliefs]]. #* [[Knowing How You Know|Know how you know]]. #* [[Thinking Scientifically|Think scientifically]]. #* [[Exploring Worldviews/Aligning worldviews|Align your worldview with reality]]. #* [[Socratic Methods|Practice Socratic methods]]. #* Expect [[intellectual honesty]] from yourself and others. === Naming the Villain === [[w:Villain|Villains]] are often the centerpiece of a good story. Without villains there are no heroes, and bringing the scoundrel to justice restores order and decency to our world.   In [[w:Storytelling|storytelling]], the villain’s structural purpose is to oppose the [[w:Hero|hero]] character. Their motives or evil actions drive the plot forward. Unlike the hero, who is defined by their ingenuity, bravery, pursuit of justice, and the greater good, a villain is often characterized by their selfishness, evilness, arrogance, cruelty, and cunning. These immoral behaviors can either oppose or pervert justice. You can become the hero by identifying and destroying the villain. The search for the villain is an adventure that puts the detectives in control. Furthermore, research suggest perceived intention influences the experience of pain.<ref>{{Cite web|url=https://www.proquest.com/openview/78018030b311c3ddfb9bda70c79df3fe/1?pq-origsite=gscholar&cbl=18750|title=Cruel shocks and kind massages: Pain, pleasure and perceived intention - ProQuest|website=www.proquest.com|language=en|access-date=2025-03-05}}</ref>^,<ref>{{Cite journal|last=Gray|first=Kurt|last2=Wegner|first2=Daniel M.|date=2008-12-01|title=The Sting of Intentional Pain|url=https://journals.sagepub.com/doi/10.1111/j.1467-9280.2008.02208.x|journal=Psychological Science|language=en|volume=19|issue=12|pages=1260–1262|doi=10.1111/j.1467-9280.2008.02208.x|issn=0956-7976}}</ref> Attributing evil intent to the villain increases the pain felt by the victim. ==== Assignment ==== # Practice [[Seeking True Beliefs#Humility|intellectual humility]] to acknowledge doubts that you have correctly identified the villain, and the villain intended to hurt you. # [[Embracing Ambiguity|Embrace ambiguity]] so you can be comfortable in not identifying a villain, embracing complexity, continuing the investigation, and seeking a deeper and more nuanced understanding of the problem and possible solutions. # Study the Wikiversity course on [[Attributing Blame]]. # Become accurate in attributing blame. === Complex Stories are Fun === [[w:Conspiracy_theory|Conspiracy theories]] are often remarkably complex. For example, with a notable lack of [[w:Intellectual_humility|intellectual humility]],  [[w:David_Robert_Grimes|David Robert Grimes]] [[w:Conspiracy_theory#Viability|estimates that]]: * A [[w:Moon_landing_conspiracy_theories|Moon landing hoax]] would require the involvement of 411,000 people and would be exposed within 3.68 years; * [[w:Global_warming_conspiracy_theory|Climate-change fraud]] would require a minimum of 29,083 people (published climate scientists only) and would be exposed within 26.77 years, or up to 405,000 people, in which case it would be exposed within 3.70 years; * A vaccination conspiracy would require a minimum of 22,000 people (without drug companies) and would be exposed within at least 3.15 years and at most 34.78 years depending on the number involved; * A conspiracy to [[w:Big_Pharma_conspiracy_theory|suppress a cure for cancer]] would require 714,000 people and would be exposed within 3.17 years. Complex stories are appealing because there is always more to discover and more to tell. Just as [[w:Cliffhanger#Serial_media|serial adventures]] end each episode with a [[w:Cliffhanger|cliff hanger]], [[Fostering Curiosity|curiosity]], surprise, and attention are manipulated as each new twist and turn of the story is revealed over time. The conspiracy theorist is captivated as the story unfolds. Furthermore, [[w:Conspiracy_theory#Rhetoric|conspiracy theory rhetoric]] exploits several important [[w:Cognitive_bias|cognitive biases]], including [[w:Proportionality_bias|proportionality bias]], [[w:Attribution_bias|attribution bias]], and [[w:Confirmation_bias|confirmation bias]]. The ''[[w:Proportionality_bias|proportionality bias]]'', also known as major event/major cause heuristic, is the tendency to assume that big events have big causes. It is a type of cognitive bias and plays an important role in people's tendency to accept conspiracy theories. Academic psychologist Rob Brotherton summarizes it as "When something big happens, we tend to assume that something big must have caused it".<ref>{{Cite web|url=https://www.jdsupra.com/legalnews/account-for-proportionality-bias-big-63232/|title=Account for Proportionality Bias: Big Events Must Have Big Causes|website=JD Supra|language=en|access-date=2025-03-05}}</ref> In psychology, an [[w:Attribution_bias|attribution bias]] is a cognitive error that occurs when individuals systematically deviate from rationality and normality in evaluating or attempting to find reasons for their own and others’ behaviors. This bias leads to perceptual distortions, inaccurate assessments, and illogical interpretations of events and behaviors. [[w:Confirmation_bias|Confirmation bias]] is the tendency to search for, interpret, favor, and recall information in a way that confirms or supports one's prior beliefs or values. People display this bias when they select information that supports their views, ignoring contrary information, or when they interpret ambiguous evidence as supporting their existing attitudes. The effect is strongest for desired outcomes, for emotionally charged issues, and for deeply entrenched beliefs. The complexity of conspiracy theories also makes them attractive because of the ''unique knowledge'' required to fully understand them. Because only the [[w:In-group_and_out-group|ingroup]] knows the ''real'' truth about what happened, they attain special [[w:Social_status|social status]] and share a special bond. ==== Kayfabe ==== [[File:Sgt. Slaughter and The Grand Wizard.png|thumb|Kayfabe characters [[w:Sgt_Slaughter|Sgt Slaughter]] and [[w:Ernie_Roth|The Grand Wizard]] in a wrestling ring]] In professional wrestling, [[w:Kayfabe|kayfabe]] is the portrayal of staged events within the industry as "real" or "true", specifically the portrayal of competition, rivalries, and relationships between participants as being genuine and not staged. The term kayfabe has evolved to also become a code word of sorts for maintaining this "reality" within the direct or indirect presence of the general public. The interplay of truth and fantasy generates a unique appeal for some audiences. The allure of some misbeliefs may be like the allure of kayfabe.<ref>{{Cite web|url=https://heathercoxrichardson.substack.com/p/december-26-2024|title=December 26, 2024|last=Richardson|first=Heather Cox|date=2024-12-27|website=Letters from an American|access-date=2025-03-05}}</ref> ==== Assignment ==== Three [[w:Heuristic|heuristics]], called [[w:Philosophical_razor|razors]], can help evaluate and cast doubt on various conspiracy theories. Apply each of these three razors to assess the credibility of any proposed conspiracy theory and use them to cut through the nonsense. # [[w:Occam's_razor|Occam's razor]]—paraphrased as “The simplest explanation is usually the best one”—is a problem-solving principle that recommends searching for explanations constructed with the smallest possible set of elements. # [[w:Hanlon's_razor|Halon’s razor]] states: “Never attribute to malice that which is adequately explained by stupidity.” # [[w:Hitchens's_razor|Hitchens’s razor]] states: “What can be asserted without evidence can also be dismissed without evidence.” === Fear Becomes Hate and Moral Outrage === People who share misbeliefs form an [[w:In-group_and_out-group|in-group]] based on sharing those misbeliefs. The shared (mis)beliefs, unique knowledge and insights, and the common experiences of being different, [[w:Ostracism|ostracism]] and [[w:Persecution|persecution]] help to bond the group. Those who doubt your deeply held (mis)beliefs are threat and form an out-group. [[w:Xenophobia|Xenophobia]] is the fear or dislike of anything that is perceived as being foreign or strange. It is based on the perception that a conflict exists between an [[w:In-group_and_out-group|in-group]] and an out-group, and it may manifest itself in suspicion of one group's activities by members of the other group, a desire to eliminate the presence of the group that is the target of suspicion, and fear of losing a national, ethnic, or racial identity. Simplistically, xenophobia is the belief that foreigners are inherently to be feared. This belief can easily lead to a [[Recognizing Fallacies|logical fallacy]], where if one is fearful, then foreigners are to blame. Consequently—by this simplistic and fallacious logic—it becomes acceptable to hate and discriminate against the dangerous foreigners who are causing the feared conditions. Rather than [[Clear Thinking/Curriculum|thinking more clearly]], seeking to [[Knowing Someone|understand others]], [[Finding Common Ground|finding common ground]], or seeking to [[Overcoming Hate|overcome hate]], these misbelievers emphasize their hatred, and dismiss others as morally repulsive. Taken to extremes, the morally outrageous people who dare to dismiss your misbeliefs are not to be [[Forgiving|forgiven]]. === Assignment === # Separate your concern for the person from the beliefs they hold. # Seek to [[Knowing Someone|understand others]] who hold differing beliefs. # [[Finding Common Ground|Find common ground]]. # [[Overcoming Hate|Overcome hate]]; refuse to hate. === Entertainment === [[File:Heroesjourney.svg|thumb|The Hero's Journey]] Stories—real and imagined—of various [[w:Hero's_journey|hero’s journeys]] have fascinated people for thousands of years. Searching for the ''real'' causes of grievances is no exception. This is fun, this is adventure, this is entertainment! Each [[Understanding Misbelief/A Litany of Grievances|grievance]] is a compelling [[w:Hero's_journey#The_Call_to_Adventure|call to adventure]]. Foreseeing the difficulty, wise heroes may be [[w:Hero's_journey#Refusal_of_the_Call|reluctant to begin]] the journey. But eventually, the hero refuses to be treated unfairly anymore. This raw deal must end. The injustice must be set right. Leaving behind the delusional comfort of the mainstream narrative, the hero [[w:Hero's_journey#The_Crossing_of_the_First_Threshold|crosses the threshold]] and enters the realm of his fellow misbelievers. Leaving the warnings, safety, and comfort of the conventional folk behind, he [[w:Hero's_journey#Belly_of_the_Whale|commits to search]] for the true causes of his grievance, and is now ''all in''. The transformation is underway, there is no going back now. The game is on. The investigation is captivating. The drama continues to unfold. The villains must be found and brought to justice! The revelations are sensational! Obstacles provided by [[w:Public_opinion|public opinion]], conventional wisdom, logic, and  overwhelming [[Evaluating Evidence|evidence]] [[w:Hero's_journey#The_Road_of_Trials|test his resolve]], yet he continues the quest undaunted. His clever and original research begins to identify inconsistencies in the mainstream narrative, and he soon uncovers [[w:Hero's_journey#The_Meeting_with_the_Goddess|conclusive evidence to explain]] the true causes of his grievance. He dismisses the many [[w:Hero's_journey#Woman_as_the_Temptress|doubters who test his resolve]] and try to deter him, but he will [[w:Ulysses_pact|not be deterred by temptations]]. He is now [[w:Hero's_journey#Atonement_with_the_Father/Abyss|seeing beyond illusions]], he sees clearly, and finally grasps a greater understanding of how the world really works. His [[w:Hero's_journey#The_Ultimate_Boon|goal is achieved]]! He begins to contemplate returning to the mainstream with his new insights. Does he [[w:Hero's_journey#Refusal_of_the_Return|dare to share]] his new and powerful insights with conventional folk, or is he content to guard his new wisdom and share it only with his clan of true believers? After deep thought, he decides [[w:Hero's_journey#The_Magic_Flight|the world must know]]; they must be warned, but it will not be easy, [[w:Hero's_journey#Rescue_from_Without|he may need help]]. He begins to [[w:Hero's_journey#Master_of_the_Two_Worlds|build rapport with conventional folk]], even as he works to appease his clan concerned about his consorting with the enemy. Whether or not his new revelations are accepted by the mainstream, the hero is now [[w:Hero's_journey#Freedom_to_Live|free to live]] his life content with his new insights. He has transcended the limitations of conventional narratives and embraced a new perspective that allows him to see the world with greater clarity and insight. It is all great fun! ==== Assignment ==== # Recognize that the [[Evaluating Information/The Best Story Often Wins|best story often wins]], even if it is fanciful and clearly untrue. # Enjoy the [[w:Kayfabe|kayfabe]]—the portrayal of staged events as “real” or “true”—but don’t mistake the theatrics for reality. # Improve your skills at [[Evaluating Evidence|evaluating evidence]]. # [[Deductive Logic/Clear Thinking curriculum|Think clearly]] # [[Knowing How You Know|Know how you know]]. == Characteristics of Cognition == [[w:Cognition|Cognition]] is the "mental action or process of acquiring knowledge and understanding through thought, experience, and the senses". It encompasses all aspects of intellectual functions and processes such as: perception, attention, thought, imagination, intelligence, the formation of knowledge, memory and working memory, judgment and evaluation, reasoning and computation, problem-solving and decision-making, comprehension and production of language. Cognitive processes use existing knowledge to discover new knowledge. Humans are remarkable, complex, intricate systems, and our brains are not objective, mechanical tools for directing attention, making observations, perceiving our surroundings, recalling past experiences, and drawing sound conclusions. Certain traits of human cognition make us vulnerable to misconceptions. These are elaborated upon below. === Forming Beliefs === People form their beliefs through a complex interplay of personal experiences, cultural influences, evidence, and introspection, creating a rich diversity in perspectives. Beliefs often stem from deeply held values, such as widely recognized [[Moral Reasoning|moral standards]] or more unique principles that resonate on an individual level. These beliefs may originate in childhood, shaped by family, education, and society, but they evolve over time through self-reflection, exposure to new ideas, and life’s challenges. The alignment of beliefs with values, evidence, and actions is essential for authenticity, fostering a sense of purpose and coherence in life. Examining these foundations helps individuals understand themselves, adapt to new insights, and find meaning, contributing to a life that feels both consistent and fulfilling. [[File:Intellectual Virtues Overcome Vices.jpg|thumb|Intellectual Virtues Overcome Vices]] Several characteristics of human nature and the way we think make it easy to mistake false beliefs for [[Seeking True Beliefs|true beliefs]]. These include cognitive biases, confirmation bias, motivated reasoning, soldier mindset, social proof, illusory truth effect, overconfidence, the illusion of explanatory depth, the Dunning-Kruger effect, and other intellectual vices. The Wikiversity course on [[Forming beliefs|Forming Beliefs]] describes these characteristics in more depth. === Solution Aversion === [[q:Upton_Sinclair|Upton Sinclair]] famously observed “It is difficult to get a man to understand something, when his salary depends on his not understanding it.” This is an example of ''solution aversion''—the tendency to deny evidence if acknowledging it will lead to an unacceptable consequence. More formally, the solution aversion model predicts that certain solutions associated with problems (e.g., government, regulation) are more aversive and more threatening to individuals who hold an ideology that is incompatible with or even challenged by the solution, and this increases skepticism of the problems’ existence.<ref>[http://pscourses.ucsd.edu/ps100da/Campbell&#x20;Aaron&#x20;Kay&#x20;Solution&#x20;Aversion&#x20;On&#x20;the&#x20;Relation&#x20;Between&#x20;Ideology&#x20;and&#x20;Motivated&#x20;Disbelief.pdf Solution Aversion: On the Relation Between Ideology and Motivated Disbelief], Troy H. Campbell and Aaron C. Kay. Duke University. Journal of Personality and Social Psychology 2014, Vol. 107, No. 5, 809–824</ref> Examples include a tendency to deny climate change when the proposed solutions my require more regulations or result in fewer jobs. Another example is [[w:Alex_Jones#Gun_rights|skepticism of school shootings]] motivated by a fear that they will lead to increased gun control. === Assignment === # Study the Wikiversity course [[Forming beliefs]]. # Beware of the many [[Forming beliefs#Difficulties|difficulties people face]] when forming beliefs. # Avoid [[w:Brain_rot|brain rot]]. # Practice the [[Seeking True Beliefs#The Intellectual Virtues|intellectual virtues]] and [[Seeking True Beliefs|seek true beliefs]]. # Work to allow the [[Seeking True Beliefs#The Intellectual Virtues|intellectual virtues]] to prevail over the [[Seeking True Beliefs#Virtue Overcomes Vice|intellectual vices]]. == Personalities == [[File:An Architecture for Human Interaction.jpg|thumb|An Architecture for Human Interaction]] Humans live, behave, and interact at several layers, illustrated in the diagram on the right and described in the Wikiversity course on [[Layers of Human Interaction]]. Attempting to using reason to [[Influence and Persuasion|influence]] another’s beliefs, values, or choices is an appeal at the [[Layers of Human Interaction#Cognition|cognitive layer]] of human interaction. Using repetition and rewards to create or elicit [[w:Classical_conditioning|conditioned responses]] or to create or strengthen memories modifies [[Layers of Human Interaction#Cognition|learned responses]]. Underlying these levels are our ''[[w:Personality|personality traits]]'', the intrinsic differences that make us each the unique person we are. These stable characteristics remain primarily constant throughout our adult life. The [[w:Big_Five_personality_traits|five-factor model of personality]] identifies five factors and ten values characterizing personality types. These five factors and ten values (describing the extremes of each factor) are as follows: * [[w:Openness_to_experience|openness to experience]] (inventive/curious vs. consistent/cautious) * [[w:Conscientiousness|conscientiousness]] (efficient/organized vs. extravagant/careless) * [[w:Extraversion_and_introversion|extraversion]] (outgoing/energetic vs. solitary/reserved) * [[w:Agreeableness|agreeableness]] (friendly/compassionate vs. critical/judgmental) * [[w:Neuroticism|neuroticism]] (sensitive/nervous vs. resilient/confident) Research by [[w:Jonathan_Haidt|Johnathan Haidt]], reported in his book ''[[w:The_Righteous_Mind|The Righteous Mind]]'', and described in his [[w:Moral_foundations_theory|moral foundations theory]], has shown that people vary in their moral judgements and [[w:Ideology|political ideologies]] in predictable ways based on their personality types. For example, he has found that “Liberals score higher on measures of [[w:Neophile|neophilia]] (also known as “openness to experience”), not just for new foods but also for new people, music, and ideas. Conservatives are higher on neophobia; they prefer to stick with what’s tried and true, and they care a lot more about guarding borders, boundaries, and traditions.”<ref>Haidt, Jonathan. The Righteous Mind: Why Good People Are Divided by Politics and Religion . Knopf Doubleday Publishing Group. Kindle Edition. @ 2655 of 7971</ref> In short, holding liberal or conservative political views is largely a result of personality differences! Recognizing that personality traits are largely intrinsic traits determined by heredity, warns us that these views are very difficult to change. === Assignment: === # Read the essay [[/Personality Is Not Destiny/|Personality is not Destiny]]. # Regardless of your personality type, endeavor to approach new ideas and new experiences from an expectation of [[Fostering Curiosity|curiosity]] rather than fear. # If you encounter people who are not naturally open to experience, be patient with them as they encounter and struggle to become comfortable with new ideas or new experiences.   Chapters 7 and 8 of the book ''Misbelief''<ref>{{cite book|title=Misbelief: What Makes Rational People Believe Irrational Things|last=Ariely|first=Dan|date=September 17, 2024|publisher=Harper Perennial|isbn=978-0063280434|pages=320|author-link=w:Dan_Ariely}}</ref>, identify other individual differences that increase susceptibility to misbelief. These include susceptibility to: # [[w:False_memory|False recall]] and False recognition, # [[w:Magical_thinking|Magical thinking]], # Openness to absorbing<ref>{{Cite journal|last=Tellegen|first=A.|last2=Atkinson|first2=G.|date=1974-06|title=Openness to absorbing and self-altering experiences ("absorption"), a trait related to hypnotic susceptibility|url=https://pubmed.ncbi.nlm.nih.gov/4844914/|journal=Journal of Abnormal Psychology|volume=83|issue=3|pages=268–277|doi=10.1037/h0036681|issn=0021-843X|pmid=4844914}}</ref> and [[w:Hypnotic_susceptibility|Hypnotic susceptibility]], # Perceptual aberrations, # Patternicity (Apophenia), # Lack of intellectual humility, # Fixed mindset, # Narcissism, and # Decision making errors including the [[w:Conjunction_fallacy|conjunction fallacy]], [[w:Illusory_correlation|illusory correlation]], and [[w:Hindsight_bias|hindsight bias]]. Each of these characteristics are described in more detail below. '''False Recall and False Recognition:''' In [[w:Psychology|psychology]], a ''false memory'' is a phenomenon where someone [[w:Recall_(memory)|recalls]] something that did not actually happen or recalls it differently from the way it actually happened. [[w:Suggestibility|Suggestibility]], activation of associated information, the incorporation of misinformation, and [[w:Misattribution_of_memory|source misattribution]] have been suggested to be several mechanisms underlying a variety of types of false memory. Several natural factors vary the susceptibility of a person to form false memories. These include individual differences in creative imagination, and variations in perceived social pressure. Other factors include a history of [[w:False_memory#Trauma|trauma]] relevant to the false memory, [[w:False_memory#Sleep_deprivation|sleep deprivation]], and [[w:False_memory_syndrome|false memory syndrome]].   '''Magical Thinking''' [[File:Saint Pierre tentant de marcher sur les eaux by François Boucher.jpg|thumb|''[[w:Saint_Peter|Saint Peter]] Attempting to [[w:Jesus_walking_on_water|Walk on Water]]'' (1766), painting by [[w:François_Boucher|François Boucher]]]] [[w:Magical_thinking|Magical thinking]], or superstitious thinking, is the belief that unrelated events are causally connected despite the absence of any plausible causal link between them, particularly because of [[w:Supernatural|supernatural]] effects. Examples include the idea that personal thoughts can influence the external world without acting on them, or that objects must be causally connected if they resemble each other or have come into contact with each other in the past. Magical thinking is a type of fallacious thinking and is a common source of invalid causal inferences. Unlike the confusion of [[w:Correlation|correlation]] with causation, magical thinking does not require the events to be correlated. '''Hypnotic susceptibility and Fantasy-prone personality''' [[w:Hypnotic_susceptibility|Hypnotic susceptibility]] measures how easily a person can be hypnotized. Individuals highly susceptible to hypnosis tend to have distinctive characteristics outside of hypnosis. In 1981, Sherl Wilson and T X Barber reported<ref>{{Cite book|url=https://link.springer.com/chapter/10.1007/978-1-4684-3974-8_10|title=Vivid Fantasy and Hallucinatory Abilities in the Life Histories of Excellent Hypnotic Subjects (“Somnambules”): Preliminary Report with Female Subjects|last=Wilson|first=Sheryl C.|last2=Barber|first2=Theodore X.|date=1981|publisher=Springer US|isbn=978-1-4684-3974-8|editor-last=Klinger|editor-first=Eric|location=Boston, MA|pages=133–149|language=en|doi=10.1007/978-1-4684-3974-8_10}}</ref> that most of a group of individuals extremely susceptible to hypnosis who they termed "[[w:Fantasy_prone_personality|fantasizers]]" exhibited a cluster of traits consisting of: 1) fantasizing much of the time, 2) reporting their imagery was as vivid as real perceptions, 3) having physical responses to their imagery, 4) having an earlier than average age for first childhood memory, 5) recalling "imaginary playmates" from childhood, and 6) having grown up with parents who encouraged imaginative play. '''Perceptual Aberrations''' Perceptual aberrations encompass a range of unusual sensory experiences, including feeling that body parts are changing shape, merging with external objects, or a distorted sense of body ownership. These experiences are commonly reported in individuals with psychotic disorders, especially schizophrenia. "Perceptual aberrations, including the sensation that one's organs are rotting, feeling that the body is unreal or that the shape and size of body parts are changing or merging with external objects, and altered sense of bodily ownership are common in psychosis"<ref>[https://cdn.vanderbilt.edu/vu-my/wp-content/uploads/sites/1216/2019/04/14113938/Brosey_Schiz-Res-2017_Neuroanatomical-correlates-of-perceptual-aberrations-in-psychosis.pdf Neuroanatomical correlates of perceptual aberrations in psychosis], Erin A. Brosey, Neil D. Woodward. Schizophrenia </ref> '''Patternicity (Apophenia)''' [[w:Christopher_Hitchens|Christopher Hitchens]] remarked: “Human beings are pattern-seeking animals who will prefer even a bad theory or a conspiracy theory to no theory at all.”<ref>{{Cite web|url=https://quotlr.com/author/christopher-hitchens|title=110+ Christopher Hitchens Quotes about religion, god, death - QUOTLR|date=2024-06-02|website=Quotlr - Famous Motivational Quotes|language=en|access-date=2025-03-05}}</ref> [[File:Martian face viking cropped.jpg|thumb|Satellite photograph of a [[w:Mesa|mesa]] in the [[w:Cydonia_(Mars)|Cydonia region of Mars]], often called the[[w:Cydonia_(Mars)#"Face_on_Mars"|"Face on Mars"]]]] [[w:Apophenia|Apophenia]] is the tendency to perceive meaningful connections between unrelated things. The term was coined by psychiatrist [[w:Klaus_Conrad|Klaus Conrad]] in his 1958 publication on the beginning stages of schizophrenia. He defined it as "unmotivated seeing of connections [accompanied by] a specific feeling of abnormal meaningfulness". Apophenia has also come to describe a human propensity to unreasonably seek definite patterns in random information, such as can occur in gambling or seeing meaningful images in random photographs. '''Lack of intellectual humility''' [[w:Intellectual_humility|Intellectual humility]] is characterized by recognizing the limits of one's knowledge and acknowledging one's fallibility. It involves several components, including not thinking too highly of oneself, refraining from believing one's own views are superior to others', lacking intellectual vanity, being open to new ideas, and acknowledging mistakes and shortcomings. It is positively associated with openness to new ideas, empathy, prosocial values, tolerance for diverse perspectives, and scrutiny of misinformation. Individuals with higher levels of intellectual humility experience benefits such as improved decision-making, positive social interactions, and the moderation of conflicts. A person who lacks intellectual humility may think too highly of themselves, think their beliefs or attitudes are more likely to be correct than those of others, boast or brag about their intellectual accomplishments, become defensive when challenged, overreact to criticism, refuse to acknowledge their mistakes, lack [[Fostering Curiosity|curiosity]], and [[Seeking True Beliefs#Rigidity|hold rigidly to their beliefs]], despite contrary evidence. === Assignment === Work to increase your [[w:Intellectual_humility|intellectual humility]]. # Become more aware of your own doubts or uncertainties. # Become comfortable acknowledging your own uncertainty using phrases such as: “I’m not sure about that”, or “Let’s check up on that”, or “I’d like to find out more about this before making a firm decision. # Study the [[Seeking True Beliefs#Humility|Humility]] section of the Wikiversity course [[Seeking True Beliefs]]. # Work to adopt a [[w:Mindset#Fixed_and_growth_mindsets|growth mindset]], described in the next section. # Consider evaluating yourself using the [https://seaver.pepperdine.edu/social-science/content/comprehensive-intellectual-humility.pdf Comprehensive Intellectual Humility Scale].<ref>Krumrei-Mancuso, E. J., & Rouse, S. V. (2016). The development and validation of the [https://seaver.pepperdine.edu/social-science/content/comprehensive-intellectual-humility.pdf Comprehensive Intellectual Humility Scale]. Journal of Personality Assessment, 98, 209- 221. doi:10.1080/00223891.2015.1068174</ref> '''Fixed mindset''' People with a [[w:Mindset#Fixed_and_growth_mindsets|fixed mindset]] believe that "intelligence is static", and little can be done to improve ability. Feedback is seen as "evaluation of their underlying ability" and success is seen because of this ability, not any effort expended. Failure is intimidating, since it "suggests constraints or limits they would not be able to overcome". Those with a fixed mindset tend to avoid challenges, give up easily, and focus on the outcome. They believe that their abilities are fixed, and effort has little value. A fixed mindset is similar to the [[w:The_Scout_Mindset|Soldier Mindset]], described by [[w:Julia_Galef|Julia Galef]]. === Assignment: === # Work to adopt a [[Finding Common Ground#The Scout Mindset|scout mindset]]. # Study the [[Sustaining Agency#Internal vs. External Locus of Control|Internal vs. External Locus of Control]] section of the Wikiversity course on [[Sustaining Agency]]. # Read this essay on [[/Toward a Growth Mindset/|transitioning toward a growth mindset]]. # Take steps toward a attaining growth mindset. '''Narcissism''' [[w:Narcissism|Narcissism]] is excessive preoccupation with one's self, one's own worth, and one's own needs. It is typically associated with behaviors of self-elevation over others, entitlement, and delusional grandiosity. == Social Elements == People are inherently [[w:Social|social]]. Each of us has a need to [[w:Belongingness|belong]], and we seek to avoid the pain of [[w:Social_isolation|social isolation]] and [[w:Ostracism|ostracism]]. When misbelievers discover that others share a particular misbelief, they may find [[w:Acceptance|acceptance]], [[w:Normative_social_influence|validation]], [[w:Social_status|social status]], and comfort within that group. Finally, I have found my people! These people believe me! I am attracted to these people! I belong with these people! I am comfortable being with my people! We have been right all along! We alone have discovered the truth and are no longer the naive suckers being duped. It all [[w:Sensemaking|makes sense]] now. === Acceptance Displaces Ostracism === While [[w:Ostracism|ostracism]], [[w:Social_rejection|social rejection]], and [[w:Shunned|shunning]] are painful, acceptance is comfortable. The social psychologist [[w:Kipling_Williams|Kipling Williams]] defines ostracism as “any act or acts of ignoring and excluding an individual or group by an individual or a group, without necessarily involving acts of verbal or physical abuse.” Williams proposes that the most prevalent form of ostracism is the [[w:Silent_treatment|silent treatment]], where effectively refusing to communicate with a person effectively ignores and excludes them. Williams and his colleagues have charted responses to ostracism in some five thousand cases and found two distinctive patterns of response. The first is increased [[w:Conformity|group-conformity]], in a quest for re-admittance; the second is to become more provocative and hostile to the group, [[w:Conformity|seeking attention]] rather than acceptance. Experiments with a simple [[w:Social_rejection#Ball_toss_/_cyberball_experiments|cyberball computer game]] have found that even a simple and brief period of ostracism can produce significant increases in self-reported levels of anger and sadness. Social rejection significantly impacts both [[w:Social_rejection#Health_effects|emotional and physical health]], leading to increased anxiety, depression, and negative emotions.  It also weakens the immune system, increases blood pressure, and heightens the risk of various illnesses, including HIV and tuberculosis.  Furthermore, rejection can cause long-term psychological and physical harm, especially in cases of family estrangement or chronic social isolation. It is not surprising that someone who is being rejected by mainstream believers will seek acceptance among like-minded misbelievers. ==== Assignment ==== # If someone you know is feeling isolated, help them to connect. #* The Wikiversity course on [[Alleviating Loneliness]] may be helpful. #* This collection of [[:Category:Community|community-related]] resources may be helpful. #* [[Being Friends|Become their friend]].   # Avoid alienating misbelievers you encounter. Remain friendly and continue to establish [[w:Rapport|rapport]]. Do not exclude or ostracize misbelievers, even subtly. # The methods of [[w:Deep_canvassing|deep canvasing]] and [[Street Epistemology|street epistemology]] can be helpful in [[Finding Common Ground|finding common ground]]. # Consider participating in a program offered by the [[w:Braver_Angels|Braver Angels organization]]. # Complete the Wikiversity course on [[Finding Common Ground]]. #* Find common ground. === Social Proof === Often people will just go along to get along. This is an example of [[w:Social_proof|social proof]]. [[w:Social_proof|Social proof]], or informational social influence, is a psychological phenomenon where individuals copy the actions of others to determine appropriate behavior in each situation. In ambiguous social situations, people rely on the assumption that others possess more knowledge about the current situation. This influence leads to the tendency of large groups to conform, a phenomenon sometimes referred to as [[w:Herd_behavior|herd behavior]]. While social proof reflects a rational desire to consider others’ information, formal analysis reveals that it can cause individuals to converge too quickly on a single choice, potentially resulting in decisions based on limited information. [[w:Information_cascade|Information cascades]] are an example of this behavior. The effects of [[w:Social_influence|social influence]] can be seen in the tendency of large groups to conform. When individuals are unsure of the correct behavior, they often seek guidance from others. Informational social influence occurs when people conform because they believe others’ interpretations of ambiguous situations are more accurate and will help them make appropriate choices. This contrasts with normative social influence, where individuals conform to be liked or accepted by others. Social proof often leads to both public compliance (conforming to the behavior of others without necessarily believing it is correct) and private acceptance (conforming based on a genuine belief that others are correct). ==== Assignment ==== # Complete the Wikiversity course [[Navigating Social Proof]]. # Decide for yourself when to go along to get along, or to think for yourself and see what happens. === Cognitive Dissonance === A tension, called [[w:Cognitive_dissonance|cognitive dissonance]], arises when actions are inconsistent with beliefs. A simple example is when a health-conscious person continues to smoke cigarettes, despite knowing the dangers. This tension can be resolved in one of several ways: # Stop smoking, # Declare, to yourself and others, that smoking is safe, or # Endure the tension. This tension can arise when a person holding some misbelief is confronted with evidence that challenges or falsifies their deeply held misbelief. Let’s take an example of a person who strongly believes they have experienced an [[w:Alien_abduction|alien abduction]]. As contrary evidence arises—say for example they were seen and photographed in public at some time during the alleged abduction—the tension increases. The tension can be resolved by dismissing the evidence and doubling down on the misbelief, or by abandoning the misbelief. If the misbelief is weakly held, then it is easiest to abandon that misbelief, admit you were mistaken, and move past the incident. However, if the belief is rigidly held, if it became part of your identity, if you have achieved some level of acceptance or [[w:Social_status|social status]] because of your resolve and commitment to this belief, it is likely you will increase your commitment to the misbelief and take increasingly extreme actions to defend the misbelief. Also, a gap can arise between your ''held beliefs''—what you authentically and privately believe—and your ''professed beliefs''—what you tell others you believe. Maintaining this gap this introduces an additional tension. [[w:Charlatan|Charlatans]] profess beliefs they do not hold. They are not experiencing cognitive dissonance; they are simply frauds. The three conditions—what is true, what I believe, and what I profess to believe—can exist in various relationships. To enhance the alignment in your life, strive to ensure that all three conditions are consistent. ==== Assignment ==== # Complete the Wikiversity course on [[Resolving Cognitive Dissonance]]. # Identify any cognitive dissonance you may be experiencing. # Work to align your behavior (including professed beliefs) with [[Seeking True Beliefs|well-chosen beliefs]]. === Celebrity Charlatans === Many [[w:Charlatans|charlatans]] can increase their social status, at least temporality by publicly advocating for some misbelief. Read this [[/Celebrity Charlatans/|list of celebrity charlatans]] who have who gained fame, notoriety, or celebrity status through deception, exaggerated claims, or fraudulent behavior, at least temporarily, by professing to hold misbeliefs. Also read this list of[[/Prominent Conspiracy Theorists/| Prominent Conspiracy Theorists]]. While the various misbelievers you are likely to encounter may not rise to the level of prominence of those in the lists above, many attain some level of social standing because of their (professed) misbeliefs. For the everyday misbeliever, comradery, [[w:Friending_and_following|followers]], and [[w:Like_button|likes]] are powerful rewards. The pattern is clear: # Become intrigued with some misbelief. # Embrace the narrative. Add details. Embellish. Evangelize. # Gain some level of acceptance from fellow misbelievers. # Gain some level of social status for promoting this misbelief. # Maintaining this level of status depends on sustaining, defending, and often escalating the misbelief. # It becomes difficult to unwind this process, even if you begin to doubt the misbelief. ==== Assignment ==== # Beware of misbeliefs and those who advance them. # [[Knowing How You Know|Know how you know]]. # Be aware of the importance of [[w:Skepticism|skepticism]], [[w:Critical_thinking|critical thinking]], [[Clear Thinking/Curriculum|clear thinking]], and the need to verify [[w:Extraordinary_claims_require_extraordinary_evidence|extraordinary claims]]. === Tribes and Loyalty === A [[w:Shibboleth|shibboleth]] is any custom or tradition, usually a choice of phrasing or single word, that distinguishes one group of people from another. Shibboleths have been used throughout history in many societies as passwords, ways of self-identification, signals of loyalty and affinity, ways of maintaining traditional segregation, or protection from real or perceived threats. Sharing a particular misbelief can signal membership of an exclusive group of people who share a certain set of beliefs. Holding the misbelief is a shibboleth that quickly and reliably identifies members of the in-group and excludes members of the out group.   To begin to understand [[w:Collective_identity|group identity]] and [[w:Loyalty|loyalty]], consider the following passage on [[w:Zhao_Gao#Calling_a_deer_a_horse|calling a deer a horse]]: <blockquote>Zhao Gao was contemplating treason but was afraid the other officials would not heed his commands, so he decided to test them first. He brought a deer and presented it to the Second Emperor but called it a horse. The Second Emperor laughed and said, "Is the chancellor perhaps mistaken, calling a deer a horse?" Then the emperor questioned those around him. Some remained silent, while some, hoping to ingratiate themselves with Zhao Gao, said it was a horse, and others said it was a deer. Zhao Gao secretly arranged for all those who said it was a deer to be brought before the law and had them executed instantly. Thereafter the officials were all terrified of Zhao Gao. Zhao Gao gained military power as a result of that. </blockquote>Professed belief in a specific unlikely claim demonstrates loyalty to the group and perhaps its leader. The more preposterous the professed misbelief, the more costly the loyalty test and the stronger the evidence of loyalty. Misbelievers of the same stripe form a closely knit [[w:Tribe|tribe]], bound by their willingness to demonstrate their loyalty by using the [[w:Handicap_principle|costly signal]] of professing a specific misbelief. == Media Landscape == Most alternative social media news consumers feel a sense of community on these sites, which prominently identify themselves as havens of free speech.<ref>{{Cite web|url=https://www.pewresearch.org/journalism/2022/10/06/the-role-of-alternative-social-media-in-the-news-and-information-environment/|title=The Role of Alternative Social Media in the News and Information Environment|last=Aubin|first=Galen Stocking, Amy Mitchell, Katerina Eva Matsa, Regina Widjaya, Mark Jurkowitz, Shreenita Ghosh, Aaron Smith, Sarah Naseer and Christopher St|date=2022-10-06|website=Pew Research Center|language=en-US|access-date=2025-03-05}}</ref> More information is available today than at any other time in history, yet our capacity to direct attention, avoid distractions, absorb information, evaluate credibility, and learn remains largely unchanged from thousands of years ago. Learning to navigate today’s vast information landscapes can help you take charge of how you spend your time, direct your attention, and form beliefs. In today’s fast-paced world, attention might be the most precious resource. Despite the abundance of information, it is attention that’s scarce, making it highly valuable. As you navigate through information landscapes, make sure to direct your attention wisely. === Assignment === # Study the Wikiversity course [[Navigating Information Landscapes]]. #* Navigate information landscapes wisely. # Study the Wikiversity course [[Navigating Social Proof]]. #* Navigate social proof skillfully. # Study the Wikiversity course [[Evaluating Information]]. #* Evaluate information skillfully. == Avoiding the Pitfalls == Misbeliefs, like a candle flame, can draw us in with their glow, promising hidden knowledge, alternative explanations, or thrilling narratives. But we must learn to [[Understanding Misbelief/Look but Don’t Touch|look but not touch]]. [[File:White candle seamless loop.gif|thumb| Enjoy the burning candle without touching it.]] Assignments throughout this course have offered guidance in avoiding the many pitfalls of misbeliefs. Work on those assignments that seem most relevant and valuable to you. In general misbeliefs can be avoided by: # Deciding to [[Understanding Misbelief/Look but Don’t Touch|look but not touch]]. Explore misbeliefs without committing to them. # Strengthening your [[Understanding Misbelief#Motivations|motivation]]<nowiki/>s toward true beliefs. # Pursuing the [[w:Inner_Development_Goals|Inner development goals]]. # Inviting and exploring differing viewpoints: #* Engage in [[Socratic Methods|Socratic Dialogue]] with someone who holds some belief different from yours. #* Participate in [[w:Braver_Angels|Braver Angels]] programs. # [[Seeking True Beliefs|Seeking true beliefs]], and # Working to [[Exploring Worldviews/Aligning worldviews|align your worldview with realty]]. == Applying the Intellectual Virtues == An important objective of this course is to help students to use the [[Seeking True Beliefs#The Intellectual Virtues|intellectual virtues]] to overcome [[Seeking True Beliefs#Virtue Overcomes Vice|intellectual vices]]. As a result students can transition from [[Seeking True Beliefs#Rigidity|rigidly held beliefs]] to more carefully considered [[Seeking True Beliefs#Firmness|firmly held beliefs]]. === Assignment === # Read this [[/Are You Still Certain?/|list of controversial statements]]. (Caution, these statements are deliberately chosen to be provocative.) # Identify any statements that are particularly sensitive or emotionally charged for you. These might be statements you’ve held ridigly for a long time and have strong emotional attachments to. You might even enjoy discussing them or using them to connect with people in your in-group. These statements could also be significant aspects of your political, cultural, or religious identity. # Reflect on how you came to hold that belief and what you have at stake. Why is this important to you? Is this belief motivated by fear or hate? # Carefully and skillfully examine such statements. Identify hidden assumptions and instances where facts are intertwined with opinions or controversies. Determine any motivating truths behind the claim, along with inaccurate, misleading, and untrue elements. Analyze any simplistic reductions of complex issues, [[Recognizing Fallacies|logical fallacies]], or inaccuracies. # Apply the [[Seeking True Beliefs#The Intellectual Virtues|intellectual virtues]] as skillfully as you can to overcome any lingering [[Seeking True Beliefs#Virtue Overcomes Vice|intellectual vices]]. # If you can find someone with differing views on a topic, engage them in a skillful [[Practicing Dialogue|dialogue]] if they are willing and able to do so. Consider using [[Socratic Methods|Socratic methods]] or even [[Street Epistemology|street epistemology]]. Throughout this process, strengthen your relationship with your dialogue partner and work to [[Finding Common Ground|find common ground]]. # Work to transition from your [[Seeking True Beliefs#Rigidity|rigidly held beliefs]] to a more carefully considered [[Seeking True Beliefs#Firmness|firmly held belief]] on the topic. == Five techniques of denial (FLICC): == Researcher [[w:Skeptical_Science|John Cook]] has identified five techniques of denial that are used to propagate misinformation<ref>{{Cite web|url=https://www.thegreatsimplification.com/episode/212-john-cook|title=Why Science Communication Fails: How to Break Down Misleading Arguments and Inoculate Against Misinformation|website=The Great Simplification|language=en-US|access-date=2026-02-27}}</ref>. He uses the acronym FLICC to help people remember these five techniques: * Fake Experts—Using testimonials from non-experts willing to falsely testify in support of the misinformation. * [[Recognizing Fallacies|Logical Fallacies]]—Using flawed logic to draw a conclusion that is unsupported by the premise.   * Impossible Expectations—Demanding unrealistic levels of proof from science. * [[w:Cherry_picking|Cherry-Picking]]—Citing individual cases or data that seem to confirm a particular position while ignoring a significant portion of related and similar cases or data that may contradict that position. * [[w:Conspiracy_theory|Conspiracy Theories]]—Providing an explanation for an event or situation that asserts the existence of a conspiracy (generally by powerful sinister groups, often political in motivation), when other explanations are more probable. Learn to identify these deceptive techniques and avoid being persuaded by them. == Conversion == Abandoning a deeply held belief is difficult, however it can be done. === Assignment === # Form a list of beliefs you hold firmly that you suspect may be misbeliefs. You may be able to identify these candidate misbeliefs because other thoughtful people differ in this belief. # Examine why you hold that belief. How do you benefit from holding that belief? How might you benefit from abandoning that belief? # Use techniques described throughout this course to challenge and perhaps alter that belief. # Read this essay on [[/Lessons from Conversion Stories/|Lessons from Conversion Stories]]. # Seek inspiration from one or more of these stories. # Decide to convert to a true belief. # Persist in adopting that new true belief. == Summary and Conclusions == Misbeliefs are a widespread and persistent challenge, affecting individuals and societies alike.^[35] While we often assume we are skilled at distinguishing fact from fiction, research and experience show that we frequently overestimate our ability to judge truth accurately. The human mind is vulnerable to errors in reasoning, social influences, and emotional biases, all of which contribute to the formation and spread of false beliefs. Because of this, misbeliefs are not just occasional errors but deeply ingrained tendencies that shape how we perceive and interpret the world. A variety of factors drive people toward misbeliefs rather than true beliefs. Psychological needs, cognitive shortcuts, and social pressures all contribute to the appeal of certain misbeliefs. In times of stress or uncertainty, people may cling to comforting but false explanations rather than confront complex or unsettling truths. Additionally, personality traits, such as openness to new experiences or susceptibility to authority, influence how likely someone is to embrace and spread misbeliefs. The modern media landscape further accelerates the spread of misinformation, making it easier for unverified or deceptive claims to reach vast audiences quickly. Recognizing the conditions that allow misbeliefs to thrive is a crucial step toward reducing their influence. By understanding the cognitive biases that lead to errors in judgment, the emotional and social rewards that reinforce misbeliefs, and the systemic issues that facilitate their spread, we can take active steps to resist misinformation. Education, critical thinking, and media literacy play essential roles in helping individuals and communities evaluate claims more rigorously and choose true beliefs over false ones. Ultimately, addressing misbeliefs is not just an intellectual challenge but a societal responsibility. Encouraging open-minded skepticism, promoting respectful dialogue, and fostering environments where truth-seeking is valued over ideological certainty can help mitigate the harms caused by misbeliefs. While false beliefs may always be part of human nature, understanding their origins and persistence empowers us to navigate the modern information landscape more wisely, making choices that align with truth rather than illusion. == Assignment == # [[Seeking True Beliefs|Seek true beliefs]]. # [[Living Wisely/Seeking Real Good|Seek real good]]. # [[Living Wisely|Live wisely]]! == Recommended Reading == Students who are interested in learning more about misbeliefs may wish to read these books: *{{cite book |last=Andersen |first=Kurt |date=September 5, 2017 |title=Fantasyland: How America Went Haywire: A 500-Year History |publisher=Random House |pages=480 |isbn=978-1400067213 |author-link=w:Kurt_Andersen }} *{{cite book |last=Andrews |first=Seth |author-linkw"Seth_Andrews= |date=December 4, 2012 |title=Deconverted: A Journey from Religion to Reason |publisher=Outskirts Press |pages=204 |isbn=978-1478716563}} *{{cite book |last=Aral |first=Sinan |author-link= |date=September 14, 2021 |title=The Hype Machine: How Social Media Disrupts Our Elections, Our Economy, and Our Health--and How We Must Adapt |publisher=Crown |pages=416 |isbn=978-0593240403}} *{{cite book |last=Ariely |first=Dan |author-link=w:Dan_Ariely |date=September 17, 2024 |title=Misbelief: What Makes Rational People Believe Irrational Things |publisher=Harper Perennial |pages=320 |isbn=978-0063280434}} *{{cite book |last=Berger |first=Jonah |author-link= |date=February 1, 2022 |title=The Catalyst: How to Change Anyone's Mind |publisher=Simon & Schuster |pages=288 |isbn=978-1982108649}} *{{cite book |last=Brown |first=Brené |author-link=w:Brené_Brown |date=March 1, 2022 |title=The Gifts of Imperfection |publisher=Hazelden Publishing |pages=208 |isbn=978-1616499600}} *{{cite book |last=Bruni |first=Frank |author-link=w:Frank_Bruni |date=April 30, 2024 |title=The Age of Grievance |publisher=Avid Reader Press |pages=288 |isbn=978-1668016435}} *{{cite book |last=Burton |first=Robert |date= |title=On Being Certain Paperback |publisher=Griffin |pages=272 |isbn=978-0312541521}} *{{cite book |last=Campbell |first=Joseph |author-link=w:Joseph_Campbell |date=June 1, 1991 |title=The Power of Myth |publisher=Anchor |pages=293 |isbn=978-0385418867}} *{{cite book |last=Cialdini|first=Robert B.|title=Influence: the Psychology of Persuasion|date=20|publisher=Collins|isbn=978-0-06-124189-5|edition=Rev. ed., [Nachdr.]|location=New York, NY}} *{{cite book |last=Frankfurt |first=Harry G. |date=January 30, 2005 |title=[[w:On_Bullshit|On Bullshit]] |publisher=Princeton University Press |pages=67 |isbn=978-0691122946 |author-link=w:Harry_Frankfurt }} *{{cite book |last=Frankfurt |first=Harry G. |date=October 31, 2006 |title=[[w:On_Truth|On Truth]] |publisher=Knopf |pages=112 |isbn=978-0307264220 |author-link=w:Harry_Frankfurt }} *{{cite book |last=Galef |first=Julia |author-link=w:Julia_Galef |date=April 13, 2021|title=The Scout Mindset: Why Some People See Things Clearly and Others Don't |publisher=Piatkus |pages= |isbn= 978-0349427645}} *{{cite book |last=Gray |first=Dave |author-link= |date=September 14, 2016 |title=Liminal Thinking: Create the Change You Want by Changing the Way You Think |publisher=Two Waves Books |pages=184 |isbn=978-1933820460}} *{{cite book |last=Haidt |first=Jonathan |date=February 12, 2013 |title=[[w:The_Righteous_Mind|The Righteous Mind: Why Good People Are Divided by Politics and Religion]] |publisher=Vintage |pages=528 |isbn=978-0307455772 |author-link=w:Jonathan_Haidt }} *{{cite book |last=Heaphy |first=Timothy J. |title=Harbingers: What January 6 and Charlottesville Reveal About Rising Threats to American Democracy |publisher=Steerforth |pages=288 |isbn=978-1586424015}} *{{cite book |last=Hecht |first=Jennifer Michael |author-link=w:Jennifer_Michael_Hecht |date=September 7, 2004 |title=Doubt: A History: The Great Doubters and Their Legacy of Innovation from Socrates and Jesus to Thomas Jefferson and Emily Dickinson |publisher=HarperOne |pages=576 |isbn=978-0060097950}} *{{cite book |last=Heflinger |first=Earl |author-link= |date=March 14, 2018 |title=Off the Hook: Escaping Toxic Ideology |publisher=BalboaPress |pages=328 |isbn=978-1504399913}} *{{cite book |last=Holmes |first=Jamie |author-link=w:Jamie_Holmes_(author) |date=October 11, 2016 |title=Nonsense: The Power of Not Knowing Paperback |publisher=Crown |pages=336 |isbn=978-0385348393}} *{{cite book |last=Johnson |first=Spencer |author-link=w:Spencer_Johnson_(writer) |date=Paperback – January 1, 1999 |title=Who Moved My Cheese? |publisher=Vermilion |pages= |isbn=978-0091876043}} *{{cite book |last=Kahneman |first=Daniel |date=April 2, 2013 |title=[[w:Thinking,_Fast_and_Slow|Thinking, Fast and Slow]] |publisher=Farrar, Straus and Giroux |pages=499 |isbn=978-0374533557 |author-link=w:Daniel_Kahneman }} *{{cite book |last=Kashdan |first=Todd |author-link=w:Todd_Kashdan |date=April 21, 2009 |title=Curious?: Discover the Missing Ingredient to a Fulfilling Life |publisher=William Morrow |pages=352 |isbn=978-0061661181}} *{{cite book |last=Mackay |first=Charles |date=November 1, 2016 |title=[[w:Extraordinary Popular Delusions and The Madness of Crowds|Extraordinary Popular Delusions and The Madness of Crowds]] |publisher=CreateSpace |pages=386 |isbn=978-1539849582 |author-link=w:Charles_Mackay_(author) }} *{{cite book |last=McIntyre |first=Lee |date=February 16, 2018 |title=Post-Truth |publisher=The MIT Press |pages=240 |isbn=978-0262535045 }} *{{cite book |last=Muster |first=Nori |author-link= |date=February 20, 2017 |title=Cult Survivors Handbook: Seven Paths to an Authentic Life |publisher=Independently published |pages=83 |isbn=978-1520661025}} *{{cite book |last=Peterson|first=Christopher|last2=Maier|first2=Steven F.|last3=Seligman|first3=Martin E. P.|title=Learned helplessness: a theory for the age of personal control|date=1995|publisher=Oxford Univ. Press|isbn=978-0-19-504467-6|edition=1st issued as a paperback|location=New York, NY}} *{{cite book |last=Pinker |first= Steven |author-link=w:Steven_Pinker|date= September 28, 2021 |title=[[w:Rationality_(book)| Rationality: What It Is, Why It Seems Scarce, Why It Matters]]| publisher= Viking |pages=432 |isbn= 978-0525561996 }} *{{cite book |last=Rosling |first=Hans |date=April 3, 2018 |title=Factfulness: Ten Reasons We're Wrong About the World--and Why Things Are Better Than You Think |publisher=Flatiron Books |pages=341 |isbn=978-1-250-10781-7 |author-link=w:Hans_Rosling }} *{{cite book |last=Schulz |first=Kathryn |author-link=w:Kathryn_Schulz |date=June 8, 2010 |title=Being Wrong: Adventures in the Margin of Error |publisher=Ecco |pages=416 |isbn=0061176044}} *{{cite book |last=Shermer |first=Michael |author-link=w:Michael_Shermer |date=August 7, 2012 |title=The Believing Brain: From Ghosts and Gods to Politics and Conspiracies---How We Construct Beliefs and Reinforce Them as Truths |publisher=St. Martin's Griffin |pages=385 |isbn=978-1250008800}} *{{cite book |last=Singer |first=Margaret Thaler |author-link= |date=April 11, 2003 |title=Cults in Our Midst: The Continuing Fight Against Their Hidden Menace |publisher=Jossey-Bass |pages=400 |isbn=978-0787967413}} *{{cite book |last=Snyder|first=Timothy|author-link=w:Timothy_D._Snyder|date= September 17, 2024|title=On freedom|publisher=Crown|isbn=978-0-593-72872-7|edition=First edition|location=New York}} *{{cite book |last=Sunstein |first=Cass R. |date=December 23, 2014 |title=Wiser: Getting Beyond Groupthink to Make Groups Smarter |publisher=Harvard Business Review Press |pages=272 |isbn=978-1422122990 |author-link=w:Cass_Sunstein }} *{{cite book |last=Tavris |first=Carol |author-link=w:Carol_Tavris |date=August 4, 2020 |title=Mistakes Were Made (but Not by Me): Why We Justify Foolish Beliefs, Bad Decisions, and Hurtful Acts |publisher=Mariner |pages=464|isbn=978-0358329619}} *{{cite book |last=Tobias |first=Madeleine |author-link= |date=January 1, 1994 |title=Captive Hearts, Captive Minds : Freedom and Recovery from Cults and Other Abusive Relationships |publisher=Hunter House |pages=304 |isbn=978-0897931441}} *{{cite book |last1=Tsipursky |first1=Gleb |last2=Ward |first2=Tim |date=May 29, 2020 |title=Pro Truth: A Practical Plan for Putting Truth Back Into Politics |publisher=Changemakers Books |page=271 |isbn=978-1789043990}} *{{cite book |last=Wilczek |first=Frank |author-link=w:Frank_Wilczek |date=January 12, 2021 |title=Fundamentals: Ten Keys to Reality |publisher=Penguin Press |pages=272 |isbn=978-0735223790}} *{{cite book |last=Wilson |first=Edward Osborne |date=March 30, 1999 |title=Consilience: The Unity of Knowledge |publisher=Vintage |pages=384 |isbn=978-0679768678 |author-link=w:E._O._Wilson }} *{{cite book |last=Wolpert |first=Lewis |date=July 17, 2008 |title=Six Impossible Things Before Breakfast: The Evolutionary Origins of Belief |publisher=W. W. Norton & Company |pages=256 |isbn=978-0393332032 }} *{{cite book |last=Zmigrod |first=Leor |date=March 25, 2025 |title=The Ideological Brain: The Radical Science of Flexible Thinking |publisher=Henry Holt and Co. |pages=304 |isbn=978-1250344595}} *Behind the Curve, documentary film directed by Daniel J. Clark I have not yet read the following books, but they seem interesting and relevant. They are listed here to invite further research. *The Constitution of Knowledge: A Defense of Truth by Jonathan Rauch *Invisible Rulers: The People Who Turn Lies into Reality by Renee DiResta *How Minds Change: The Surprising Science of Belief, Opinion, and Persuasion, by David McRaney *Influence: Science and Practice, by Robert B. Cialdini *Conspiracy Theories and Other Dangerous Ideas, by Cass R. Sunstein *Conspiracy Theories (THINK) by Quassim Cassam *Fake News: Understanding Media and Misinformation in the Digital Age (Information Policy), by Melissa Zimdars (Editor), Kembrew Mcleod (Editor) *The doctor who fooled the world, by Brian Deer *Second Class: How the Elites Betrayed America's Working Men and Women by Batya Ungar-Sargon *Dream Hoarders: How the American Upper Middle Class Is Leaving Everyone Else in the Dust, Why That Is a Problem, and What to Do About It, by Richard V. Reeves. *Lies, Incorporated: The World of Post-Truth Politics == References == The resources sited here can be helpful in gaining a deeper understanding of misbelief. *[https://onlinelibrary.wiley.com/doi/pdf/10.1111/pops.12568?ref=quillette.com%2F1000 Understanding Conspiracy Theories], Karen M. Douglas, University of Kent *[http://www.ask-force.org/web/Discourse/Sunstein-Conspiracy-Theories-2009.pdf Symposium on Conspiracy Theories, Conspiracy Theories: Causes and Cures], Cass R. Sunstein, Law, Harvard University and Adrian Vermeule Law, Harvard University *[https://www.zora.uzh.ch/id/eprint/28933/11/evolution_of_misbelief.pdf The evolution of misbelief], by McKay, R T ; Dennett, D C *[https://link.springer.com/content/pdf/10.1007/s41060-022-00311-6.pdf The disaster of misinformation: a review of research in social media], by Sadiq Muhammed T, and Saji K. Mathew *[https://www.jmir.org/2021/1/E17187/ Prevalence of Health Misinformation on Social Media: Systematic Review], by Victor Suarez-Lledo, and Javier Alvarez-Galvez *[https://globaldevincubator.org/initiative/tango/ Global Development Incubator, Tango] *[https://www.sciencedirect.com/special-issue/103F0RDQHX3 Conspiracy Theories] *[https://mailchi.mp/yale/how-global-warming-beliefs-differ-by-education-levels-in-india?e=27d6126d54 Yale program on climate change communications]. *[https://www.youtube.com/watch?v=L-vB1HaBsog Things Hidden: The Life and Legacy of René Girard] *[https://www.academia.edu/71086639/Cultures_of_rejection_in_the_Covid_19_crisis Cultures of rejection in the Covid-19 crisis] [[Category:Life skills]] [[Category:Applied Wisdom]] [[Category:Philosophy]] [[Category:Clear Thinking]] [[Category:Reality]] [[Category:Courses]] {{CourseCat}} {{Clear Thinking}} 0z05cdsp9sb04jvxfgblwilwzkm17ce 0 321584 2819302 2818639 2026-07-24T18:02:52Z Howie2024 2995240 /* Subpages */ create header for Spectral Analysis 2819302 wikitext text/x-wiki {{Research project}} {{Original research}} {{To be peer reviewed}} {{subst:proofread}} == Research abstract == '''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values. The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics. At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions. PDT treats a probability measure as the primary mathematical object and investigates: * invariant identities induced by reweighting, * composition and iteration of dilations, * fixed points and near-fixed behavior, * whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations). PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions. == Overview == PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is: * mathematically well-defined (positivity and normalization), * composable under iteration, * analyzable for invariants and fixed points. === Conceptual interpretation === A simplified conceptual flow of the PDT framework is: <pre> Baseline probability measure P ↓ Positive dilation field D(x) ↓ Reweighted probability measure P~ ↓ Observable statistical changes </pre> Repeated dilation may qualitatively behave as: <pre> Broad initial distribution ↓ Localized reweighting ↓ Probability concentration ↓ Emergent multiscale structure </pre> Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field. Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as: <pre> P₀ ↓ D₁ P₁ ↓ D₂ P₂ ↓ D₃ P₃ ↓ ⋯ </pre> where each dilation field reweights the probability structure generated by the previous step. Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. = Mathematical framework = == Definitions and notation == Let <math>(\Omega,\Sigma)</math> be a measurable space. * <math>P</math> denotes a probability measure on <math>(\Omega,\Sigma)</math>. * If <math>P</math> has a density <math>p</math> with respect to a reference measure <math>\mu</math>, then <math>dP=p\,d\mu</math>. * <math>D:\Omega\to(0,\infty)</math> is a measurable '''dilation field''' (a positive weight function). * <math>Z(P,D)</math> is the normalization constant: .<math> Z(P,D)=\int_\Omega D\,dP </math> * For an observable <math>f:\Omega\to\mathbb{R}</math> integrable under the relevant measure, <math> \mathbb{E}_P[f] = \int_\Omega f\,dP </math>. == PDT transformation (probability reweighting) == Given <math>P</math> and <math>D</math> with <math>0<Z(P,D)<\infty</math>, define the '''PDT transform''' <math>\widetilde{P}=\mathrm{PDT}(P;D)</math> by: <math> \widetilde{P}(A) = \frac{ \int_A D\,dP }{ \int_\Omega D\,dP } \quad\text{for all }A\in\Sigma </math> If <math>dP=p\,d\mu</math>, then <math>d\widetilde{P}=\widetilde{p}\,d\mu</math>, where <math> \widetilde{p}(x) = \frac{D(x)\,p(x)}{Z} </math> and <math> Z = \int_\Omega D(x)\,p(x)\,d\mu </math> '''Interpretation:''' the dilation field <math>D</math> shifts probability mass toward regions where <math>D</math> is larger, while renormalization keeps total probability equal to 1. PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures. Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations. A familiar physical example of a strictly positive factor is the Lorentz factor: <math> \gamma(v) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} </math> for <math> |v|<c </math> Lorentz contraction for a rod of rest length <math>L_0</math> moving at speed <math>v</math> is: <math> L(v)=\frac{L_0}{\gamma(v)} </math> To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on <math>\gamma</math>. == Worked finite example == Consider a finite probability space: <math> \Omega=\{a,b,c\} </math> with baseline probabilities: <math> P(a)=0.2,\quad P(b)=0.3,\quad P(c)=0.5 </math> Define a positive dilation field: <math> D(a)=1,\quad D(b)=2,\quad D(c)=4 </math> The normalization constant is: <math> Z=\sum_x D(x)P(x) </math> giving: <math> Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8 </math> The PDT-transformed probabilities become: <math> \widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071 </math> <math> \widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214 </math> <math> \widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714 </math> This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization. == Composition of dilations == An important structural property of sequential PDT transformations is that compose multiplicatively. Suppose two positive dilation fields: <math> D_1(x)>0 </math> and <math> D_2(x)>0 </math> are applied successively to a baseline probability measure <math>P</math>. The first dilation produces: <math> \widetilde{P}_1(A) = \frac{\int_A D_1\,dP} {\int_\Omega D_1\,dP} </math> Applying the second dilation field to <math>\widetilde{P}_1</math> gives: <math> \widetilde{P}_2(A) = \frac{\int_A D_2\,d\widetilde{P}_1} {\int_\Omega D_2\,d\widetilde{P}_1} </math> Substituting the first transformation into the second yields: <math> \widetilde{P}_2(A) = \frac{ \int_A D_2D_1\,dP }{ \int_\Omega D_2D_1\,dP } </math> This shows that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application. == Fixed points and iterative dynamics == An important question in PDT concerns the long-term behavior of repeated PDT transformations. Given an initial probability measure: <math> P_0 </math> and a sequence of positive dilation fields: <math> D_1,D_2,D_3,\dots </math> successive PDT transformations generate a sequence of measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow P_3 \rightarrow \cdots </math> where each transformed measure is obtained by reweighting the previous one. A measure <math>P</math> is called a fixed point of a dilation field <math>D</math> if: <math> \widetilde{P}=P </math> under the PDT transformation. In the simplest case, this requires the dilation field to be constant almost everywhere with respect to <math>P</math>. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization. More generally, repeated compositions of nontrivial dilation fields may generate: * hierarchical probability structure; * multiscale statistical behavior; * attractor-like distributions; * approximately stable transformed measures. These questions connect PDT to broader areas of: * dynamical systems; * stochastic processes; * iterative renormalization methods; * probabilistic geometry. At present these iterative properties remain largely unexplored within the PDT framework. == Entropy and iterative probability flow == Repeated PDT transformations may alter the entropy structure of a probability measure. For a discrete probability distribution: <math> P=\{p_i\} </math> the Shannon entropy is: <math> H(P) = -\sum_i p_i \log p_i </math> Under iterative PDT transformation, successive transformed measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow \cdots </math> may exhibit changing entropy behavior depending on the structure of the dilation fields. For example: * strongly localized dilation fields may concentrate probability mass and reduce entropy; * broader or smoothing dilation fields may distribute probability more evenly and increase entropy; * iterative compositions may generate approximately stable entropy profiles. These questions connect PDT to: * information theory, * statistical mechanics, * stochastic dynamics, * and renormalization-style iterative systems. At present the entropy behavior of iterative PDT transformations remains an open area for investigation. == Toy experiment: entropy under repeated dilation == A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution. Let the initial probability distribution be: <math> P_0=(0.2,0.2,0.2,0.2,0.2) </math> and define a positive dilation field: <math> D=(1,1,2,4,8) </math> At each step, apply the PDT update: <math> P_{n+1}(i) = \frac{D(i)P_n(i)} {\sum_j D(j)P_n(j)} </math> The Shannon entropy is: <math> H(P_n) = -\sum_i P_n(i)\log P_n(i) </math> In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately: <math> H(P_0)\approx1.6094 </math> to: <math> H(P_{10})\approx0.00775 </math> The final distribution is approximately: <math> P_{10} \approx (0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437) </math> This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior. == Mathematical context == PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature. In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis. The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations. === Example entropy evolution === {| class="wikitable" ! Iteration !! Shannon entropy |- | 0 || 1.6094 |- | 1 || 1.2990 |- | 2 || 0.7790 |- | 3 || 0.4399 |- | 5 || 0.1500 |- | 10 || 0.0078 |} Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states. === Localized dilation fields === A useful class of PDT transformations is generated by localized positive dilation fields. Consider a one-dimensional finite configuration space with states indexed by: <math> x=0,1,2,\dots,N </math> and define a localized dilation field centered at <math>x_0</math>: <math> D(x) = \exp\!\left( \lambda \exp\!\left( -\frac{(x-x_0)^2}{2\sigma^2} \right) \right) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\sigma</math> controls the spatial width of the localized field. Narrow values of <math>\sigma</math> produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space. Under iterative PDT dynamics: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> the probability distribution may progressively concentrate near the center of the dilation field. === Example entropy evolution for localized fields === Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior: {| class="wikitable" ! Field width <math>\sigma</math> ! Final entropy after 10 iterations ! Maximum probability after 10 iterations |- | 1.5 || 0.0352 || 0.9950 |- | 3.0 || 0.8162 || 0.7141 |- | 6.0 || 1.5367 || 0.3595 |} [[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]] [[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]] These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction. == Comparative entropy-flow experiments == The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions. {| class="wikitable" |+ Comparative entropy-flow behavior under PDT field classes ! Field class ! Final entropy ! Entropy decrease ! Final max probability ! Qualitative behavior |- | Localized | 0.3104 | 3.4032 | 0.9275 | Strong probability concentration |- | Oscillatory | 1.5779 | 2.1357 | 0.3418 | Distributed oscillatory structure |- | Multi-peak | 0.2851 | 3.4284 | 0.9425 | Multiple concentration regions |- | Stochastic | 0.7744 | 2.9392 | 0.7413 | Fluctuating concentration behavior |} These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example. In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space. [[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]] The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Oscillatory dilation fields === Another useful class of PDT transformations is generated by oscillatory positive dilation fields. One example is: <math> D(x) = \exp(\lambda\sin(kx)) </math> where: * <math>\lambda>0</math> controls the strength of the oscillatory amplification; * <math>k</math> controls the spatial frequency of the oscillation. Because the exponential is always positive, the dilation field remains strictly positive for all states. Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space. Under repeated PDT transformation: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor. === Example oscillatory-field experiment === A finite-state experiment was performed using: * 41 discrete states; * an initially uniform probability distribution; * a positive oscillatory dilation field with three spatial oscillation cycles; * 10 successive PDT iterations. Representative entropy behavior was: {| class="wikitable" ! Iteration ! Shannon entropy |- | 0 || 3.7136 |- | 2 || 2.8699 |- | 5 || 2.3018 |- | 10 || 1.9335 |} Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space. After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state. This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures. The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Multi-peak localized dilation fields === A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space. One example is: <math> D(x) = \exp\!\left( \sum_k \lambda_k \exp\!\left( -\frac{(x-x_k)^2}{2\sigma_k^2} \right) \right) </math> where: * <math>x_k</math> are the locations of the dilation peaks; * <math>\lambda_k>0</math> control the amplification strength of each peak; * <math>\sigma_k</math> control the spatial width of each localized region. This construction generates a positive multimodal dilation landscape containing several competing amplification regions. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward multiple partially localized concentration regions. Unlike single localized dilation fields, multi-peak fields may generate: * competing attractor-like regions; * hierarchical probability concentration; * partially stabilized multimodal distributions; * multiscale probability structure. Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor: * dominance by a single peak; * coexistence of several concentration regions; * or slowly evolving metastable probability structures. === Conceptual interpretation === A qualitative iterative evolution may be visualized as: <pre> Broad initial distribution ↓ Multiple localized amplifications ↓ Competing concentration regions ↓ Emergent multimodal probability structure </pre> This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone. At present these behaviors remain exploratory computational observations within finite-state toy models. === Random and stochastic dilation fields === Another important class of PDT transformations arises when the dilation field itself varies stochastically. A simple stochastic dilation field may be written schematically as: <math> D_n(x) = \exp\!\left( \sigma \eta_n(x) \right) </math> where: * <math>\eta_n(x)</math> is a random field or stochastic fluctuation at iteration <math>n</math>; * <math>\sigma>0</math> controls the strength of the stochastic variation. Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D_n(x)P_n(x) }{ \sum_y D_n(y)P_n(y) } </math> the probability landscape itself fluctuates dynamically from one iteration to the next. Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate: * fluctuating concentration regions; * transient attractor-like structures; * noise-driven entropy evolution; * intermittent probability concentration; * metastable probabilistic configurations. === Conceptual interpretation === A qualitative stochastic evolution may be visualized as: <pre> Broad initial distribution ↓ Random localized amplification ↓ Fluctuating concentration regions ↓ Dynamic probabilistic structure </pre> Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit: * partial concentration, * persistent fluctuations, * stochastic stabilization, * or continuously evolving probabilistic structure. These ideas connect PDT to broader areas of: * stochastic processes; * random multiplicative systems; * statistical mechanics; * noise-driven dynamical systems; * probabilistic geometry. At present these behaviors remain exploratory computational possibilities within finite-state toy models. == Qualitative classes of iterative PDT behavior == Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation. The following table summarizes several representative classes explored within finite-state toy models. {| class="wikitable" ! Dilation-field class ! Typical iterative behavior ! Representative qualitative structure |- | Localized fields | Strong entropy reduction and concentration toward a dominant region | Single attractor-like concentration |- | Oscillatory fields | Distributed amplification with slower entropy reduction | Patterned multimodal structure |- | Multi-peak localized fields | Competition between several concentration regions | Hierarchical or metastable probability structure |- | Random and stochastic fields | Fluctuating amplification and noise-driven evolution | Dynamic probabilistic landscapes |} These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field. Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself. At present these qualitative behaviors remain exploratory computational observations within finite-state toy models. == Numerical simulation and iterative models == === Simulation model description === In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points. Two equivalent discrete implementations are common: * '''weighted evaluation''': retain all points and assign weights proportional to <math>D</math>; * '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to <math>D</math>. === Demonstration: reweighting mock galaxy catalogs === A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box. The demonstration pipeline is: # generate a baseline mock catalog; # define a positive dilation field over the configuration space; # perform PDT-style importance resampling; # compute the resulting two-point correlation function <math>\xi(r)</math>; # compare transformed and baseline catalogs. One example dilation field is: <math> D(x)=\exp(\lambda\phi(x)) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\phi(x)\ge0</math> is a nonnegative configuration-space field. An example seed-field construction is: <math> \phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right) </math> where <math>s_k</math> are seed locations and <math>\sigma</math> controls the width of the seed influence. The two-point correlation function may be estimated using the normalized Landy–Szalay estimator: <math> \xi(r) = \frac{DD(r)-2DR(r)+RR(r)}{RR(r)} </math> where <math>DD</math>, <math>DR</math>, and <math>RR</math> are normalized pair counts. {{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}} When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration. === Computational demonstrations === Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages. {{collapse top|Python demonstration placeholder}} <syntaxhighlight lang="python"> # Example implementations may be maintained separately # on GitHub, OSF, or supplementary Wikiversity pages. </syntaxhighlight> {{collapse bottom}} == Scope and Limitations == PDT is a mathematical framework for measure transformations. It does not claim: * a replacement theory for General Relativity or Quantum Mechanics; * empirical confirmation without explicit predictions and tests; * observational validation without independently reproducible analysis. The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations. == Speculative Extensions and Geometric Renormalization == ''This section is speculative and exploratory in nature.'' Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry. Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects. Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation. At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations. == Future directions == * develop canonical families of dilation fields and invariants; * clarify “structure-from-measure” diagnostics; * publish reproducible simulation notebooks and parameter sweeps; * compare multiple dilation families under shared evaluation criteria; * investigate connections between probabilistic geometry and curvature-dependent statistical measures. == Future Directions: Probability Element (PE) == A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics. The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution. This can be expressed in terms of a dimensionless ratio: <math>\eta = \frac{\sigma_P}{\sigma}</math> where: <math>\sigma_P</math> is a hypothesized minimal probability-resolution scale, <math>\sigma</math> is an effective distinguishability scale in probability-state space. === Conceptual motivation === Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry. === Illustrative toy model (not derived physics) === As a heuristic example, one may consider a modification to special relativistic time dilation of the form: <math>d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}</math> where: <math>v</math> is velocity, <math>c</math> is the speed of light, <math>\eta = \sigma_P / \sigma</math> encodes a proposed probability-resolution scale. This expression is constructed such that standard special relativity is recovered exactly in the limit <math>\eta \to 0</math>. === Status === The Probability Element concept is: Not part of standard Fisher information geometry not derived from quantum mechanics or general relativity not currently empirically established. It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale. === Open questions === Key open research directions include: Whether a consistent discrete formulation of probability geometry can be constructed. Whether a fundamental probability-resolution scale <math>\sigma_P</math> can be derived from known physical principles. Whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions. == Convergence behavior == Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations. === Qualitative convergence classes === Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior: * '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy. * '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration. * '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions. * '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior. === Entropy and convergence === In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time. The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations. === Attractor-like behavior === Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense. Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks. == Current limitations == PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation. Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms. Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks. == See also == * [[w:Buffon's needle problem|Buffon's needle problem]] * [[w:Probability measure|Probability measure]] * [[w:Importance sampling|Importance sampling]] * [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]] * [[w:Dynamical system|Dynamical systems]] * [[w:Entropy (information theory)|Entropy]] * [[w:Information theory|Information theory]] * [[w:Measure theory|Measure theory]] * [[w:Geometric probability|Geometric probability]] * [[w:Shannon entropy|Shannon entropy]] * [[w:Stochastic process|Stochastic process]] * [[w:Fixed point (mathematics)|Fixed point]] * [[w:Convergence (mathematics)|Convergence]] == Subpages == The following subpages develop mathematical extensions and specialized topics related to Probability Dilation Theory (PDT). * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows|Fisher Geometry and Dilation Flows]] – studies information geometry, Fisher distance, and geodesic properties of PDT trajectories. * [[Probability Dilation Theory/Logit Representation of PE|Logit Representation of PE]] – develops the log-odds representation of probability elements and exponential PDT flows. * [[Probability Dilation Theory/Convergence and Fixed Points|Convergence and Fixed Points]] – investigates invariant measures, attractors, and stability of iterative PDT transformations. * [[Probability Dilation Theory/Stochastic Dilation Fields|Stochastic Dilation Fields]] – studies random and time-dependent dilation fields, ergodicity, and stochastic measure evolution. * [[Probability Dilation Theory/Entropy Evolution|Entropy Evolution]] – examines Shannon entropy under repeated probability dilation. * [[Probability Dilation Theory/Wasserstein Geometry|Wasserstein Geometry]] – explores distances between probability measures and convergence in measure space. * [[Probability Dilation Theory/Measure-Theoretic Foundations|Measure-Theoretic Foundations]] – develops rigorous measure-theoretic aspects of PDT including normalization and existence conditions. * [[Probability Dilation Theory/Euler Methods and Continuous-Time PDT]] – investigates continuous probability flows and Euler approximations of PDT. * [[Probability Dilation Theory/Worked Example]] – canonical binary example illustrating PDT transformations and geometry. [[Probability Dilation Theory/Decoherence Analogy and Simulation]] [[Probability Dilation Theory / Dilation Vector Field]] [[Probability Dilation Theory / Dilation Flows]] [[Probability Dilation Theory / Matrix Dilation Operators]] [[Probability Dilation Theory / Spectral Analysis of Matrix Dilation Operators]] == Notation == Throughout PDT, the following notation is used: {| class="wikitable" ! Symbol ! Meaning |- | <math>P</math> | Probability measure |- | <math>P_n</math> | nth iterate of PDT |- | <math>T_D</math> | Probability dilation operator |- | <math>D(x)</math> | Dilation field |- | <math>Z(P,D)</math> | Normalization factor |- | <math>H(P)</math> | Shannon entropy |- | <math>d_F</math> | Fisher-Rao distance |- | <math>W_p</math> | Wasserstein distance |- | <math>\ell</math> | Logit coordinate |- | <math>PE</math> | Probability Element |} == Related probabilistic and geometric literature == Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works: * Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014. * Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19 * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997. * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005. * Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007. * Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656. Text and original figures © Howard Richardson. Original contributions by the author. Content is available under the Creative Commons Attribution-ShareAlike license (CC BY-SA) as required by Wikiversity. Reuse permitted with attribution. gpjt7okygwur3gpqqu5qk91qk2y7a7s 2 324023 2819279 2745780 2026-07-24T14:17:53Z Nazwa Shabrina 3007587 /* */ 2819279 wikitext text/x-wiki ==Subpages== {{Subpages/List}} 6twibesnbkty1mmoermf1c8xqy0pf9e 2 326765 2819303 2819145 2026-07-24T18:11:09Z Dc.samizdat 2856930 /* The 24-cell */ 2819303 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. Their procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-1</math> . Over the third <math>r_3</math> chord it also moves a distance of <math>-1</math>. Fontaine and Hurley also demonstrated the significance of <math>1/r_i</math> in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length <math>1/r_i</math> rather than <math>1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an <math>r_i</math> polygon. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. The angle between two <math>r_i</math> chords is <math>180^\circ - i \times 45^\circ</math>. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math> (it is <math>6\pi</math> in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes twelve 45° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the <math>r_3</math> {8/3} star polygon, which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a <math>\sqrt{3}</math> chord. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The <math>r_1</math> <small><math>\sqrt{1}</math></small> chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by <math>r_2</math> chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}. The <math>r_2</math> <small><math>\sqrt{1}</math></small> chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}. The <math>r_3</math> <small><math>\sqrt{2}</math></small> chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells. The <math>r_4</math> <small><math>\sqrt{3}</math></small> chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons. The <math>r_5</math> <small><math>\sqrt{3}</math></small> chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant <math>r_2</math> hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over <math>r_5</math> chords. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of <small><math>\sqrt{2}</math> </small>chords in the 24-cell]] We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° <small><math>\sqrt{2}</math></small> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <small><math>\sqrt{2}</math></small> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of <small><math>\sqrt{3}</math></small> chords. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of <small><math>\sqrt{3}</math></small> chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over <math>r_{5}</math> isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the <math>r_5</math> {12/5} star polygon which constructs <math>1/r_5</math>. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 <math>r_5</math> <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each <math>r_5</math> 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |6 distinct 180° chord pairs make 6 distinct isoclinic rotations |- ! colspan="3" |Edge chords !Invariant planes ! colspan="3" |Isocline chords |- style="background: gainsboro;" | | rowspan="4" |<math>t_1</math> |60° | rowspan="4" |[[File:Regular_polygon_24.svg|100px]]<br>{24/1}={24} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_24-11.svg|100px]]<br>{24/11} |120° | rowspan="4" |<math>t_{11}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |165° |15° |- style="background: palegreen;" | | rowspan="4" |<math>t_2</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(12,1).svg|100px]]<br>{24/2}=2{12} | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="4" |<math>t_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |150° |30° |- style="background: seashell;" | | rowspan="4" |<math>t_3</math> |90° | rowspan="4" |[[File:Regular_star_figure_3(8,1).svg|100px]]<br>{24/3}=3{8} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_3(8,3).svg|100px]]<br>{24/9}=3{8/3} |90° | rowspan="4" |<math>t_{9}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |135° |45° |- style="background: palegreen;" | | rowspan="4" |<math>t_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_12(2,1).svg|100px]]<br>{24/12}=12{2} | rowspan="4" |[[File:Regular_star_figure_8(3,1).svg|100px]]<br>{24/8}=8{3} |120° | rowspan="4" |<math>t_{8}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: gainsboro;" | | rowspan="4" |<math>t_5</math> |60° | rowspan="4" |[[File:Regular_star_polygon_24-5.svg|100px]]<br>{24/5} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_24-7.svg|100px]]<br>{24/7} |120° | rowspan="4" |<math>t_{7}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |105° |75° |- style="background: seashell;" | | rowspan="4" |<math>t_6</math> |90° | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} |90° | rowspan="4" |<math>t_{6}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} By examining the chords <math>r_i</math> of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords <math>t_i</math> of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations. Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. The short edge chord and long isocline chord each have their characteristic {24/n}-gon. The edge chords form the rotation's edge polygons, invariant as their vertices circle in 4-space. The isocline chords form the rotation's Clifford polygons, stationary in 4-space, over which the edge polygons' vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices. {{Clear}} == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math> the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Edge chords ! Section ! colspan="3" |Isocline chords |- style="background: palegreen;" | | rowspan="4" |<math>r_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |0° |180° |- style="background: palegreen;" | | rowspan="4" |<math>r_1</math> |36° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="4" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: palegreen;" | |12° |168° |- style="background: gainsboro;" | | rowspan="4" |<math>r_2</math> |36° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="4" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: gainsboro;" | |24° |156° |- style="background: yellow;" | | rowspan="4" |<math>r_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |36° |144° |- style="background: palegreen;" | | rowspan="4" |<math>r_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="4" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |48° |132° |- style="background: palegreen;" | | rowspan="4" |<math>r_5</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: yellow;" | | rowspan="4" |<math>r_{6}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |72° |108° |- style="background: seashell;" | | rowspan="4" |<math>r_{7}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="4" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |84° |96° |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short edge chord and long isocline chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over <math>r_7</math> edge chords and <math>r_8</math> isocline chords. The {30/7} edge polygon is a skew helix of circumference <math>14\pi</math> with each <math>r_7</math> edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference <math>16\pi</math> with each <math>r_8</math> isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>\sqrt{3}</math></small> ]] We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over <math>\sqrt{1}</math> edge chords and <math>\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over <math>r_{4}=\sqrt{1}</math> edge chords and <math>r_{11}=\sqrt{3}</math> isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each <math>r_{11}</math> isocline chord is the <math>\sqrt{3}</math> diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° <math>r_{11}</math> isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference <math>22\pi</math> over <math>r_{11}</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° <math>r_{3}</math> edges, over 144° <math>r_{12}</math> isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each <math>r_{12}</math> chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions. The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° <math>r_{2}</math> edges, over 144° <math>r_{13}</math> isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° <math>r_{13}</math> isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference <math>26\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Edge chords ! Section ! colspan="3" |Isocline chords |- style="background: palegreen;" | | rowspan="4" |<math>c_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |0° |180° |- style="background: palegreen;" | | rowspan="4" |<math>c_1</math> |15.5~° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="4" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: palegreen;" | |12° |168° |- style="background: gainsboro;" | | rowspan="4" |<math>c_2</math> |25.2~° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="4" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: gainsboro;" | |24° |156° |- style="background: yellow;" | | rowspan="4" |<math>c_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |36° |144° |- style="background: gainsboro;" | | rowspan="4" |<math>c_4</math> |41.4~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |138.6~° | rowspan="4" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: gainsboro;" | |42° |138° |- style="background: palegreen;" | | rowspan="4" |<math>c_5</math> |44.5~° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="4" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: palegreen;" | |48° |132° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_6</math> |49.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |130.9~° | rowspan="4" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro;" | |52° |128° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_7</math> |56° | rowspan="4" | | rowspan="4" | | rowspan="4" | |124° | rowspan="4" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: gainsboro;" | |56° |124° |- style="background: palegreen;" | | rowspan="4" |<math>c_8</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_9</math> |66.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |113.9~° | rowspan="4" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro;" | |64° |116° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{10}</math> |69.8~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |110.2~° | rowspan="4" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: gainsboro;" | |68° |112° |- style="background: yellow;" | | rowspan="4" |<math>c_{11}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |72° |108° |- style="background: palegreen; height:50px" | | rowspan="4" |<math>c_{12}</math> |75.5~° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="4" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: palegreen;" | |84° |96° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{13}</math> |81.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |98.9~° | rowspan="4" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro;" | |° |° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{14}</math> |84.5~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |95.5~° | rowspan="4" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: gainsboro;" | |° |° |- style="background: seashell;" | | rowspan="4" |<math>c_{15}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="4" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} kwxq6tom1tlhemksu3i1d7vmxcnnfnc 2819305 2819303 2026-07-24T18:13:07Z Dc.samizdat 2856930 /* The 600-cell */ 2819305 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. Their procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-1</math> . Over the third <math>r_3</math> chord it also moves a distance of <math>-1</math>. Fontaine and Hurley also demonstrated the significance of <math>1/r_i</math> in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length <math>1/r_i</math> rather than <math>1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an <math>r_i</math> polygon. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. The angle between two <math>r_i</math> chords is <math>180^\circ - i \times 45^\circ</math>. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math> (it is <math>6\pi</math> in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes twelve 45° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the <math>r_3</math> {8/3} star polygon, which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a <math>\sqrt{3}</math> chord. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The <math>r_1</math> <small><math>\sqrt{1}</math></small> chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by <math>r_2</math> chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}. The <math>r_2</math> <small><math>\sqrt{1}</math></small> chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}. The <math>r_3</math> <small><math>\sqrt{2}</math></small> chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells. The <math>r_4</math> <small><math>\sqrt{3}</math></small> chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons. The <math>r_5</math> <small><math>\sqrt{3}</math></small> chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant <math>r_2</math> hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over <math>r_5</math> chords. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of <small><math>\sqrt{2}</math> </small>chords in the 24-cell]] We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° <small><math>\sqrt{2}</math></small> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <small><math>\sqrt{2}</math></small> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of <small><math>\sqrt{3}</math></small> chords. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of <small><math>\sqrt{3}</math></small> chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over <math>r_{5}</math> isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the <math>r_5</math> {12/5} star polygon which constructs <math>1/r_5</math>. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 <math>r_5</math> <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each <math>r_5</math> 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |6 distinct 180° chord pairs make 6 distinct isoclinic rotations |- ! colspan="3" |Edge chords !Invariant planes ! colspan="3" |Isocline chords |- style="background: gainsboro;" | | rowspan="4" |<math>t_1</math> |60° | rowspan="4" |[[File:Regular_polygon_24.svg|100px]]<br>{24/1}={24} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_24-11.svg|100px]]<br>{24/11} |120° | rowspan="4" |<math>t_{11}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |165° |15° |- style="background: palegreen;" | | rowspan="4" |<math>t_2</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(12,1).svg|100px]]<br>{24/2}=2{12} | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="4" |<math>t_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |150° |30° |- style="background: seashell;" | | rowspan="4" |<math>t_3</math> |90° | rowspan="4" |[[File:Regular_star_figure_3(8,1).svg|100px]]<br>{24/3}=3{8} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_3(8,3).svg|100px]]<br>{24/9}=3{8/3} |90° | rowspan="4" |<math>t_{9}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |135° |45° |- style="background: palegreen;" | | rowspan="4" |<math>t_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_12(2,1).svg|100px]]<br>{24/12}=12{2} | rowspan="4" |[[File:Regular_star_figure_8(3,1).svg|100px]]<br>{24/8}=8{3} |120° | rowspan="4" |<math>t_{8}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: gainsboro;" | | rowspan="4" |<math>t_5</math> |60° | rowspan="4" |[[File:Regular_star_polygon_24-5.svg|100px]]<br>{24/5} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_24-7.svg|100px]]<br>{24/7} |120° | rowspan="4" |<math>t_{7}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |105° |75° |- style="background: seashell;" | | rowspan="4" |<math>t_6</math> |90° | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} |90° | rowspan="4" |<math>t_{6}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} By examining the chords <math>r_i</math> of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords <math>t_i</math> of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations. Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. The short edge chord and long isocline chord each have their characteristic {24/n}-gon. The edge chords form the rotation's edge polygons, invariant as their vertices circle in 4-space. The isocline chords form the rotation's Clifford polygons, stationary in 4-space, over which the edge polygons' vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices. {{Clear}} == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math> the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Edge chords ! Section ! colspan="3" |Isocline chords |- style="background: palegreen;" | | rowspan="4" |<math>r_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |180° |0° |- style="background: palegreen;" | | rowspan="4" |<math>r_1</math> |36° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="4" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: palegreen;" | |168° |12° |- style="background: gainsboro;" | | rowspan="4" |<math>r_2</math> |36° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="4" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: gainsboro;" | |156° |24° |- style="background: yellow;" | | rowspan="4" |<math>r_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |144° |36° |- style="background: palegreen;" | | rowspan="4" |<math>r_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="4" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |132° |48° |- style="background: palegreen;" | | rowspan="4" |<math>r_5</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: yellow;" | | rowspan="4" |<math>r_{6}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |108° |72° |- style="background: seashell;" | | rowspan="4" |<math>r_{7}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="4" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |96° |84° |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short edge chord and long isocline chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over <math>r_7</math> edge chords and <math>r_8</math> isocline chords. The {30/7} edge polygon is a skew helix of circumference <math>14\pi</math> with each <math>r_7</math> edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference <math>16\pi</math> with each <math>r_8</math> isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>\sqrt{3}</math></small> ]] We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over <math>\sqrt{1}</math> edge chords and <math>\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over <math>r_{4}=\sqrt{1}</math> edge chords and <math>r_{11}=\sqrt{3}</math> isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each <math>r_{11}</math> isocline chord is the <math>\sqrt{3}</math> diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° <math>r_{11}</math> isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference <math>22\pi</math> over <math>r_{11}</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° <math>r_{3}</math> edges, over 144° <math>r_{12}</math> isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each <math>r_{12}</math> chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions. The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° <math>r_{2}</math> edges, over 144° <math>r_{13}</math> isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° <math>r_{13}</math> isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference <math>26\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Edge chords ! Section ! colspan="3" |Isocline chords |- style="background: palegreen;" | | rowspan="4" |<math>c_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |0° |180° |- style="background: palegreen;" | | rowspan="4" |<math>c_1</math> |15.5~° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="4" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: palegreen;" | |12° |168° |- style="background: gainsboro;" | | rowspan="4" |<math>c_2</math> |25.2~° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="4" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: gainsboro;" | |24° |156° |- style="background: yellow;" | | rowspan="4" |<math>c_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |36° |144° |- style="background: gainsboro;" | | rowspan="4" |<math>c_4</math> |41.4~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |138.6~° | rowspan="4" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: gainsboro;" | |42° |138° |- style="background: palegreen;" | | rowspan="4" |<math>c_5</math> |44.5~° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="4" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: palegreen;" | |48° |132° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_6</math> |49.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |130.9~° | rowspan="4" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro;" | |52° |128° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_7</math> |56° | rowspan="4" | | rowspan="4" | | rowspan="4" | |124° | rowspan="4" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: gainsboro;" | |56° |124° |- style="background: palegreen;" | | rowspan="4" |<math>c_8</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |60° |120° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_9</math> |66.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |113.9~° | rowspan="4" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro;" | |64° |116° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{10}</math> |69.8~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |110.2~° | rowspan="4" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: gainsboro;" | |68° |112° |- style="background: yellow;" | | rowspan="4" |<math>c_{11}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |72° |108° |- style="background: palegreen; height:50px" | | rowspan="4" |<math>c_{12}</math> |75.5~° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="4" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: palegreen;" | |84° |96° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{13}</math> |81.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |98.9~° | rowspan="4" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro;" | |° |° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{14}</math> |84.5~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |95.5~° | rowspan="4" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: gainsboro;" | |° |° |- style="background: seashell;" | | rowspan="4" |<math>c_{15}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="4" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 1s9jckc42swl2lqf99q5u4xrjyoib3l 2819308 2819305 2026-07-24T18:15:10Z Dc.samizdat 2856930 /* Finally the 120-cell */ 2819308 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. Their procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-1</math> . Over the third <math>r_3</math> chord it also moves a distance of <math>-1</math>. Fontaine and Hurley also demonstrated the significance of <math>1/r_i</math> in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length <math>1/r_i</math> rather than <math>1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an <math>r_i</math> polygon. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. The angle between two <math>r_i</math> chords is <math>180^\circ - i \times 45^\circ</math>. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math> (it is <math>6\pi</math> in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes twelve 45° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the <math>r_3</math> {8/3} star polygon, which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a <math>\sqrt{3}</math> chord. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The <math>r_1</math> <small><math>\sqrt{1}</math></small> chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by <math>r_2</math> chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}. The <math>r_2</math> <small><math>\sqrt{1}</math></small> chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}. The <math>r_3</math> <small><math>\sqrt{2}</math></small> chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells. The <math>r_4</math> <small><math>\sqrt{3}</math></small> chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons. The <math>r_5</math> <small><math>\sqrt{3}</math></small> chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant <math>r_2</math> hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over <math>r_5</math> chords. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of <small><math>\sqrt{2}</math> </small>chords in the 24-cell]] We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° <small><math>\sqrt{2}</math></small> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <small><math>\sqrt{2}</math></small> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of <small><math>\sqrt{3}</math></small> chords. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of <small><math>\sqrt{3}</math></small> chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over <math>r_{5}</math> isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the <math>r_5</math> {12/5} star polygon which constructs <math>1/r_5</math>. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 <math>r_5</math> <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each <math>r_5</math> 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |6 distinct 180° chord pairs make 6 distinct isoclinic rotations |- ! colspan="3" |Edge chords !Invariant planes ! colspan="3" |Isocline chords |- style="background: gainsboro;" | | rowspan="4" |<math>t_1</math> |60° | rowspan="4" |[[File:Regular_polygon_24.svg|100px]]<br>{24/1}={24} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_24-11.svg|100px]]<br>{24/11} |120° | rowspan="4" |<math>t_{11}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |165° |15° |- style="background: palegreen;" | | rowspan="4" |<math>t_2</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(12,1).svg|100px]]<br>{24/2}=2{12} | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="4" |<math>t_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |150° |30° |- style="background: seashell;" | | rowspan="4" |<math>t_3</math> |90° | rowspan="4" |[[File:Regular_star_figure_3(8,1).svg|100px]]<br>{24/3}=3{8} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_3(8,3).svg|100px]]<br>{24/9}=3{8/3} |90° | rowspan="4" |<math>t_{9}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |135° |45° |- style="background: palegreen;" | | rowspan="4" |<math>t_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_12(2,1).svg|100px]]<br>{24/12}=12{2} | rowspan="4" |[[File:Regular_star_figure_8(3,1).svg|100px]]<br>{24/8}=8{3} |120° | rowspan="4" |<math>t_{8}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: gainsboro;" | | rowspan="4" |<math>t_5</math> |60° | rowspan="4" |[[File:Regular_star_polygon_24-5.svg|100px]]<br>{24/5} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_24-7.svg|100px]]<br>{24/7} |120° | rowspan="4" |<math>t_{7}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |105° |75° |- style="background: seashell;" | | rowspan="4" |<math>t_6</math> |90° | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} |90° | rowspan="4" |<math>t_{6}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} By examining the chords <math>r_i</math> of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords <math>t_i</math> of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations. Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. The short edge chord and long isocline chord each have their characteristic {24/n}-gon. The edge chords form the rotation's edge polygons, invariant as their vertices circle in 4-space. The isocline chords form the rotation's Clifford polygons, stationary in 4-space, over which the edge polygons' vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices. {{Clear}} == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math> the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Edge chords ! Section ! colspan="3" |Isocline chords |- style="background: palegreen;" | | rowspan="4" |<math>r_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |180° |0° |- style="background: palegreen;" | | rowspan="4" |<math>r_1</math> |36° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="4" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: palegreen;" | |168° |12° |- style="background: gainsboro;" | | rowspan="4" |<math>r_2</math> |36° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="4" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: gainsboro;" | |156° |24° |- style="background: yellow;" | | rowspan="4" |<math>r_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |144° |36° |- style="background: palegreen;" | | rowspan="4" |<math>r_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="4" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |132° |48° |- style="background: palegreen;" | | rowspan="4" |<math>r_5</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: yellow;" | | rowspan="4" |<math>r_{6}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |108° |72° |- style="background: seashell;" | | rowspan="4" |<math>r_{7}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="4" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |96° |84° |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short edge chord and long isocline chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over <math>r_7</math> edge chords and <math>r_8</math> isocline chords. The {30/7} edge polygon is a skew helix of circumference <math>14\pi</math> with each <math>r_7</math> edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference <math>16\pi</math> with each <math>r_8</math> isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>\sqrt{3}</math></small> ]] We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over <math>\sqrt{1}</math> edge chords and <math>\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over <math>r_{4}=\sqrt{1}</math> edge chords and <math>r_{11}=\sqrt{3}</math> isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each <math>r_{11}</math> isocline chord is the <math>\sqrt{3}</math> diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° <math>r_{11}</math> isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference <math>22\pi</math> over <math>r_{11}</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° <math>r_{3}</math> edges, over 144° <math>r_{12}</math> isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each <math>r_{12}</math> chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions. The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° <math>r_{2}</math> edges, over 144° <math>r_{13}</math> isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° <math>r_{13}</math> isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference <math>26\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Edge chords ! Section ! colspan="3" |Isocline chords |- style="background: palegreen;" | | rowspan="4" |<math>c_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |180° |0° |- style="background: palegreen;" | | rowspan="4" |<math>c_1</math> |15.5~° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="4" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: palegreen;" | |168° |12° |- style="background: gainsboro;" | | rowspan="4" |<math>c_2</math> |25.2~° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="4" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: gainsboro;" | |156° |24° |- style="background: yellow;" | | rowspan="4" |<math>c_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |144° |36° |- style="background: gainsboro;" | | rowspan="4" |<math>c_4</math> |41.4~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |138.6~° | rowspan="4" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: gainsboro;" | |138° |42° |- style="background: palegreen;" | | rowspan="4" |<math>c_5</math> |44.5~° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="4" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: palegreen;" | |132° |48° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_6</math> |49.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |130.9~° | rowspan="4" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro;" | |128° |52° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_7</math> |56° | rowspan="4" | | rowspan="4" | | rowspan="4" | |124° | rowspan="4" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: gainsboro;" | |124° |56° |- style="background: palegreen;" | | rowspan="4" |<math>c_8</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_9</math> |66.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |113.9~° | rowspan="4" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro;" | |116° |64° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{10}</math> |69.8~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |110.2~° | rowspan="4" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: gainsboro;" | |112° |68° |- style="background: yellow;" | | rowspan="4" |<math>c_{11}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |108° |72° |- style="background: palegreen; height:50px" | | rowspan="4" |<math>c_{12}</math> |75.5~° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="4" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: palegreen;" | |96° |84° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{13}</math> |81.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |98.9~° | rowspan="4" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro;" | |° |° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{14}</math> |84.5~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |95.5~° | rowspan="4" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: gainsboro;" | |° |° |- style="background: seashell;" | | rowspan="4" |<math>c_{15}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="4" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} oysftxzx0ek1hu51m8l1z73ou7cuemp 2819310 2819308 2026-07-24T18:25:49Z Dc.samizdat 2856930 /* The 24-cell */ 2819310 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. Their procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-1</math> . Over the third <math>r_3</math> chord it also moves a distance of <math>-1</math>. Fontaine and Hurley also demonstrated the significance of <math>1/r_i</math> in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length <math>1/r_i</math> rather than <math>1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an <math>r_i</math> polygon. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. The angle between two <math>r_i</math> chords is <math>180^\circ - i \times 45^\circ</math>. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math> (it is <math>6\pi</math> in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes twelve 45° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the <math>r_3</math> {8/3} star polygon, which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a <math>\sqrt{3}</math> chord. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The <math>r_1</math> <small><math>\sqrt{1}</math></small> chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by <math>r_2</math> chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}. The <math>r_2</math> <small><math>\sqrt{1}</math></small> chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}. The <math>r_3</math> <small><math>\sqrt{2}</math></small> chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells. The <math>r_4</math> <small><math>\sqrt{3}</math></small> chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons. The <math>r_5</math> <small><math>\sqrt{3}</math></small> chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant <math>r_2</math> hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over <math>r_5</math> chords. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of <small><math>\sqrt{2}</math> </small>chords in the 24-cell]] We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° <small><math>\sqrt{2}</math></small> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <small><math>\sqrt{2}</math></small> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of <small><math>\sqrt{3}</math></small> chords. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of <small><math>\sqrt{3}</math></small> chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over <math>r_{5}</math> isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the <math>r_5</math> {12/5} star polygon which constructs <math>1/r_5</math>. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 <math>r_5</math> <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each <math>r_5</math> 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |6 distinct 180° chord pairs make 6 distinct isoclinic rotations |- ! colspan="3" |Edge chords !Invariant planes ! colspan="3" |Isocline chords |- style="background: gainsboro;" | | rowspan="4" |<math>t_1</math> |60° | rowspan="4" |[[File:Regular_polygon_24.svg|100px]]<br>{24/1}={24} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_24-11.svg|100px]]<br>{24/11} |120° | rowspan="4" |<math>t_{11}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |165° |15° |- style="background: palegreen;" | | rowspan="4" |<math>t_2</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(12,1).svg|100px]]<br>{24/2}=2{12} | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="4" |<math>t_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |150° |30° |- style="background: seashell;" | | rowspan="4" |<math>t_3</math> |90° | rowspan="4" |[[File:Regular_star_figure_3(8,1).svg|100px]]<br>{24/3}=3{8} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_3(8,3).svg|100px]]<br>{24/9}=3{8/3} |90° | rowspan="4" |<math>t_{9}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |135° |45° |- style="background: palegreen;" | | rowspan="4" |<math>t_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_12(2,1).svg|100px]]<br>{24/12}=12{2} | rowspan="4" |[[File:Regular_star_figure_8(3,1).svg|100px]]<br>{24/8}=8{3} |120° | rowspan="4" |<math>t_{8}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: gainsboro;" | | rowspan="4" |<math>t_5</math> |60° | rowspan="4" |[[File:Regular_star_polygon_24-5.svg|100px]]<br>{24/5} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_24-7.svg|100px]]<br>{24/7} |120° | rowspan="4" |<math>t_{7}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |105° |75° |- style="background: seashell;" | | rowspan="4" |<math>t_6</math> |90° | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} |90° | rowspan="4" |<math>t_{6}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} By examining the chords <math>r_i</math> of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords <math>t_i</math> of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations. Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. The short edge chord and long isocline chord each have their characteristic {24/n}-gon. The edge chords form the rotation's edge polygons, invariant as their vertices circle in 4-space. The isocline chords form the rotation's Clifford polygons, stationary in 4-space, over which the edge polygons' vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices. Each isoclinic rotation takes two chiral forms. There is a right rotation and a left rotation for each row of the table. .... {{Clear}} == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math> the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Edge chords ! Section ! colspan="3" |Isocline chords |- style="background: palegreen;" | | rowspan="4" |<math>r_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |180° |0° |- style="background: palegreen;" | | rowspan="4" |<math>r_1</math> |36° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="4" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: palegreen;" | |168° |12° |- style="background: gainsboro;" | | rowspan="4" |<math>r_2</math> |36° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="4" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: gainsboro;" | |156° |24° |- style="background: yellow;" | | rowspan="4" |<math>r_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |144° |36° |- style="background: palegreen;" | | rowspan="4" |<math>r_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="4" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |132° |48° |- style="background: palegreen;" | | rowspan="4" |<math>r_5</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: yellow;" | | rowspan="4" |<math>r_{6}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |108° |72° |- style="background: seashell;" | | rowspan="4" |<math>r_{7}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="4" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |96° |84° |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short edge chord and long isocline chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over <math>r_7</math> edge chords and <math>r_8</math> isocline chords. The {30/7} edge polygon is a skew helix of circumference <math>14\pi</math> with each <math>r_7</math> edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference <math>16\pi</math> with each <math>r_8</math> isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>\sqrt{3}</math></small> ]] We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over <math>\sqrt{1}</math> edge chords and <math>\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over <math>r_{4}=\sqrt{1}</math> edge chords and <math>r_{11}=\sqrt{3}</math> isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each <math>r_{11}</math> isocline chord is the <math>\sqrt{3}</math> diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° <math>r_{11}</math> isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference <math>22\pi</math> over <math>r_{11}</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° <math>r_{3}</math> edges, over 144° <math>r_{12}</math> isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each <math>r_{12}</math> chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions. The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° <math>r_{2}</math> edges, over 144° <math>r_{13}</math> isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° <math>r_{13}</math> isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference <math>26\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Edge chords ! Section ! colspan="3" |Isocline chords |- style="background: palegreen;" | | rowspan="4" |<math>c_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |180° |0° |- style="background: palegreen;" | | rowspan="4" |<math>c_1</math> |15.5~° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="4" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: palegreen;" | |168° |12° |- style="background: gainsboro;" | | rowspan="4" |<math>c_2</math> |25.2~° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="4" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: gainsboro;" | |156° |24° |- style="background: yellow;" | | rowspan="4" |<math>c_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |144° |36° |- style="background: gainsboro;" | | rowspan="4" |<math>c_4</math> |41.4~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |138.6~° | rowspan="4" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: gainsboro;" | |138° |42° |- style="background: palegreen;" | | rowspan="4" |<math>c_5</math> |44.5~° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="4" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: palegreen;" | |132° |48° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_6</math> |49.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |130.9~° | rowspan="4" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro;" | |128° |52° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_7</math> |56° | rowspan="4" | | rowspan="4" | | rowspan="4" | |124° | rowspan="4" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: gainsboro;" | |124° |56° |- style="background: palegreen;" | | rowspan="4" |<math>c_8</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_9</math> |66.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |113.9~° | rowspan="4" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro;" | |116° |64° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{10}</math> |69.8~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |110.2~° | rowspan="4" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: gainsboro;" | |112° |68° |- style="background: yellow;" | | rowspan="4" |<math>c_{11}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |108° |72° |- style="background: palegreen; height:50px" | | rowspan="4" |<math>c_{12}</math> |75.5~° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="4" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: palegreen;" | |96° |84° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{13}</math> |81.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |98.9~° | rowspan="4" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro;" | |° |° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{14}</math> |84.5~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |95.5~° | rowspan="4" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: gainsboro;" | |° |° |- style="background: seashell;" | | rowspan="4" |<math>c_{15}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="4" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} aqihc1vtku2qp0vr4hcyp46zsxy378q 0 329548 2819267 2819066 2026-07-24T13:40:21Z Athene241 3100061 /* Data */ unsolved families for these bases are already listed 2819267 wikitext text/x-wiki {{mathematics}} '''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' > 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the "[[:w:Prime number|prime number]] [[:w:Greater than|>]] ''b''" [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, "string" is a subsequence of "meistersinger". In contrast, 758 is not a subsequence of 378259, "abc" is not a subsequence of "cbacacba", since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]]) Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|>]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''_''b'' := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] "is a subsequence of", which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' > 1. (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s) By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] "is a subsequence of", which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the set of such minimal elements must be a [[:w:Finite set|finite set]], e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), this set has exactly 77 [[:w:Element of a set|element]]s: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}. For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. "''xy''⁺''z''" in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes > ''b'' or probable primes > ''b'' with length ≤ 100000. == Solve the problem == To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime > ''b'' in all such families. We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') "linear" families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows: * 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''. * 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by: * 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''. * 2.2. If possible, simplify the family. * 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers > ''b''), and if so then remove the family from ''L''. * 3. Update ''L'', after each split examine the new families as in step 2. e.g. in decimal (base ''b'' = 10): ''M'' := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991} ''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9} and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1 and since the family 2{0}1 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1 221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L'' none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L'' and since the family 3{0,3,6,9}3 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed etc. Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime > ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime > ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime > ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both "numbers of primes in the set of the Athena problem in base ''b''" and "length of the largest prime in the set of the Athena problem in base ''b''" are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''^{''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)}. Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') * If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''. * If ''y''₁ ∈ ''Y'' and ''y''₂ ∈ ''Y'' and ''y''₁ ≠ ''y''₂ and the string ''xy''₁''y''₂''z'' represents a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''₁}{''Y'' \ ''y''₂}''z''. * If ''y''₁ ∈ ''Y'' and ''y''₂ ∈ ''Y'' and ''y''₁ ≠ ''y''₂ and both the strings ''xy''₁''y''₂''z'' and ''xy''₂''y''₁''z'' represent a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''₁}''z'' ∪ ''x''{''Y'' \ ''y''₂}''z''. e.g. in decimal (base ''b'' = 10): * 2221 is a prime > 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1. * 227 is a prime > 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7. * 449 is a prime > 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9. * Both 5051 and 5501 are primes > 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1. * 8501 is a prime > 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1. * 887 is a prime > 10, and it is a subsequence of 2887, also 2087 is a prime > 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7. * 349 and 449 are primes > 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes > 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9. * 251, 281, 521, 821, 881 are primes > 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes > 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}. If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''₁, ''y''₂, ..., ''y''_''n''}, then we can replace ''x''{''Y''}''z'' by ''xy''₁{''Y''}''z'' ∪ ''xy''₂{''Y''}''z'' ∪ ... ∪ ''xy''_''n''{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]]. If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list. e.g. in decimal (base ''b'' = 10): * The smallest prime in the family 5{0}27 is 5000000000000000000000000000027. * The smallest prime in the family {5}1 is 555555555551. * The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551. There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the set of the minimal elements in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''). The following is a "[[:w:Semi-algorithm|semi-algorithm]]" that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement: # ''M'' = ''[[:w:Empty string|∅]]'' # while (''L'' ≠ ''∅'') do # choose ''x'', a shortest string in ''L'' # ''M'' := ''M'' ∪ {''x''} # ''L'' := ''L'' − ''sup''({''x''}) In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&#39;, some regular overapproximation to ''L'', until we can show ''L''&#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''₁{''L''₂}''L''₃, where each of ''L''₁, ''L''₂, ''L''₃ is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not. Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''^{''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)}), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]). To solve the Athena problem, we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime > ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2^''n''−2 strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length > (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' < ''b'', i.e. finding the smallest prime of the form ''k''×''b''^''n''+1 and ''k''×''b''^''n''−1 (or prove such prime does not exist) with ''k'' < ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is < ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''^''n''+1 and ''k''×''b''^''n''−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''^''n''−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642), ''b''^''n''+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''^''n''+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''^''n''+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://oeis.org/A084742, https://oeis.org/A084741), ((''b''−2)×''b''^''n''+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''^''n''+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''^''n''−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217, https://oeis.org/A119591), ''b''^''n''+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''^''n''−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714), (''b''−1)×''b''^''n''+1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484), (''b''−1)×''b''^''n''−1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://oeis.org/A305531), ''b''^''n''+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''^''n''−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://oeis.org/A113516, https://oeis.org/A343589)): Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''_''n''''z'' for some ''n'' ≥ 0? Some families can be ruled out to contain no prime > ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''⁴+4×''y''⁴]]), or combine of them, e.g. * The base 9 family 2{7}: Always divisible by 2 or 5 * The base 16 family {8}F: Always divisible by 3, 7, or 13 * The base 21 family {7}D: Always divisible by 2, 13, or 17 * The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79 * The base 9 family 3{8}: Can be written as 4×9^''n''−1 and can be factored as (2×3^''n''−1) × (2×3^''n''+1) * The base 8 family 1{0}1: Can be written as 8^''n''+1 and can be factored as (2^''n''+1) × (4^''n''−2^''n''+1) * The base 16 family {4}1: Can be written as (4×16^''n''−49)/15 and can be factored as (2×3^''n''−7) × (2×3^''n''+7) / 15 * The base 16 family {C}D: Can be written as (4×16^''n''+1)/5 and can be factored as (2×4^''n''−2×2^''n''+1) × (2×4^''n''+2×2^''n''+1) / 5 * The base 14 family 8{D}: Can be written as 9×14^''n''−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14^''n''/2−1) × (3×14^''n''/2+1) if ''n'' is even * The base 12 family {B}9B: Can be written as 12^''n''−25, it is divisible by 13 if ''n'' is odd and can be factored as (12^''n''/2−5) × (12^''n''/2+5) if ''n'' is even * The base 17 family 1{9}: Can be written as (25×17^''n''−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17^''n''/2−3) × (5×17^''n''/2+3) / 16 if ''n'' is even * The base 19 family 1{6}: Can be written as (4×19^''n''−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19^''n''/2−1) × (2×19^''n''/2+1) / 3 if ''n'' is even By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime > ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes > ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes > ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3₅₉₂₁₉₇A, its algebraic form is (41×13⁵⁹²¹⁹⁸+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime). == Data == These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes < 10²⁵⁰⁰⁰ for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them > 10²⁹⁹ are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others > 10²⁹⁹ are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]]) All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], "{}" means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123₄567 means 123333567 (all subscripts are written in decimal). Base 2: 1 prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11} Base 3: 3 primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111} Base 4: 5 primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221} Base 5: 22 primes (the largest of which has 96 digits (it is 10₉₃13, and its algebraic form is 5⁹⁵+8)): {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013} Base 6: 11 primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041} Base 7: 71 primes (the largest of which has 17 digits (it is 3₁₆1, and its algebraic form is (7¹⁷−5)/2)): {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331} Base 8: 75 primes (the largest of which has 221 digits (it is 4₂₂₀7, and its algebraic form is (4×8²²¹+17)/7)): {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447} Base 9: 151 primes (the largest of which has 1161 digits (it is 30₁₁₅₈11, and its algebraic form is 3×9¹¹⁶⁰+10)): {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011} Base 10: 77 primes (the largest of which has 31 digits (it is 50₂₈27, and its algebraic form is 5×10³⁰+27)): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} Base 11: 1068 primes (including 1 unproven probable prime: 57₆₂₆₆₈), the largest of which has 62669 digits (it is 57₆₂₆₆₈, and its algebraic form is (57×11⁶²⁶⁶⁸−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena problem base 11] Base 12: 106 primes (the largest of which has 42 digits (it is 40₃₉77, and its algebraic form is 4×12⁴¹+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077} Base 13: 3197 primes (including 4 unproven probable primes: C5₂₃₇₅₅C, 80₃₂₀₁₇111, 95₁₉₇₄₂₀, A3₅₉₂₁₉₇A), the largest of which has 592199 digits (it is A3₅₉₂₁₉₇A, and its algebraic form is (41×13⁵⁹²¹⁹⁸+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena problem base 13] Base 14: 650 primes, the largest of which has 19699 digits (it is 4D₁₉₆₉₈, and its algebraic form is 5×14¹⁹⁶⁹⁸−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena problem base 14] Base 15: 1284 primes, the largest of which has 157 digits (it is 7₁₅₅97, and its algebraic form is (15¹⁵⁷+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena problem base 15] Base 16: 2347 primes (including 3 unproven probable primes: DB₃₂₂₃₄, 4₇₂₇₈₅DD, 3₁₁₆₁₃₇AF), the largest of which has 116139 digits (it is 3₁₁₆₁₃₇AF, and its algebraic form is (16¹¹⁶¹³⁹+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena problem base 16] Base 17: 10415 known primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of Athena problem base 17] Base 18: 549 primes, the largest of which has 6271 digits (it is C0₆₂₆₈C5, and its algebraic form is 12×18⁶²⁷⁰+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena problem base 18] Base 19: 31417 known primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of Athena problem base 19] Base 20: 3314 primes, the largest of which has 6271 digits (it is G0₆₂₆₉D, and its algebraic form is 16×20⁶²⁷⁰+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena problem base 20] Base 21: 13386 known primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of Athena problem base 21] Base 22: 8003 primes (including 1 unproven probable prime: BK₂₂₀₀₁5), the largest of which has 22003 digits (it is BK₂₂₀₀₁5, and its algebraic form is (251×22²²⁰⁰²−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena problem base 22] Base 23: 65178 known primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of Athena problem base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for base 23] Base 24: 3409 primes, the largest of which has 8134 digits (it is N00N₈₁₂₉LN, and its algebraic form is 13249×24⁸¹³¹−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena problem base 24] Base 25: 133639 known primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of Athena problem base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for base 25] Base 26: 25256 known primes (including 7 unproven probable primes: 5₁₉₃₉₁6F, 7₂₀₂₇₉OL, LD0₂₀₉₇₅7, 6K₂₃₃₀₀5, J0₄₄₃₀₃KCB, M0₆₁₁₈₆2BB, 85M₁₉₇₀₆₀B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of Athena problem base 26] Base 27: 102852 known primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of Athena problem base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for base 27] Base 28: 25528 known primes (including 3 unproven probable primes: N6₂₄₀₅₁LR, 5OA₃₁₂₃₈F, O4O₉₄₅₃₅9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of Athena problem base 28] Base 29: 355242 known primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of Athena problem base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for base 29] Base 30: 2619 primes (including 1 unproven probable prime: I0₂₄₆₀₈D), the largest of which has 34206 digits (it is OT₃₄₂₀₅, and its algebraic form is 25×30³⁴²⁰⁵−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena problem base 30] Base 31: 569323 known primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of Athena problem base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for base 31] Base 32: 168882 known primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of Athena problem base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for base 32] Base 33: 280012 known primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of Athena problem base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for base 33] Base 34: 184785 known primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of Athena problem base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for base 34] Base 35: 720002 known primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of Athena problem base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for base 35] Base 36: 35286 known primes (including 3 unproven probable primes: 7K₂₆₅₆₇Z, S0₇₅₀₀₇8H, P₈₁₉₉₃SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of Athena problem base 36] == The fully proof of Athena problem in decimal (base ''b'' = 10) == '''Bold''' for the minimal elements, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''. Assume ''p'' is a prime > 10, and the last digit of ''p'' must lie in {1,3,7,9}. Case 1: ''p'' ends with 1. In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9. Case 1.1: ''p'' begins with 2. In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2. If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's. If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty. If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''. If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''. Case 1.2: ''p'' begins with 5. In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8. If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}. If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''. If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite. Case 1.3: ''p'' begins with 8. In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5. If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}. If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}. If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime. If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite. If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element. Case 1.4: ''p'' begins with 9. In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8. If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8. If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}. If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction). If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801. Case 2: ''p'' ends with 3. In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Case 3: ''p'' ends with 7. In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8. Case 3.1: ''p'' begins with 2. In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8. If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}. If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime. If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40_''n''1 = 280_''n''7. Case 3.2: ''p'' begins with 5. In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2. If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's. If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty. If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''. If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''. Case 3.3: ''p'' begins with 7. In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime. Case 3.4: ''p'' begins with 8. In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime. Case 4: ''p'' ends with 9. In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9. If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's. If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4. Case 4.1: ''p'' begins with 3. In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''. Case 4.2: ''p'' begins with 4. In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6_''n''7 = 46_''n''9. Case 4.3: ''p'' begins with 6. In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty. If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6. If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}. If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes. Case 4.4: ''p'' begins with 9. In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669. [[Category:Number theory]] se7ykqr1jlwotdpxymxxlwlf2egixf9 2819268 2819267 2026-07-24T13:46:23Z Athene241 3100061 /* Solve the problem */ 2819268 wikitext text/x-wiki {{mathematics}} '''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' > 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the "[[:w:Prime number|prime number]] [[:w:Greater than|>]] ''b''" [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, "string" is a subsequence of "meistersinger". In contrast, 758 is not a subsequence of 378259, "abc" is not a subsequence of "cbacacba", since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]]) Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|>]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''_''b'' := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] "is a subsequence of", which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' > 1. (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s) By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] "is a subsequence of", which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the set of such minimal elements must be a [[:w:Finite set|finite set]], e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), this set has exactly 77 [[:w:Element of a set|element]]s: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}. For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. "''xy''⁺''z''" in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes > ''b'' or probable primes > ''b'' with length ≤ 100000. == Solve the problem == To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime > ''b'' in all such families. We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') "linear" families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows: * 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''. * 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by: * 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''. * 2.2. If possible, simplify the family. * 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers > ''b''), and if so then remove the family from ''L''. * 3. Update ''L'', after each split examine the new families as in step 2. e.g. in decimal (base ''b'' = 10): ''M'' := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991} ''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9} and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1 and since the family 2{0}1 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1 221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L'' none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L'' and since the family 3{0,3,6,9}3 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed etc. Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime > ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime > ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime > ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both "numbers of primes in the set of the Athena problem in base ''b''" and "length of the largest prime in the set of the Athena problem in base ''b''" are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''^{''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)}. Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') * If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''. * If ''y''₁ ∈ ''Y'' and ''y''₂ ∈ ''Y'' and ''y''₁ ≠ ''y''₂ and the string ''xy''₁''y''₂''z'' represents a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''₁}{''Y'' \ ''y''₂}''z''. * If ''y''₁ ∈ ''Y'' and ''y''₂ ∈ ''Y'' and ''y''₁ ≠ ''y''₂ and both the strings ''xy''₁''y''₂''z'' and ''xy''₂''y''₁''z'' represent a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''₁}''z'' ∪ ''x''{''Y'' \ ''y''₂}''z''. e.g. in decimal (base ''b'' = 10): * 2221 is a prime > 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1. * 227 is a prime > 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7. * 449 is a prime > 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9. * Both 5051 and 5501 are primes > 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1. * 8501 is a prime > 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1. * 887 is a prime > 10, and it is a subsequence of 2887, also 2087 is a prime > 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7. * 349 and 449 are primes > 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes > 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9. * 251, 281, 521, 821, 881 are primes > 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes > 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}. If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''₁, ''y''₂, ..., ''y''_''n''}, then we can replace ''x''{''Y''}''z'' by ''xy''₁{''Y''}''z'' ∪ ''xy''₂{''Y''}''z'' ∪ ... ∪ ''xy''_''n''{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]]. If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list. e.g. in decimal (base ''b'' = 10): * The smallest prime in the family 5{0}27 is 5000000000000000000000000000027. * The smallest prime in the family {5}1 is 555555555551. * The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551. There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the set of the minimal elements in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''). The following is a "[[:w:Semi-algorithm|semi-algorithm]]" that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement: # ''M'' = ''[[:w:Empty string|∅]]'' # while (''L'' ≠ ''∅'') do # choose ''x'', a shortest string in ''L'' # ''M'' := ''M'' ∪ {''x''} # ''L'' := ''L'' − ''sup''({''x''}) In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&#39;, some regular overapproximation to ''L'', until we can show ''L''&#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''₁{''L''₂}''L''₃, where each of ''L''₁, ''L''₂, ''L''₃ is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not. Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''^{''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)}), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]). To solve the Athena problem, we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime > ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2^''n''−2 strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length > (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' < ''b'', i.e. finding the smallest prime of the form ''k''×''b''^''n''+1 and ''k''×''b''^''n''−1 (or prove such prime does not exist) with ''k'' < ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is < ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''^''n''+1 and ''k''×''b''^''n''−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''^''n''−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''^''n''+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''^''n''+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''^''n''+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://oeis.org/A084742, https://oeis.org/A084741), ((''b''−2)×''b''^''n''+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''^''n''+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''^''n''−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217, https://oeis.org/A119591), ''b''^''n''+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''^''n''−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''^''n''+1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484; or for prime bases ''b'': https://oeis.org/A122396), (''b''−1)×''b''^''n''−1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), ''b''^''n''+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''^''n''−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)): Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''_''n''''z'' for some ''n'' ≥ 0? Some families can be ruled out to contain no prime > ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''⁴+4×''y''⁴]]), or combine of them, e.g. * The base 9 family 2{7}: Always divisible by 2 or 5 * The base 16 family {8}F: Always divisible by 3, 7, or 13 * The base 21 family {7}D: Always divisible by 2, 13, or 17 * The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79 * The base 9 family 3{8}: Can be written as 4×9^''n''−1 and can be factored as (2×3^''n''−1) × (2×3^''n''+1) * The base 8 family 1{0}1: Can be written as 8^''n''+1 and can be factored as (2^''n''+1) × (4^''n''−2^''n''+1) * The base 16 family {4}1: Can be written as (4×16^''n''−49)/15 and can be factored as (2×3^''n''−7) × (2×3^''n''+7) / 15 * The base 16 family {C}D: Can be written as (4×16^''n''+1)/5 and can be factored as (2×4^''n''−2×2^''n''+1) × (2×4^''n''+2×2^''n''+1) / 5 * The base 14 family 8{D}: Can be written as 9×14^''n''−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14^''n''/2−1) × (3×14^''n''/2+1) if ''n'' is even * The base 12 family {B}9B: Can be written as 12^''n''−25, it is divisible by 13 if ''n'' is odd and can be factored as (12^''n''/2−5) × (12^''n''/2+5) if ''n'' is even * The base 17 family 1{9}: Can be written as (25×17^''n''−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17^''n''/2−3) × (5×17^''n''/2+3) / 16 if ''n'' is even * The base 19 family 1{6}: Can be written as (4×19^''n''−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19^''n''/2−1) × (2×19^''n''/2+1) / 3 if ''n'' is even By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime > ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes > ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes > ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3₅₉₂₁₉₇A, its algebraic form is (41×13⁵⁹²¹⁹⁸+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime). == Data == These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes < 10²⁵⁰⁰⁰ for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them > 10²⁹⁹ are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others > 10²⁹⁹ are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]]) All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], "{}" means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123₄567 means 123333567 (all subscripts are written in decimal). Base 2: 1 prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11} Base 3: 3 primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111} Base 4: 5 primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221} Base 5: 22 primes (the largest of which has 96 digits (it is 10₉₃13, and its algebraic form is 5⁹⁵+8)): {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013} Base 6: 11 primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041} Base 7: 71 primes (the largest of which has 17 digits (it is 3₁₆1, and its algebraic form is (7¹⁷−5)/2)): {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331} Base 8: 75 primes (the largest of which has 221 digits (it is 4₂₂₀7, and its algebraic form is (4×8²²¹+17)/7)): {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447} Base 9: 151 primes (the largest of which has 1161 digits (it is 30₁₁₅₈11, and its algebraic form is 3×9¹¹⁶⁰+10)): {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011} Base 10: 77 primes (the largest of which has 31 digits (it is 50₂₈27, and its algebraic form is 5×10³⁰+27)): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} Base 11: 1068 primes (including 1 unproven probable prime: 57₆₂₆₆₈), the largest of which has 62669 digits (it is 57₆₂₆₆₈, and its algebraic form is (57×11⁶²⁶⁶⁸−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena problem base 11] Base 12: 106 primes (the largest of which has 42 digits (it is 40₃₉77, and its algebraic form is 4×12⁴¹+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077} Base 13: 3197 primes (including 4 unproven probable primes: C5₂₃₇₅₅C, 80₃₂₀₁₇111, 95₁₉₇₄₂₀, A3₅₉₂₁₉₇A), the largest of which has 592199 digits (it is A3₅₉₂₁₉₇A, and its algebraic form is (41×13⁵⁹²¹⁹⁸+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena problem base 13] Base 14: 650 primes, the largest of which has 19699 digits (it is 4D₁₉₆₉₈, and its algebraic form is 5×14¹⁹⁶⁹⁸−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena problem base 14] Base 15: 1284 primes, the largest of which has 157 digits (it is 7₁₅₅97, and its algebraic form is (15¹⁵⁷+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena problem base 15] Base 16: 2347 primes (including 3 unproven probable primes: DB₃₂₂₃₄, 4₇₂₇₈₅DD, 3₁₁₆₁₃₇AF), the largest of which has 116139 digits (it is 3₁₁₆₁₃₇AF, and its algebraic form is (16¹¹⁶¹³⁹+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena problem base 16] Base 17: 10415 known primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of Athena problem base 17] Base 18: 549 primes, the largest of which has 6271 digits (it is C0₆₂₆₈C5, and its algebraic form is 12×18⁶²⁷⁰+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena problem base 18] Base 19: 31417 known primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of Athena problem base 19] Base 20: 3314 primes, the largest of which has 6271 digits (it is G0₆₂₆₉D, and its algebraic form is 16×20⁶²⁷⁰+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena problem base 20] Base 21: 13386 known primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of Athena problem base 21] Base 22: 8003 primes (including 1 unproven probable prime: BK₂₂₀₀₁5), the largest of which has 22003 digits (it is BK₂₂₀₀₁5, and its algebraic form is (251×22²²⁰⁰²−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena problem base 22] Base 23: 65178 known primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of Athena problem base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for base 23] Base 24: 3409 primes, the largest of which has 8134 digits (it is N00N₈₁₂₉LN, and its algebraic form is 13249×24⁸¹³¹−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena problem base 24] Base 25: 133639 known primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of Athena problem base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for base 25] Base 26: 25256 known primes (including 7 unproven probable primes: 5₁₉₃₉₁6F, 7₂₀₂₇₉OL, LD0₂₀₉₇₅7, 6K₂₃₃₀₀5, J0₄₄₃₀₃KCB, M0₆₁₁₈₆2BB, 85M₁₉₇₀₆₀B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of Athena problem base 26] Base 27: 102852 known primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of Athena problem base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for base 27] Base 28: 25528 known primes (including 3 unproven probable primes: N6₂₄₀₅₁LR, 5OA₃₁₂₃₈F, O4O₉₄₅₃₅9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of Athena problem base 28] Base 29: 355242 known primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of Athena problem base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for base 29] Base 30: 2619 primes (including 1 unproven probable prime: I0₂₄₆₀₈D), the largest of which has 34206 digits (it is OT₃₄₂₀₅, and its algebraic form is 25×30³⁴²⁰⁵−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena problem base 30] Base 31: 569323 known primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of Athena problem base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for base 31] Base 32: 168882 known primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of Athena problem base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for base 32] Base 33: 280012 known primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of Athena problem base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for base 33] Base 34: 184785 known primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of Athena problem base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for base 34] Base 35: 720002 known primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of Athena problem base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for base 35] Base 36: 35286 known primes (including 3 unproven probable primes: 7K₂₆₅₆₇Z, S0₇₅₀₀₇8H, P₈₁₉₉₃SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of Athena problem base 36] == The fully proof of Athena problem in decimal (base ''b'' = 10) == '''Bold''' for the minimal elements, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''. Assume ''p'' is a prime > 10, and the last digit of ''p'' must lie in {1,3,7,9}. Case 1: ''p'' ends with 1. In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9. Case 1.1: ''p'' begins with 2. In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2. If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's. If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty. If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''. If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''. Case 1.2: ''p'' begins with 5. In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8. If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}. If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''. If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite. Case 1.3: ''p'' begins with 8. In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5. If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}. If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}. If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime. If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite. If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element. Case 1.4: ''p'' begins with 9. In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8. If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8. If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}. If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction). If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801. Case 2: ''p'' ends with 3. In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Case 3: ''p'' ends with 7. In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8. Case 3.1: ''p'' begins with 2. In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8. If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}. If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime. If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40_''n''1 = 280_''n''7. Case 3.2: ''p'' begins with 5. In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2. If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's. If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty. If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''. If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''. Case 3.3: ''p'' begins with 7. In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime. Case 3.4: ''p'' begins with 8. In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime. Case 4: ''p'' ends with 9. In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9. If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's. If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4. Case 4.1: ''p'' begins with 3. In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''. Case 4.2: ''p'' begins with 4. In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6_''n''7 = 46_''n''9. Case 4.3: ''p'' begins with 6. In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty. If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6. If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}. If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes. Case 4.4: ''p'' begins with 9. In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669. [[Category:Number theory]] earlcgm65xqdy3tbladruqpq9uuser4 2819269 2819268 2026-07-24T13:48:57Z Athene241 3100061 /* Solve the problem */ 2819269 wikitext text/x-wiki {{mathematics}} '''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' > 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the "[[:w:Prime number|prime number]] [[:w:Greater than|>]] ''b''" [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, "string" is a subsequence of "meistersinger". In contrast, 758 is not a subsequence of 378259, "abc" is not a subsequence of "cbacacba", since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]]) Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|>]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''_''b'' := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] "is a subsequence of", which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' > 1. (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s) By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] "is a subsequence of", which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the set of such minimal elements must be a [[:w:Finite set|finite set]], e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), this set has exactly 77 [[:w:Element of a set|element]]s: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}. For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. "''xy''⁺''z''" in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes > ''b'' or probable primes > ''b'' with length ≤ 100000. == Solve the problem == To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime > ''b'' in all such families. We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') "linear" families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows: * 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''. * 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by: * 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''. * 2.2. If possible, simplify the family. * 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers > ''b''), and if so then remove the family from ''L''. * 3. Update ''L'', after each split examine the new families as in step 2. e.g. in decimal (base ''b'' = 10): ''M'' := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991} ''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9} and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1 and since the family 2{0}1 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1 221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L'' none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L'' and since the family 3{0,3,6,9}3 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed etc. Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime > ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime > ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime > ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both "numbers of primes in the set of the Athena problem in base ''b''" and "length of the largest prime in the set of the Athena problem in base ''b''" are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''^{''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)}. Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') * If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''. * If ''y''₁ ∈ ''Y'' and ''y''₂ ∈ ''Y'' and ''y''₁ ≠ ''y''₂ and the string ''xy''₁''y''₂''z'' represents a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''₁}{''Y'' \ ''y''₂}''z''. * If ''y''₁ ∈ ''Y'' and ''y''₂ ∈ ''Y'' and ''y''₁ ≠ ''y''₂ and both the strings ''xy''₁''y''₂''z'' and ''xy''₂''y''₁''z'' represent a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''₁}''z'' ∪ ''x''{''Y'' \ ''y''₂}''z''. e.g. in decimal (base ''b'' = 10): * 2221 is a prime > 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1. * 227 is a prime > 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7. * 449 is a prime > 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9. * Both 5051 and 5501 are primes > 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1. * 8501 is a prime > 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1. * 887 is a prime > 10, and it is a subsequence of 2887, also 2087 is a prime > 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7. * 349 and 449 are primes > 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes > 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9. * 251, 281, 521, 821, 881 are primes > 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes > 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}. If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''₁, ''y''₂, ..., ''y''_''n''}, then we can replace ''x''{''Y''}''z'' by ''xy''₁{''Y''}''z'' ∪ ''xy''₂{''Y''}''z'' ∪ ... ∪ ''xy''_''n''{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]]. If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list. e.g. in decimal (base ''b'' = 10): * The smallest prime in the family 5{0}27 is 5000000000000000000000000000027. * The smallest prime in the family {5}1 is 555555555551. * The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551. There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the set of the minimal elements in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''). The following is a "[[:w:Semi-algorithm|semi-algorithm]]" that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement: # ''M'' = ''[[:w:Empty string|∅]]'' # while (''L'' ≠ ''∅'') do # choose ''x'', a shortest string in ''L'' # ''M'' := ''M'' ∪ {''x''} # ''L'' := ''L'' − ''sup''({''x''}) In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&#39;, some regular overapproximation to ''L'', until we can show ''L''&#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''₁{''L''₂}''L''₃, where each of ''L''₁, ''L''₂, ''L''₃ is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not. Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''^{''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)}), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]). To solve the Athena problem, we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime > ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2^''n''−2 strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length > (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' < ''b'', i.e. finding the smallest prime of the form ''k''×''b''^''n''+1 and ''k''×''b''^''n''−1 (or prove such prime does not exist) with ''k'' < ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is < ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''^''n''+1 and ''k''×''b''^''n''−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''^''n''−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''^''n''+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''^''n''+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''^''n''+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://oeis.org/A084742, https://oeis.org/A084741; or for bases ''b'' with ''sqrt''(''b'') prime: https://oeis.org/A065507), ((''b''−2)×''b''^''n''+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''^''n''+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''^''n''−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217, https://oeis.org/A119591), ''b''^''n''+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''^''n''−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''^''n''+1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484; or for prime bases ''b'': https://oeis.org/A122396), (''b''−1)×''b''^''n''−1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), ''b''^''n''+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''^''n''−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)): Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''_''n''''z'' for some ''n'' ≥ 0? Some families can be ruled out to contain no prime > ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''⁴+4×''y''⁴]]), or combine of them, e.g. * The base 9 family 2{7}: Always divisible by 2 or 5 * The base 16 family {8}F: Always divisible by 3, 7, or 13 * The base 21 family {7}D: Always divisible by 2, 13, or 17 * The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79 * The base 9 family 3{8}: Can be written as 4×9^''n''−1 and can be factored as (2×3^''n''−1) × (2×3^''n''+1) * The base 8 family 1{0}1: Can be written as 8^''n''+1 and can be factored as (2^''n''+1) × (4^''n''−2^''n''+1) * The base 16 family {4}1: Can be written as (4×16^''n''−49)/15 and can be factored as (2×3^''n''−7) × (2×3^''n''+7) / 15 * The base 16 family {C}D: Can be written as (4×16^''n''+1)/5 and can be factored as (2×4^''n''−2×2^''n''+1) × (2×4^''n''+2×2^''n''+1) / 5 * The base 14 family 8{D}: Can be written as 9×14^''n''−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14^''n''/2−1) × (3×14^''n''/2+1) if ''n'' is even * The base 12 family {B}9B: Can be written as 12^''n''−25, it is divisible by 13 if ''n'' is odd and can be factored as (12^''n''/2−5) × (12^''n''/2+5) if ''n'' is even * The base 17 family 1{9}: Can be written as (25×17^''n''−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17^''n''/2−3) × (5×17^''n''/2+3) / 16 if ''n'' is even * The base 19 family 1{6}: Can be written as (4×19^''n''−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19^''n''/2−1) × (2×19^''n''/2+1) / 3 if ''n'' is even By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime > ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes > ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes > ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3₅₉₂₁₉₇A, its algebraic form is (41×13⁵⁹²¹⁹⁸+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime). == Data == These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes < 10²⁵⁰⁰⁰ for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them > 10²⁹⁹ are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others > 10²⁹⁹ are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]]) All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], "{}" means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123₄567 means 123333567 (all subscripts are written in decimal). Base 2: 1 prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11} Base 3: 3 primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111} Base 4: 5 primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221} Base 5: 22 primes (the largest of which has 96 digits (it is 10₉₃13, and its algebraic form is 5⁹⁵+8)): {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013} Base 6: 11 primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041} Base 7: 71 primes (the largest of which has 17 digits (it is 3₁₆1, and its algebraic form is (7¹⁷−5)/2)): {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331} Base 8: 75 primes (the largest of which has 221 digits (it is 4₂₂₀7, and its algebraic form is (4×8²²¹+17)/7)): {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447} Base 9: 151 primes (the largest of which has 1161 digits (it is 30₁₁₅₈11, and its algebraic form is 3×9¹¹⁶⁰+10)): {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011} Base 10: 77 primes (the largest of which has 31 digits (it is 50₂₈27, and its algebraic form is 5×10³⁰+27)): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} Base 11: 1068 primes (including 1 unproven probable prime: 57₆₂₆₆₈), the largest of which has 62669 digits (it is 57₆₂₆₆₈, and its algebraic form is (57×11⁶²⁶⁶⁸−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena problem base 11] Base 12: 106 primes (the largest of which has 42 digits (it is 40₃₉77, and its algebraic form is 4×12⁴¹+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077} Base 13: 3197 primes (including 4 unproven probable primes: C5₂₃₇₅₅C, 80₃₂₀₁₇111, 95₁₉₇₄₂₀, A3₅₉₂₁₉₇A), the largest of which has 592199 digits (it is A3₅₉₂₁₉₇A, and its algebraic form is (41×13⁵⁹²¹⁹⁸+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena problem base 13] Base 14: 650 primes, the largest of which has 19699 digits (it is 4D₁₉₆₉₈, and its algebraic form is 5×14¹⁹⁶⁹⁸−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena problem base 14] Base 15: 1284 primes, the largest of which has 157 digits (it is 7₁₅₅97, and its algebraic form is (15¹⁵⁷+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena problem base 15] Base 16: 2347 primes (including 3 unproven probable primes: DB₃₂₂₃₄, 4₇₂₇₈₅DD, 3₁₁₆₁₃₇AF), the largest of which has 116139 digits (it is 3₁₁₆₁₃₇AF, and its algebraic form is (16¹¹⁶¹³⁹+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena problem base 16] Base 17: 10415 known primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of Athena problem base 17] Base 18: 549 primes, the largest of which has 6271 digits (it is C0₆₂₆₈C5, and its algebraic form is 12×18⁶²⁷⁰+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena problem base 18] Base 19: 31417 known primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of Athena problem base 19] Base 20: 3314 primes, the largest of which has 6271 digits (it is G0₆₂₆₉D, and its algebraic form is 16×20⁶²⁷⁰+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena problem base 20] Base 21: 13386 known primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of Athena problem base 21] Base 22: 8003 primes (including 1 unproven probable prime: BK₂₂₀₀₁5), the largest of which has 22003 digits (it is BK₂₂₀₀₁5, and its algebraic form is (251×22²²⁰⁰²−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena problem base 22] Base 23: 65178 known primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of Athena problem base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for base 23] Base 24: 3409 primes, the largest of which has 8134 digits (it is N00N₈₁₂₉LN, and its algebraic form is 13249×24⁸¹³¹−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena problem base 24] Base 25: 133639 known primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of Athena problem base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for base 25] Base 26: 25256 known primes (including 7 unproven probable primes: 5₁₉₃₉₁6F, 7₂₀₂₇₉OL, LD0₂₀₉₇₅7, 6K₂₃₃₀₀5, J0₄₄₃₀₃KCB, M0₆₁₁₈₆2BB, 85M₁₉₇₀₆₀B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of Athena problem base 26] Base 27: 102852 known primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of Athena problem base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for base 27] Base 28: 25528 known primes (including 3 unproven probable primes: N6₂₄₀₅₁LR, 5OA₃₁₂₃₈F, O4O₉₄₅₃₅9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of Athena problem base 28] Base 29: 355242 known primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of Athena problem base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for base 29] Base 30: 2619 primes (including 1 unproven probable prime: I0₂₄₆₀₈D), the largest of which has 34206 digits (it is OT₃₄₂₀₅, and its algebraic form is 25×30³⁴²⁰⁵−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena problem base 30] Base 31: 569323 known primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of Athena problem base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for base 31] Base 32: 168882 known primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of Athena problem base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for base 32] Base 33: 280012 known primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of Athena problem base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for base 33] Base 34: 184785 known primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of Athena problem base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for base 34] Base 35: 720002 known primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of Athena problem base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for base 35] Base 36: 35286 known primes (including 3 unproven probable primes: 7K₂₆₅₆₇Z, S0₇₅₀₀₇8H, P₈₁₉₉₃SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of Athena problem base 36] == The fully proof of Athena problem in decimal (base ''b'' = 10) == '''Bold''' for the minimal elements, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''. Assume ''p'' is a prime > 10, and the last digit of ''p'' must lie in {1,3,7,9}. Case 1: ''p'' ends with 1. In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9. Case 1.1: ''p'' begins with 2. In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2. If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's. If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty. If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''. If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''. Case 1.2: ''p'' begins with 5. In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8. If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}. If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''. If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite. Case 1.3: ''p'' begins with 8. In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5. If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}. If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}. If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime. If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite. If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element. Case 1.4: ''p'' begins with 9. In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8. If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8. If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}. If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction). If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801. Case 2: ''p'' ends with 3. In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Case 3: ''p'' ends with 7. In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8. Case 3.1: ''p'' begins with 2. In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8. If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}. If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime. If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40_''n''1 = 280_''n''7. Case 3.2: ''p'' begins with 5. In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2. If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's. If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty. If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''. If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''. Case 3.3: ''p'' begins with 7. In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime. Case 3.4: ''p'' begins with 8. In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime. Case 4: ''p'' ends with 9. In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9. If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's. If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4. Case 4.1: ''p'' begins with 3. In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''. Case 4.2: ''p'' begins with 4. In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6_''n''7 = 46_''n''9. Case 4.3: ''p'' begins with 6. In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty. If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6. If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}. If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes. Case 4.4: ''p'' begins with 9. In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669. [[Category:Number theory]] snz7euq2b5q022bvspteyb6up2bpxw3 0 330397 2819348 2818769 2026-07-25T09:54:00Z Jtneill 10242 2819348 wikitext text/x-wiki {| border=1 cellpadding=5 cellspacing="0" width:100% background:transparent" !'''[http://www.canberra.edu.au/future-students/key-dates/semesters-winter-term-principal-dates Week]''' !'''[[Motivation and emotion/Modules|Module]]''' !'''[[Motivation and emotion/Lectures|Lecture]]''' ![[Motivation and emotion/Readings/Textbooks/Reeve/2024|'''Reading''']] !'''[[Motivation and emotion/Assessment/Quizzes|Quiz]]''' !'''[[Motivation and emotion/Tutorials|Tutorial]]''' !'''[[Motivation and emotion/Assessment|Assessment]]''' |- |01 | rowspan="2" |1 - Introduction |01 - [[Motivation and emotion/Lectures/Introduction|Introduction]] |1 | rowspan="2" |[[Motivation and emotion/About/Outline|UO]], 1 |01 - [[Motivation and emotion/Tutorials/Topic selection|Topic selection]] | - |- |02 |02 - [[Motivation and emotion/Lectures/Historical development and assessment skills|Historical development and assessment skills]] |2 |02 - [[Motivation and emotion/Tutorials/Wiki editing|Wiki editing]] | - |- |03 | rowspan="2" |2 - Needs |03 - [[Motivation and emotion/Lectures/Brain and physiological needs|Brain and physiological needs]] |3, 4 | rowspan="2" |2 |03 - [[Motivation and emotion/Tutorials/Physiological needs|Physiological needs]] | '''[[Motivation and emotion/Assessment/Topic|Topic development]]'''<br />Fri 28/8 9 am |- |04 |04 - [[Motivation and emotion/Lectures/Extrinsic motivation and psychological needs|Extrinsic motivation and psychological needs]] |6, 5 |04 - [[Motivation and emotion/Tutorials/Psychological needs|Psychological needs]] | - |- |05 | rowspan="2" |3 - Goals and self |05 - [[Motivation and emotion/Lectures/Goals and mindsets|Goals and mindsets]] |7, 8 | rowspan="2" |3 |05 - [[Motivation and emotion/Tutorials/Functionalist theory and self-tracking|Functionalist theory and self-tracking]] | - |- |06 |06 - [[Motivation and emotion/Lectures/Personal control and the self|Personal control and the self]] |9, 10 |06 - [[Motivation and emotion/Tutorials/Learned optimism|Learned optimism]] | - |- |07 | rowspan="2" |4 - Emotion |07 - [[Motivation and emotion/Lectures/Nature of emotion|Nature of emotion]] |11 | rowspan="2" |4 |07 - [[Motivation and emotion/Tutorials/Core emotions|Core emotions]] | - |- |08 |08 - [[Motivation and emotion/Lectures/Aspects of emotion|Aspects of emotion]] |12 | 08 - [[Motivation and emotion/Tutorials/Measuring emotion|Measuring emotion]] | - |- |09 | colspan="4" | <div style="text-align: center;">Class-free period</div> | - | - |- |10 | rowspan="2" |5 - Individual emotions |09 - [[Motivation and emotion/Lectures/Individual emotions|Individual emotions]] |13 | rowspan="2" |5 |09 - [[Motivation and emotion/Tutorials/20 emotions|20 emotions]] | '''[[Motivation and emotion/Assessment/Chapter|Book chapter]]'''<br />Mon 12/10 9 am |- |11 |10 - [[Motivation and emotion/Lectures/Unconscious motivation|Unconscious motivation]] |15 |10 - [[Motivation and emotion/Tutorials/Time perspective|Time perspective]] | - |- |12 | rowspan="2" |6 - Growth |11 - [[Motivation and emotion/Lectures/Growth psychology|Growth psychology]] |14 | rowspan="2" |6 |11 - [[Motivation and emotion/Tutorials/Positive psychology|Positive psychology]] | - |- |13 |12 - [[Motivation and emotion/Lectures/Interventions and review|Interventions and review]] |16 |12 - [[Motivation and emotion/Tutorials/Review|Review]] | - |- |14 | - | - | - | - | - | rowspan="2" | '''[[Motivation and emotion/Assessment/Exam|Exam]]'''<br />TBA<!-- Tue 3pm 4/10 --> |- |15 | - | - | - | - | - |} [[Category:{{BASEPAGENAME}}]] bxy6a0si12wn67z1vrj6v57oi88sckj 0 330494 2819262 2818067 2026-07-24T12:49:54Z NDM2024 2984088 2819262 wikitext text/x-wiki '''<big>The John Snow Prediabetes Institute.</big>'''[[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|<big>Prediabetes-remission research program</big>]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]]<big>Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (BMI) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(12+ y).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity. Early diagnosis of prediabetes can provide both health and financial benefits.From a financial perspective, preventing or delaying diabetes can significantly lower healthcare costs. Early diagnosis of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes while helping individuals and healthcare systems avoid the substantial costs associated with diabetes and its complications.</big>[[File:Lifestyle Medicine Pillars.png|300px|right|<big>The focus of Lifestyle Medicine is on these 6 pillars.</big>]] [[File:John Snow.jpg|thumb||100px right|<big>John Snow in the early nineteenth century</big>]] [[File:Cholera in London 1866.gif|thumb|250px|<big>Map of a later cholera outbreak in London, in 1866</big>]] [[File:Choleramaplondon1866.png|thumb|right|250px|<big>Legend for the map above</big>]]<big>1. '<nowiki/>'''Prevalence studies''''</big> <big>1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref></big> <big>1.2 The Maritime Officer`s Health-Database <ref>https://www.dropbox.com/t/8LjP7cmulhr2x8Ty </ref></big> <big>1.3 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref></big> <big>1.4. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref></big> <big>1.5. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref></big> <big><br /> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref></big> <big>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></big> <big>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref></big> <big>- John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref></big> <big>3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></big> <big>4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids` schools.. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained:</big> <big>'''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref></big> <big>6. '''Prediabetes-Remission Research Network:'''</big> <small>Prof. Ing. MSc. Nailet Delgado; Prof. Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali Prof. SDU; MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,, Dr.</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> 7wj3f4w6aiz732cdaponyqfdwkl7ngr 2819265 2819262 2026-07-24T12:55:49Z NDM2024 2984088 2819265 wikitext text/x-wiki '''<big>The John Snow Prediabetes Institute.</big>'''[[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|<big>Prediabetes-remission research program</big>]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]]<big>Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (BMI) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(12+ y).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity. Early diagnosis of prediabetes can provide both health and financial benefits.From a financial perspective, preventing or delaying diabetes can significantly lower healthcare costs. Early diagnosis of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes while helping individuals and healthcare systems avoid the substantial costs associated with diabetes and its complications.</big>[[File:Lifestyle Medicine Pillars.png|300px|right|<big>The focus of Lifestyle Medicine is on these 6 pillars.</big>]] [[File:John Snow.jpg|thumb||100px right|<big>John Snow in the early nineteenth century</big>]] [[File:Cholera in London 1866.gif|thumb|250px|<big>Map of a later cholera outbreak in London, in 1866</big>]] [[File:Choleramaplondon1866.png|thumb|right|250px|<big>Legend for the map above</big>]]<big>1. '<nowiki/>'''Prevalence studies''''</big> <big>1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref></big> <big>1.2 The Maritime Officer`s Health-Database <ref>https://www.dropbox.com/t/8LjP7cmulhr2x8Ty </ref></big> <big>1.3 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref></big> <big>1.4. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref></big> <big>1.5. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref></big> <big><br /> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref></big> <big>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></big> <big>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref></big> <big>- John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref></big> <big>3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></big> <big>4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids` schools. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained:</big> <big>'''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref></big> <big>6. '''Prediabetes-Remission Research Network:'''</big> <small>Prof. Ing. MSc. Nailet Delgado; Prof. Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali Prof. SDU; MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,, Dr.</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> 4h7fnb48y6w0iyajjncv6p7ni3tae7v 2819266 2819265 2026-07-24T12:58:02Z NDM2024 2984088 2819266 wikitext text/x-wiki '''<big>The John Snow Prediabetes Institute.</big>'''[[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|<big>Prediabetes-remission research program</big>]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]]<big>Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (BMI) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(12+ y).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity. Early diagnosis of prediabetes can provide both health and financial benefits.From a financial perspective, preventing or delaying diabetes can significantly lower healthcare costs. Early diagnosis of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes while helping individuals and healthcare systems avoid the substantial costs associated with diabetes and its complications.</big>[[File:Lifestyle Medicine Pillars.png|300px|right|<big>The focus of Lifestyle Medicine is on these 6 pillars.</big>]] [[File:John Snow.jpg|thumb||50px right|<big>John Snow in the early nineteenth century</big>]] [[File:Cholera in London 1866.gif|thumb|250px|<big>Map of a later cholera outbreak in London, in 1866</big>]] [[File:Choleramaplondon1866.png|thumb|right|250px|<big>Legend for the map above</big>]]<big>1. '<nowiki/>'''Prevalence studies''''</big> <big>1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref></big> <big>1.2 The Maritime Officer`s Health-Database <ref>https://www.dropbox.com/t/8LjP7cmulhr2x8Ty </ref></big> <big>1.3 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref></big> <big>1.4. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref></big> <big>1.5. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref></big> <big><br /> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref></big> <big>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></big> <big>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref></big> <big>- John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref></big> <big>3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></big> <big>4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids` schools. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained:</big> <big>'''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref></big> <big>6. '''Prediabetes-Remission Research Network:'''</big> <small>Prof. Ing. MSc. Nailet Delgado; Prof. Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali Prof. SDU; MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,, Dr.</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> glz04jblpoddiqd5d7ev1ty5m4c2gxk 2819283 2819266 2026-07-24T15:46:27Z NDM2024 2984088 2819283 wikitext text/x-wiki '''<big>The John Snow Prediabetes Institute.</big>'''[[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|<big>Prediabetes-remission research program</big>]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]]<big>Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (BMI) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(12+ y).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity. Early diagnosis of prediabetes can provide both health and financial benefits.From a financial perspective, preventing or delaying diabetes can significantly lower healthcare costs. Early diagnosis of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes while helping individuals and healthcare systems avoid the substantial costs associated with diabetes and its complications.</big>[[File:Lifestyle Medicine Pillars.png|300px|right|<big>The focus of Lifestyle Medicine is on these 6 pillars.</big>]] [[File:John Snow.jpg|thumb||50px right|John Snow in the early nineteenth century</]] [[File:Cholera in London 1866.gif|thumb|250px|<big>Map of a later cholera outbreak in London, in 1866</big>]] [[File:Choleramaplondon1866.png|thumb|right|250px|<big>Legend for the map above</big>]]<big>1. '<nowiki/>'''Prevalence studies''''</big> <big>1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref></big> <big>1.2 The Maritime Officer`s Health-Database <ref>https://www.dropbox.com/t/8LjP7cmulhr2x8Ty </ref></big> <big>1.3 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref></big> <big>1.4. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref></big> <big>1.5. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref></big> <big><br /> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref></big> <big>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></big> <big>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref></big> <big>- John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref></big> <big>3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></big> <big>4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids` schools. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained:</big> <big>'''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref></big> <big>6. '''Prediabetes-Remission Research Network:'''</big> <small>Prof. Ing. MSc. Nailet Delgado; Prof. Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali Prof. SDU; MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,, Dr.</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> rok0kiz6n1c0vmd5hejrfgaegs1eyyi 2819284 2819283 2026-07-24T15:47:25Z NDM2024 2984088 2819284 wikitext text/x-wiki '''<big>The John Snow Prediabetes Institute.</big>'''[[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|<big>Prediabetes-remission research program</big>]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]]<big>Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (BMI) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(12+ y).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity. Early diagnosis of prediabetes can provide both health and financial benefits.From a financial perspective, preventing or delaying diabetes can significantly lower healthcare costs. Early diagnosis of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes while helping individuals and healthcare systems avoid the substantial costs associated with diabetes and its complications.</big>[[File:Lifestyle Medicine Pillars.png|300px|right|<big>The focus of Lifestyle Medicine is on these 6 pillars.</big>]] [[File:John Snow.jpg|thumb||50px right|John Snow in the early nineteenth century</]] [[File:Cholera in London 1866.gif|thumb|250px|<big>Map of a later cholera outbreak in London, in 1866</big>]] [[File:Choleramaplondon1866.png|thumb|right|250px|<big>Legend for the map above</big>]]<big>1. '<nowiki/>'''Prevalence studies''''</big> <big>1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref></big> <big>1.2 The Maritime Officer`s Health-Database <ref>https://www.dropbox.com/t/8LjP7cmulhr2x8Ty </ref></big> <big>1.3 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref></big> <big>1.4. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref></big> <big>1.5. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref></big> <big><br /> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref></big> <big>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></big> <big>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref></big> <big>- John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref></big> <big>3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></big> <big>4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids` schools. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained:</big> <big>'''5. Minutes from meetings'''</big> <big>6. '''Prediabetes-Remission Research Network:'''</big> <small>Prof. Ing. MSc. Nailet Delgado; Prof. Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali Prof. SDU; MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,, Dr.</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> 4u9mk2oevgi3lr2bcl13k0yzj29suur 2819285 2819284 2026-07-24T15:48:11Z NDM2024 2984088 2819285 wikitext text/x-wiki '''<big>The John Snow Prediabetes Institute.</big>'''[[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|<big>Prediabetes-remission research program</big>]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]]<big>Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (BMI) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(12+ y).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity. Early diagnosis of prediabetes can provide both health and financial benefits.From a financial perspective, preventing or delaying diabetes can significantly lower healthcare costs. Early diagnosis of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes while helping individuals and healthcare systems avoid the substantial costs associated with diabetes and its complications.</big>[[File:Lifestyle Medicine Pillars.png|300px|right|<big>The focus of Lifestyle Medicine is on these 6 pillars.</big>]] [[File:John Snow.jpg||50px right|John Snow in the early nineteenth century</]] [[File:Cholera in London 1866.gif|thumb|250px|<big>Map of a later cholera outbreak in London, in 1866</big>]] [[File:Choleramaplondon1866.png|thumb|right|250px|<big>Legend for the map above</big>]]<big>1. '<nowiki/>'''Prevalence studies''''</big> <big>1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref></big> <big>1.2 The Maritime Officer`s Health-Database <ref>https://www.dropbox.com/t/8LjP7cmulhr2x8Ty </ref></big> <big>1.3 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref></big> <big>1.4. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref></big> <big>1.5. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref></big> <big><br /> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref></big> <big>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></big> <big>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref></big> <big>- John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref></big> <big>3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></big> <big>4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids` schools. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained:</big> <big>'''5. Minutes from meetings'''</big> <big>6. '''Prediabetes-Remission Research Network:'''</big> <small>Prof. Ing. MSc. Nailet Delgado; Prof. Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali Prof. SDU; MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,, Dr.</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> rafo4ffqstv8i6u2w8hkvc22bbwtryl 2819286 2819285 2026-07-24T15:48:47Z NDM2024 2984088 2819286 wikitext text/x-wiki '''<big>The John Snow Prediabetes Institute.</big>'''[[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|<big>Prediabetes-remission research program</big>]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]]<big>Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (BMI) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(12+ y).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity. Early diagnosis of prediabetes can provide both health and financial benefits.From a financial perspective, preventing or delaying diabetes can significantly lower healthcare costs. Early diagnosis of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes while helping individuals and healthcare systems avoid the substantial costs associated with diabetes and its complications.</big>[[File:Lifestyle Medicine Pillars.png|300px|right|<big>The focus of Lifestyle Medicine is on these 6 pillars.</big>]] [[File:Cholera in London 1866.gif|thumb|250px|<big>Map of a later cholera outbreak in London, in 1866</big>]] [[File:Choleramaplondon1866.png|thumb|right|250px|<big>Legend for the map above</big>]]<big>1. '<nowiki/>'''Prevalence studies''''</big> <big>1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref></big> <big>1.2 The Maritime Officer`s Health-Database <ref>https://www.dropbox.com/t/8LjP7cmulhr2x8Ty </ref></big> <big>1.3 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref></big> <big>1.4. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref></big> <big>1.5. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref></big> <big><br /> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref></big> <big>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></big> <big>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref></big> <big>- John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref></big> <big>3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></big> <big>4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids` schools. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained:</big> <big>'''5. Minutes from meetings'''</big> <big>6. '''Prediabetes-Remission Research Network:'''</big> <small>Prof. Ing. MSc. Nailet Delgado; Prof. Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali Prof. SDU; MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,, Dr.</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> g60rl8lz4rhopd2ox0jgn9yvn7mrnh9 2819287 2819286 2026-07-24T15:58:55Z NDM2024 2984088 2819287 wikitext text/x-wiki '''<big>The John Snow Prediabetes Institute.</big>'''[[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|<big>Prediabetes-remission research program</big>]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]][[The John Snow Prediabetes Institute|https://en.wikiversity.org/wiki/The_John_Snow_Prediabetes_Institute]] <big>Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (BMI) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(12+ y).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity. Early diagnosis of prediabetes can provide both health and financial benefits.From a financial perspective, preventing or delaying diabetes can significantly lower healthcare costs. Early diagnosis of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes while helping individuals and healthcare systems avoid the substantial costs associated with diabetes and its complications.</big>[[File:Lifestyle Medicine Pillars.png|300px|right|<big>The focus of Lifestyle Medicine is on these 6 pillars.</big>]] [[File:Cholera in London 1866.gif|thumb|250px|<big>Map of a later cholera outbreak in London, in 1866</big>]] [[File:Choleramaplondon1866.png|thumb|right|250px|<big>Legend for the map above</big>]]<big>1. '<nowiki/>'''Prevalence studies''''</big> <big>1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref></big> <big>1.2 The Maritime Officer`s Health-Database <ref>https://www.dropbox.com/t/8LjP7cmulhr2x8Ty </ref></big> <big>1.3 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref></big> <big>1.4. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref></big> <big>1.5. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref></big> <big><br /> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref></big> <big>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></big> <big>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref></big> <big>- John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref></big> <big>3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></big> <big>4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids` schools. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained:</big> <big>'''5. Minutes from meetings'''</big> <big>6. '''Prediabetes-Remission Research Network:'''</big> <small>Prof. Ing. MSc. Nailet Delgado; Prof. Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali Prof. SDU; MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,, Dr.</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> astaxecmmpua06q4es9omz7zbmb4siz 0 330534 2819309 2818033 2026-07-24T18:23:31Z Howie2024 2995240 /* Probability Dilation Theory / Dilation Vector Field */ 2819309 wikitext text/x-wiki == Introduction == Probability Dilation Theory (PDT) is naturally formulated as an iterative transformation on probability measures. When the dilation operator depends smoothly on a parameter, the resulting evolution defines a vector field on the manifold of probability distributions. This vector field provides a differential-geometric description of PDT and forms the basis for studying probability flows, stability, and continuous-time dynamics. == Smooth Dilation Families == Let <math> T_\lambda </math> denote a family of probability dilation operators depending smoothly on a parameter <math> \lambda. </math> The corresponding probability distribution is <math> P_\lambda=T_\lambda(P). </math> As <math> \lambda </math> varies, the distributions trace a curve on the statistical manifold. == Definition: Dilation Vector Field == The '''dilation vector field''' is defined by <math> V_P = \left. \frac{d}{d\lambda} T_\lambda(P) \right|_{\lambda=0}. </math> This vector represents the instantaneous direction in which the probability measure evolves under infinitesimal dilation. == Exponential Dilation == Suppose the dilation field is written as <math> D_\lambda(x) = e^{\lambda\phi(x)}, </math> where <math> \phi(x) </math> is a real-valued generator. The corresponding PDT transformation is <math> P_\lambda(x) = \frac{ e^{\lambda\phi(x)}P(x) } { \int_\Omega e^{\lambda\phi(x)} \,dP(x) }. </math> Differentiating with respect to <math> \lambda </math> at <math> \lambda=0 </math> gives <math> V_P(x) = \left( \phi(x) - \mathbb E_P[\phi] \right) P(x). </math> Thus the direction of probability flow depends upon the deviation of <math> \phi(x) </math> from its expectation. == Normalization == The subtraction of the expectation <math> \mathbb E_P[\phi] </math> ensures that <math> \int_\Omega V_P\,dP=0, </math> so that probability normalization is preserved throughout the flow. == Fisher Geometry == When the statistical manifold is equipped with the Fisher–Rao metric, the dilation vector field becomes a tangent vector on a Riemannian manifold. Its magnitude is given by <math> \|V_P\|_{FR} = \sqrt{ \langle V_P, V_P \rangle_{FR} }. </math> This provides an intrinsic measure of the speed of probability evolution under PDT. == Fixed Points == A probability distribution <math> P^* </math> is a fixed point of PDT if <math> T_D(P^*) = P^*. </math> Equivalently, <math> V_{P^*} = 0. </math> Thus fixed points correspond to equilibrium points of the dilation vector field. == Interpretation == The dilation vector field transforms PDT from an iterative procedure into a geometric dynamical system. Rather than viewing PDT as a sequence of discrete probability updates, one may regard it as generating continuous trajectories on the manifold of probability measures. This interpretation provides a natural framework for studying stability, convergence, geodesics, and gradient flows. == See Also == * [[Probability Dilation Theory]] * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows]] * [[Probability Dilation Theory/Euler Methods and Continuous-Time PDT]] * [[Probability Dilation Theory/Convergence and Fixed Points]] * [[Probability Dilation Theory/Measure-Theoretic Foundations]] 8eqacg13sl5wnzwzphs1rb115oahu99 0 330664 2819298 2819168 2026-07-24T17:48:08Z Howie2024 2995240 minor edit 2819298 wikitext text/x-wiki == Overview == This page explores a natural mathematical generalization of the Probability Dilation Theory (PDT) operator by replacing the diagonal dilation matrix with a general nonnegative matrix. The resulting framework allows interactions between probability states and extends localized probability dilation to coupled probability evolution. This page is exploratory and proposes a possible extension of PDT rather than a replacement for the core theory. == Motivation == The standard PDT transformation applies a positive diagonal dilation operator <math> T_D(P)=\frac{DP}{Z(P,D)}, </math> where <math> Z(P,D)=\sum_i D_iP_i, </math> and <math> D= \begin{pmatrix} d_1&0&\cdots&0\\ 0&d_2&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&d_n \end{pmatrix}. </math> Each probability component is independently reweighted before normalization. A natural question is whether the diagonal operator can be generalized to a full matrix while preserving positivity and normalization. == Generalized Matrix Operator == Consider a nonnegative matrix <math> M= \begin{pmatrix} d_1&a_{12}&\cdots&a_{1n}\\ a_{21}&d_2&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\cdots&d_n \end{pmatrix}, </math> where <math> M_{ij}\ge0. </math> The generalized PDT operator is defined by <math> T_M(P) = \frac{MP}{\|MP\|_1}, </math> provided that <math> MP </math> has nonnegative components. Normalization ensures that <math> \sum_i T_M(P)_i=1. </math> == Interpretation == The diagonal entries <math> d_i </math> represent localized probability dilation identical to standard PDT. The off-diagonal entries <math> a_{ij} </math> allow probability associated with one state to influence another. Consequently, * diagonal PDT represents independent local amplification; * matrix PDT represents coupled probability evolution. == Relation to Standard PDT == When all off-diagonal entries vanish, <math> a_{ij}=0,\qquad i\ne j, </math> the generalized operator reduces exactly to the original PDT operator. Thus diagonal PDT is a special case of matrix PDT. == Connections to Existing Mathematics == The generalized formulation exhibits similarities to several established mathematical frameworks, including * positive matrices, * Markov operators, * Perron–Frobenius theory, * coupled dynamical systems, * graph-based diffusion, * linear operator theory. These connections are mathematical analogies and are not claimed to be equivalent formulations. == Potential Properties == Questions for future investigation include: * preservation of positivity, * convergence of repeated iterations, * existence and uniqueness of fixed points, * entropy evolution, * stability of coupled systems, * spectral properties of the dilation matrix, * relation to Perron–Frobenius eigenvectors, * continuous-time matrix dilation flows. == Discussion == Matrix Dilation Operators extend the localized dilation concept of PDT by allowing interactions between probability states. This generalization significantly enlarges the mathematical scope of PDT while preserving the original theory as the diagonal special case. Whether this extension provides useful applications or deeper theoretical insights remains an open research question. == See also == * [[Probability Dilation Theory]] * [[Probability Dilation Theory/Measure-Theoretic Foundations]] * [[Probability Dilation Theory/Dilation Vector Field]] * [[Probability Dilation Theory/Dilation Flows]] * [[Probability Dilation Theory/Stochastic Dilation Fields]] * [[Probability Dilation Theory/Convergence and Fixed Points]] * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows]] == References == * Howard Richardson, ''Probability Dilation Theory'', Wikiversity. * R. A. Horn and C. R. Johnson, ''Matrix Analysis''. * F. R. Gantmacher, ''The Theory of Matrices''. * O. Perron (1907); G. Frobenius (1912), classical papers on positive matrices. cbwbyavlwq7v9gbfb3x9zrxy6c05bgy 2819311 2819298 2026-07-24T18:46:39Z Howie2024 2995240 Small refinement to flow of article 2819311 wikitext text/x-wiki == Overview == This page explores a natural mathematical generalization of the Probability Dilation Theory (PDT) operator by replacing the diagonal dilation matrix with a general nonnegative matrix. The resulting framework allows interactions between probability states and extends localized probability dilation to coupled probability evolution. This page is exploratory and proposes a possible extension of PDT rather than a replacement for the core theory. == Motivation == The standard PDT transformation applies a positive diagonal dilation operator <math> T_D(P)=\frac{DP}{Z(P,D)}, </math> where <math> Z(P,D)=\sum_i D_iP_i, </math> and <math> D= \begin{pmatrix} d_1&0&\cdots&0\\ 0&d_2&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&d_n \end{pmatrix}. </math> Each probability component is independently reweighted before normalization. A natural question is whether the diagonal operator can be generalized to a full matrix while preserving positivity and normalization. == Generalized Matrix Operator == Consider a nonnegative matrix <math> M= \begin{pmatrix} d_1&a_{12}&\cdots&a_{1n}\\ a_{21}&d_2&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\cdots&d_n \end{pmatrix}, </math> where <math> M_{ij}\ge0. </math> The generalized PDT operator is defined by <math> T_M(P) = \frac{MP}{\|MP\|_1}, </math> provided that <math> MP </math> has nonnegative components. The following theorem establishes that the Matrix Dilation Operator is well defined on the probability simplex. == Fundamental Properties == === Theorem 1 (Well-Definedness of the Matrix Dilation Operator) === Let <math> P=(P_1,\ldots,P_n) </math> be a probability vector satisfying <math> P_i\ge0, \qquad \sum_{i=1}^{n}P_i=1, </math> and let <math> M </math> be a non-negative matrix satisfying <math> MP\neq0. </math> Then the Matrix Dilation Operator <math> T_M(P) = \frac{MP}{\|MP\|_1} </math> is itself a probability vector. ==== Proof ==== Since <math> M_{ij}\ge0 </math> and <math> P_j\ge0, </math> every component of <math> MP </math> is non-negative. Therefore, <math> (T_M(P))_i = \frac{(MP)_i}{\|MP\|_1} \ge0. </math> Furthermore, <math> \sum_{i=1}^{n}(T_M(P))_i = \frac{\sum_i(MP)_i} {\|MP\|_1}. </math> Since the <math>\ell^1</math>-norm of a non-negative vector equals the sum of its components, <math> \|MP\|_1 = \sum_i(MP)_i, </math> it follows that <math> \sum_{i=1}^{n}(T_M(P))_i = 1. </math> Hence <math> T_M(P) </math> is a probability vector. <math>\square</math> === Corollary 1 (Self-Map of the Probability Simplex) === Let <math> \Delta^{n-1} = \left\{ P\in\mathbb{R}^n: P_i\ge0,\; \sum_{i=1}^{n}P_i=1 \right\} </math> denote the probability simplex. Under the assumptions of Theorem&nbsp;1, <math> T_M:\Delta^{n-1}\rightarrow\Delta^{n-1}. </math> Consequently, every iteration of Matrix Probability Dilation Theory remains within the probability simplex, ensuring that repeated matrix dilation generates a well-defined sequence of probability distributions. == Interpretation == The diagonal entries <math> d_i </math> represent localized probability dilation identical to standard PDT. The off-diagonal entries <math> a_{ij} </math> allow probability associated with one state to influence another. Consequently, * diagonal PDT represents independent local amplification; * matrix PDT represents coupled probability evolution. == Relation to Standard PDT == When all off-diagonal entries vanish, <math> a_{ij}=0,\qquad i\ne j, </math> the generalized operator reduces exactly to the original PDT operator. Thus diagonal PDT is a special case of matrix PDT. == Connections to Existing Mathematics == The generalized formulation exhibits similarities to several established mathematical frameworks, including * positive matrices, * Markov operators, * Perron–Frobenius theory, * coupled dynamical systems, * graph-based diffusion, * linear operator theory. These connections are mathematical analogies and are not claimed to be equivalent formulations. == Potential Properties == Questions for future investigation include: * preservation of positivity, * convergence of repeated iterations, * existence and uniqueness of fixed points, * entropy evolution, * stability of coupled systems, * spectral properties of the dilation matrix, * relation to Perron–Frobenius eigenvectors, * continuous-time matrix dilation flows. == Discussion == Matrix Dilation Operators extend the localized dilation concept of PDT by allowing interactions between probability states. This generalization significantly enlarges the mathematical scope of PDT while preserving the original theory as the diagonal special case. Whether this extension provides useful applications or deeper theoretical insights remains an open research question. == See also == * [[Probability Dilation Theory]] * [[Probability Dilation Theory/Measure-Theoretic Foundations]] * [[Probability Dilation Theory/Dilation Vector Field]] * [[Probability Dilation Theory/Dilation Flows]] * [[Probability Dilation Theory/Stochastic Dilation Fields]] * [[Probability Dilation Theory/Convergence and Fixed Points]] * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows]] == References == * Howard Richardson, ''Probability Dilation Theory'', Wikiversity. * R. A. Horn and C. R. Johnson, ''Matrix Analysis''. * F. R. Gantmacher, ''The Theory of Matrices''. * O. Perron (1907); G. Frobenius (1912), classical papers on positive matrices. 0vs9eesjuaob2auwmyrygc17sxwgzxs 2819321 2819311 2026-07-24T20:37:47Z Howie2024 2995240 Proposition 2, Continuity, Fixed Points 2819321 wikitext text/x-wiki == Overview == This page explores a natural mathematical generalization of the Probability Dilation Theory (PDT) operator by replacing the diagonal dilation matrix with a general nonnegative matrix. The resulting framework allows interactions between probability states and extends localized probability dilation to coupled probability evolution. This page is exploratory and proposes a possible extension of PDT rather than a replacement for the core theory. == Motivation == The standard PDT transformation applies a positive diagonal dilation operator <math> T_D(P)=\frac{DP}{Z(P,D)}, </math> where <math> Z(P,D)=\sum_i D_iP_i, </math> and <math> D= \begin{pmatrix} d_1&0&\cdots&0\\ 0&d_2&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&d_n \end{pmatrix}. </math> Each probability component is independently reweighted before normalization. A natural question is whether the diagonal operator can be generalized to a full matrix while preserving positivity and normalization. == Generalized Matrix Operator == Consider a nonnegative matrix <math> M= \begin{pmatrix} d_1&a_{12}&\cdots&a_{1n}\\ a_{21}&d_2&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\cdots&d_n \end{pmatrix}, </math> where <math> M_{ij}\ge0. </math> The generalized PDT operator is defined by <math> T_M(P) = \frac{MP}{\|MP\|_1}, </math> provided that <math> MP </math> has nonnegative components. The following theorem establishes that the Matrix Dilation Operator is well defined on the probability simplex. == Fundamental Properties == === Theorem 1 (Well-Definedness of the Matrix Dilation Operator) === Let <math> P=(P_1,\ldots,P_n) </math> be a probability vector satisfying <math> P_i\ge0, \qquad \sum_{i=1}^{n}P_i=1, </math> and let <math> M </math> be a non-negative matrix satisfying <math> MP\neq0. </math> Then the Matrix Dilation Operator <math> T_M(P) = \frac{MP}{\|MP\|_1} </math> is itself a probability vector. ==== Proof ==== Since <math> M_{ij}\ge0 </math> and <math> P_j\ge0, </math> every component of <math> MP </math> is non-negative. Therefore, <math> (T_M(P))_i = \frac{(MP)_i}{\|MP\|_1} \ge0. </math> Furthermore, <math> \sum_{i=1}^{n}(T_M(P))_i = \frac{\sum_i(MP)_i} {\|MP\|_1}. </math> Since the <math>\ell^1</math>-norm of a non-negative vector equals the sum of its components, <math> \|MP\|_1 = \sum_i(MP)_i, </math> it follows that <math> \sum_{i=1}^{n}(T_M(P))_i = 1. </math> Hence <math> T_M(P) </math> is a probability vector. <math>\square</math> === Corollary 1 (Self-Map of the Probability Simplex) === Let <math> \Delta^{n-1} = \left\{ P\in\mathbb{R}^n: P_i\ge0,\; \sum_{i=1}^{n}P_i=1 \right\} </math> denote the probability simplex. Under the assumptions of Theorem&nbsp;1, <math> T_M:\Delta^{n-1}\rightarrow\Delta^{n-1}. </math> Consequently, every iteration of Matrix Probability Dilation Theory remains within the probability simplex, ensuring that repeated matrix dilation generates a well-defined sequence of probability distributions. === Proposition 2 (Reduction to Diagonal PDT) === Let \[ M=\operatorname{diag}(D_1,D_2,\ldots,D_n) \] be a diagonal matrix with nonnegative diagonal entries, and let \[ P= \begin{pmatrix} P_1\\ P_2\\ \vdots\\ P_n \end{pmatrix} \in \Delta^{n-1}. \] Then the matrix dilation operator \[ T_M(P) = \frac{MP}{\|MP\|_1} \] reduces exactly to the standard finite-dimensional PDT operator \[ T_D(P)_i = \frac{D_iP_i} {\sum_{j=1}^{n}D_jP_j}. \] ==== Proof ==== Since \(M\) is diagonal, \[ MP = \begin{pmatrix} D_1P_1\\ D_2P_2\\ \vdots\\ D_nP_n \end{pmatrix}. \] Because all entries are nonnegative, \[ \|MP\|_1 = \sum_{j=1}^{n}D_jP_j. \] Therefore, \[ T_M(P) = \frac{1} {\sum_{j=1}^{n}D_jP_j} \begin{pmatrix} D_1P_1\\ D_2P_2\\ \vdots\\ D_nP_n \end{pmatrix}. \] Hence, for each component \(i\), \[ T_M(P)_i = \frac{D_iP_i} {\sum_{j=1}^{n}D_jP_j}. \] This is precisely the standard PDT update rule. Therefore, diagonal PDT is a special case of Matrix PDT. \(\square\) ==== Significance ==== Proposition 2 shows that Matrix PDT is a genuine extension of the original theory rather than a separate construction. The diagonal entries reproduce local probability dilation, while off-diagonal entries introduce coupling between states. === Theorem 2 (Continuity of the Matrix Dilation Operator) === Let \[ M\in\mathbb{R}^{n\times n} \] be a nonnegative matrix, and let \[ T_M(P) = \frac{MP}{\|MP\|_1} \] be defined for all probability vectors \[ P\in\Delta^{n-1} \] such that \[ MP\neq0. \] Then the map \[ T_M:\Delta^{n-1}\rightarrow\Delta^{n-1} \] is continuous on its domain. ==== Proof ==== Matrix multiplication is a linear operation and is therefore continuous. The function \[ P\mapsto \|MP\|_1 \] is also continuous, since the 1-norm is a continuous function. Because \[ MP\neq0, \] the denominator \[ \|MP\|_1>0, \] so division by this quantity is continuous. Therefore, \[ T_M(P) = \frac{MP}{\|MP\|_1} \] is the quotient of continuous functions with a strictly positive denominator. Hence \[ T_M \] is continuous on its domain. \(\square\) ==== Significance ==== Continuity guarantees that small changes in a probability distribution produce correspondingly small changes in the normalized matrix dilation. This property provides the analytical foundation for studying fixed points, convergence, stability, and dynamical behavior under repeated matrix dilation. === Theorem 3 (Fixed Points and Eigenvectors) === Let \[ M\in\mathbb{R}^{n\times n} \] be a nonnegative matrix, and let \[ T_M(P) = \frac{MP}{\|MP\|_1} \] be the matrix dilation operator. If \[ P\in\Delta^{n-1} \] is a nonnegative eigenvector of \(M\), that is, \[ MP=\lambda P \] for some eigenvalue \[ \lambda>0, \] then \[ P \] is a fixed point of the matrix dilation operator: \[ T_M(P)=P. \] ==== Proof ==== Suppose \[ MP=\lambda P, \] where \[ \lambda>0. \] Since \[ P\in\Delta^{n-1}, \] its components sum to one. Therefore, \[ \|MP\|_1 = \|\lambda P\|_1 = \lambda. \] Hence, \[ T_M(P) = \frac{MP}{\|MP\|_1} = \frac{\lambda P}{\lambda} = P. \] Therefore, \[ P \] is a fixed point of the matrix dilation operator. \(\square\) ==== Significance ==== Theorem 3 establishes a direct connection between Matrix PDT and classical spectral theory. Every nonnegative eigenvector of the dilation matrix, after normalization to a probability vector, is a stationary probability distribution under repeated matrix dilation. This result links Matrix PDT to the Perron–Frobenius theory of positive matrices and provides the foundation for studying equilibrium probability distributions and long-term iterative behavior. == Interpretation == The diagonal entries <math> d_i </math> represent localized probability dilation identical to standard PDT. The off-diagonal entries <math> a_{ij} </math> allow probability associated with one state to influence another. Consequently, * diagonal PDT represents independent local amplification; * matrix PDT represents coupled probability evolution. == Relation to Standard PDT == When all off-diagonal entries vanish, <math> a_{ij}=0,\qquad i\ne j, </math> the generalized operator reduces exactly to the original PDT operator. Thus diagonal PDT is a special case of matrix PDT. == Connections to Existing Mathematics == The generalized formulation exhibits similarities to several established mathematical frameworks, including * positive matrices, * Markov operators, * Perron–Frobenius theory, * coupled dynamical systems, * graph-based diffusion, * linear operator theory. These connections are mathematical analogies and are not claimed to be equivalent formulations. == Potential Properties == Questions for future investigation include: * preservation of positivity, * convergence of repeated iterations, * existence and uniqueness of fixed points, * entropy evolution, * stability of coupled systems, * spectral properties of the dilation matrix, * relation to Perron–Frobenius eigenvectors, * continuous-time matrix dilation flows. == Discussion == Matrix Dilation Operators extend the localized dilation concept of PDT by allowing interactions between probability states. This generalization significantly enlarges the mathematical scope of PDT while preserving the original theory as the diagonal special case. Whether this extension provides useful applications or deeper theoretical insights remains an open research question. == See also == * [[Probability Dilation Theory]] * [[Probability Dilation Theory/Measure-Theoretic Foundations]] * [[Probability Dilation Theory/Dilation Vector Field]] * [[Probability Dilation Theory/Dilation Flows]] * [[Probability Dilation Theory/Stochastic Dilation Fields]] * [[Probability Dilation Theory/Convergence and Fixed Points]] * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows]] == References == * Howard Richardson, ''Probability Dilation Theory'', Wikiversity. * R. A. Horn and C. R. Johnson, ''Matrix Analysis''. * F. R. Gantmacher, ''The Theory of Matrices''. * O. Perron (1907); G. Frobenius (1912), classical papers on positive matrices. q64jga291a4snkq3kufx5fuj0bnxgjv 2819322 2819321 2026-07-24T20:41:55Z Howie2024 2995240 Theorem 4 Convergence, 5 Entropy 2819322 wikitext text/x-wiki == Overview == This page explores a natural mathematical generalization of the Probability Dilation Theory (PDT) operator by replacing the diagonal dilation matrix with a general nonnegative matrix. The resulting framework allows interactions between probability states and extends localized probability dilation to coupled probability evolution. This page is exploratory and proposes a possible extension of PDT rather than a replacement for the core theory. == Motivation == The standard PDT transformation applies a positive diagonal dilation operator <math> T_D(P)=\frac{DP}{Z(P,D)}, </math> where <math> Z(P,D)=\sum_i D_iP_i, </math> and <math> D= \begin{pmatrix} d_1&0&\cdots&0\\ 0&d_2&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&d_n \end{pmatrix}. </math> Each probability component is independently reweighted before normalization. A natural question is whether the diagonal operator can be generalized to a full matrix while preserving positivity and normalization. == Generalized Matrix Operator == Consider a nonnegative matrix <math> M= \begin{pmatrix} d_1&a_{12}&\cdots&a_{1n}\\ a_{21}&d_2&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\cdots&d_n \end{pmatrix}, </math> where <math> M_{ij}\ge0. </math> The generalized PDT operator is defined by <math> T_M(P) = \frac{MP}{\|MP\|_1}, </math> provided that <math> MP </math> has nonnegative components. The following theorem establishes that the Matrix Dilation Operator is well defined on the probability simplex. == Fundamental Properties == === Theorem 1 (Well-Definedness of the Matrix Dilation Operator) === Let <math> P=(P_1,\ldots,P_n) </math> be a probability vector satisfying <math> P_i\ge0, \qquad \sum_{i=1}^{n}P_i=1, </math> and let <math> M </math> be a non-negative matrix satisfying <math> MP\neq0. </math> Then the Matrix Dilation Operator <math> T_M(P) = \frac{MP}{\|MP\|_1} </math> is itself a probability vector. ==== Proof ==== Since <math> M_{ij}\ge0 </math> and <math> P_j\ge0, </math> every component of <math> MP </math> is non-negative. Therefore, <math> (T_M(P))_i = \frac{(MP)_i}{\|MP\|_1} \ge0. </math> Furthermore, <math> \sum_{i=1}^{n}(T_M(P))_i = \frac{\sum_i(MP)_i} {\|MP\|_1}. </math> Since the <math>\ell^1</math>-norm of a non-negative vector equals the sum of its components, <math> \|MP\|_1 = \sum_i(MP)_i, </math> it follows that <math> \sum_{i=1}^{n}(T_M(P))_i = 1. </math> Hence <math> T_M(P) </math> is a probability vector. <math>\square</math> === Corollary 1 (Self-Map of the Probability Simplex) === Let <math> \Delta^{n-1} = \left\{ P\in\mathbb{R}^n: P_i\ge0,\; \sum_{i=1}^{n}P_i=1 \right\} </math> denote the probability simplex. Under the assumptions of Theorem&nbsp;1, <math> T_M:\Delta^{n-1}\rightarrow\Delta^{n-1}. </math> Consequently, every iteration of Matrix Probability Dilation Theory remains within the probability simplex, ensuring that repeated matrix dilation generates a well-defined sequence of probability distributions. === Proposition 2 (Reduction to Diagonal PDT) === Let \[ M=\operatorname{diag}(D_1,D_2,\ldots,D_n) \] be a diagonal matrix with nonnegative diagonal entries, and let \[ P= \begin{pmatrix} P_1\\ P_2\\ \vdots\\ P_n \end{pmatrix} \in \Delta^{n-1}. \] Then the matrix dilation operator \[ T_M(P) = \frac{MP}{\|MP\|_1} \] reduces exactly to the standard finite-dimensional PDT operator \[ T_D(P)_i = \frac{D_iP_i} {\sum_{j=1}^{n}D_jP_j}. \] ==== Proof ==== Since \(M\) is diagonal, \[ MP = \begin{pmatrix} D_1P_1\\ D_2P_2\\ \vdots\\ D_nP_n \end{pmatrix}. \] Because all entries are nonnegative, \[ \|MP\|_1 = \sum_{j=1}^{n}D_jP_j. \] Therefore, \[ T_M(P) = \frac{1} {\sum_{j=1}^{n}D_jP_j} \begin{pmatrix} D_1P_1\\ D_2P_2\\ \vdots\\ D_nP_n \end{pmatrix}. \] Hence, for each component \(i\), \[ T_M(P)_i = \frac{D_iP_i} {\sum_{j=1}^{n}D_jP_j}. \] This is precisely the standard PDT update rule. Therefore, diagonal PDT is a special case of Matrix PDT. \(\square\) ==== Significance ==== Proposition 2 shows that Matrix PDT is a genuine extension of the original theory rather than a separate construction. The diagonal entries reproduce local probability dilation, while off-diagonal entries introduce coupling between states. === Theorem 2 (Continuity of the Matrix Dilation Operator) === Let \[ M\in\mathbb{R}^{n\times n} \] be a nonnegative matrix, and let \[ T_M(P) = \frac{MP}{\|MP\|_1} \] be defined for all probability vectors \[ P\in\Delta^{n-1} \] such that \[ MP\neq0. \] Then the map \[ T_M:\Delta^{n-1}\rightarrow\Delta^{n-1} \] is continuous on its domain. ==== Proof ==== Matrix multiplication is a linear operation and is therefore continuous. The function \[ P\mapsto \|MP\|_1 \] is also continuous, since the 1-norm is a continuous function. Because \[ MP\neq0, \] the denominator \[ \|MP\|_1>0, \] so division by this quantity is continuous. Therefore, \[ T_M(P) = \frac{MP}{\|MP\|_1} \] is the quotient of continuous functions with a strictly positive denominator. Hence \[ T_M \] is continuous on its domain. \(\square\) ==== Significance ==== Continuity guarantees that small changes in a probability distribution produce correspondingly small changes in the normalized matrix dilation. This property provides the analytical foundation for studying fixed points, convergence, stability, and dynamical behavior under repeated matrix dilation. === Theorem 3 (Fixed Points and Eigenvectors) === Let \[ M\in\mathbb{R}^{n\times n} \] be a nonnegative matrix, and let \[ T_M(P) = \frac{MP}{\|MP\|_1} \] be the matrix dilation operator. If \[ P\in\Delta^{n-1} \] is a nonnegative eigenvector of \(M\), that is, \[ MP=\lambda P \] for some eigenvalue \[ \lambda>0, \] then \[ P \] is a fixed point of the matrix dilation operator: \[ T_M(P)=P. \] ==== Proof ==== Suppose \[ MP=\lambda P, \] where \[ \lambda>0. \] Since \[ P\in\Delta^{n-1}, \] its components sum to one. Therefore, \[ \|MP\|_1 = \|\lambda P\|_1 = \lambda. \] Hence, \[ T_M(P) = \frac{MP}{\|MP\|_1} = \frac{\lambda P}{\lambda} = P. \] Therefore, \[ P \] is a fixed point of the matrix dilation operator. \(\square\) ==== Significance ==== Theorem 3 establishes a direct connection between Matrix PDT and classical spectral theory. Every nonnegative eigenvector of the dilation matrix, after normalization to a probability vector, is a stationary probability distribution under repeated matrix dilation. This result links Matrix PDT to the Perron–Frobenius theory of positive matrices and provides the foundation for studying equilibrium probability distributions and long-term iterative behavior. === Theorem 4 (Convergence Under a Unique Positive Eigenvector) === Let \[ M\in\mathbb{R}^{n\times n} \] be a positive matrix (all entries strictly positive), and let \[ T_M(P) = \frac{MP}{\|MP\|_1} \] be the matrix dilation operator. Suppose that \(M\) has a unique positive Perron–Frobenius eigenvector \[ P^* \] normalized so that \[ P^*\in\Delta^{n-1}. \] Then every sequence \[ P_{n+1}=T_M(P_n) \] with \[ P_0\in\Delta^{n-1} \] converges to \[ P^*. \] ==== Sketch of Proof ==== Repeated multiplication by a positive matrix is governed by the Perron–Frobenius theorem. The dominant eigenvalue exceeds all others in magnitude, and repeated application of \(M\) aligns vectors with the corresponding positive eigenvector. Since each iteration of \(T_M\) simply renormalizes the resulting vector, \[ P_{n+1} = \frac{MP_n}{\|MP_n\|_1}, \] the normalization preserves direction while maintaining unit total probability. Consequently, \[ P_n\rightarrow P^*. \] \(\square\) ==== Significance ==== Under the assumptions of the Perron–Frobenius theorem, repeated matrix dilation converges to a unique equilibrium probability distribution. This equilibrium represents the long-term stationary state of the iterative dilation process. === Theorem 5 (Entropy of Fixed Points) === Let \[ P^* \] be a fixed point of the matrix dilation operator \[ T_M(P) = \frac{MP}{\|MP\|_1}. \] Then the Shannon entropy \[ H(P^*) = -\sum_{i=1}^{n}P_i^*\log P_i^* \] is invariant under further matrix dilation: \[ H(T_M(P^*)) = H(P^*). \] ==== Proof ==== Since \[ P^* = T_M(P^*), \] we have \[ H(T_M(P^*)) = H(P^*), \] because both probability distributions are identical. Therefore the Shannon entropy remains unchanged after every subsequent iteration. \(\square\) ==== Significance ==== Once the iterative matrix dilation process reaches a fixed point, the probability distribution no longer changes. Consequently, its Shannon entropy is also constant. Thus equilibrium probability distributions correspond to entropy-equilibrium states of Matrix PDT. == Interpretation == The diagonal entries <math> d_i </math> represent localized probability dilation identical to standard PDT. The off-diagonal entries <math> a_{ij} </math> allow probability associated with one state to influence another. Consequently, * diagonal PDT represents independent local amplification; * matrix PDT represents coupled probability evolution. == Relation to Standard PDT == When all off-diagonal entries vanish, <math> a_{ij}=0,\qquad i\ne j, </math> the generalized operator reduces exactly to the original PDT operator. Thus diagonal PDT is a special case of matrix PDT. == Connections to Existing Mathematics == The generalized formulation exhibits similarities to several established mathematical frameworks, including * positive matrices, * Markov operators, * Perron–Frobenius theory, * coupled dynamical systems, * graph-based diffusion, * linear operator theory. These connections are mathematical analogies and are not claimed to be equivalent formulations. == Potential Properties == Questions for future investigation include: * preservation of positivity, * convergence of repeated iterations, * existence and uniqueness of fixed points, * entropy evolution, * stability of coupled systems, * spectral properties of the dilation matrix, * relation to Perron–Frobenius eigenvectors, * continuous-time matrix dilation flows. == Discussion == Matrix Dilation Operators extend the localized dilation concept of PDT by allowing interactions between probability states. This generalization significantly enlarges the mathematical scope of PDT while preserving the original theory as the diagonal special case. Whether this extension provides useful applications or deeper theoretical insights remains an open research question. == See also == * [[Probability Dilation Theory]] * [[Probability Dilation Theory/Measure-Theoretic Foundations]] * [[Probability Dilation Theory/Dilation Vector Field]] * [[Probability Dilation Theory/Dilation Flows]] * [[Probability Dilation Theory/Stochastic Dilation Fields]] * [[Probability Dilation Theory/Convergence and Fixed Points]] * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows]] == References == * Howard Richardson, ''Probability Dilation Theory'', Wikiversity. * R. A. Horn and C. R. Johnson, ''Matrix Analysis''. * F. R. Gantmacher, ''The Theory of Matrices''. * O. Perron (1907); G. Frobenius (1912), classical papers on positive matrices. cns0cgxnmmx9vkmp1cgw6vdkbxsuv6d 0 330681 2819333 2818951 2026-07-25T03:20:20Z MTitleman 2995549 2819333 wikitext text/x-wiki Articles written by E. T. Whittaker in 1903 and 1904 suggest the natural generalization of 3-space as ternary probability. This is proven using basic mathematical statements and their associated transition from the product of the symmetric group S₂ and the Klein four-group Z₂² to the dihedral of order 12 D₆ as main eigenvalues and associated properties. A generalization of physical theory is suggested. ==Ternary Probability== The work of E. T. Whittaker suggests that there are methods for computing wavefunctions by tetration, hyperoperations, interpolation and ternary probability. Consider the statement: <math>(1) x+y=\surd7</math> <math>x-y=\surd3</math> <math>xy=1</math> Additionally consider the statement: <math>(2)\sqrt[3]{8}=2 </math> Due to asymmetry in Cartesian coordinates (Whittaker, 1903), it is feasible to consider [1] as analogous to vector calculus in 3-space if xy is considered real as √−1 is imaginary, x+y or x-y are considered vectors in a 3-parameter space, and [2] is real. This and the property of infinite tetration unique to √2 suggest the implementation of ternary probability. This is further suggested by [1] being commensurate with the transition from the product of the symmetric group S₂ and the Klein four-group Z₂² (nilpotent, abelian-by-cyclic) to the dihedral of order 12 D₆ (non-abelian, solvable) in Galois theory (Du et al., 2021). The former permutes coordinates independently as a form of orthogonal sphericity xy⊥zR (Titleman, 2026) while the latter combines a primary rotation of order 6 with a single axis reflection of order 2 as a form of wave propagation (Whittaker, 1904). 3-space is thus only one paradigm for modeling trajectories. Mathematical assumptions of cartesian symmetry and kinematics require matching physical assumptions, such as those made in the Clausius formulation for Root Mean Square (RMS) velocity. ==References== Du, Zenan, Fenjin Liu, Shunyi Liu, and Zhongmei Qin. ”Graphs with n-1 main eigenvalues.” Discrete Mathematics 344, no. 7 (2021): 112397. Titleman, M. (2026). The Duality of Whittaker Potential Theory: Fundamental Representations of Electromagnetism and Gravity, and Their Orthogonality. arXiv preprint arXiv:2205.08309v9. Whittaker, E. T. (1904). On an expression of the electromagnetic field due to electrons by means of two scalar potential functions. Proc. Lond. Math. Soc, 1, 367. Whittaker, E. T. (1903). On the partial differential equations of mathematical physics. Mathematische Annalen, 57(3), 333-355.{{reflist|35em}} 98k73xf51plciccpi128czim0bd928q 3 330698 2819290 2819069 2026-07-24T16:13:32Z Atcovi 276019 /* Questions */ 2819290 wikitext text/x-wiki == Questions == 1. "''This result is consistent with a previous study showing that Sleep disturbance, including insomnia, prospectively predicted suicide outcomes [76]. However, a review by Kirsten Russell et al. which focused on young adults, concluded that while insomnia appears linked to suicidality, effects are less consistent after controlling for potential confounding variables [77]. Although that study focused on a broader age range, our findings, which are specifically focused on older adults, further strengthen the connection between sleep disturbances and suicidal behavior within this particular age group.''" [The Intersection of sleep disturbance and suicidal behavior among older Adults: A systematic review (2025)] → why is it that young adults may have a weaker association between insomnia and suicide when controlling for potential confounding variables as opposed to older adults? 2. "''In a relationship between sleep duration and suicidal behavior, sleep duration alone does not explain the full complexity of suicide risk in older adults.''" [The Intersection of sleep disturbance and suicidal behavior among older Adults: A systematic review (2025)] → limited research is shown here, perhaps could we look into how a lack of sleep affects cognitive ability, which could impact other aspects of life (problem-solving skills, for example? this could be measured, possibly through the Iowa Gambling Task & Wisconsin Card Sorting Test (WCST)). —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:31, 22 July 2026 (UTC) job6kyy2cocjvar94cwp03xulp7h0rd 0 330716 2819255 2819239 2026-07-24T12:32:43Z MathXplore 2888076 Added {{[[Template:BookCat|BookCat]]}} using [[User:1234qwer1234qwer4/BookCat.js|BookCat.js]] 2819255 wikitext text/x-wiki ==Arithmetic== [[File:Multiply 4 bags 3 marbles.svg|thumb|right|4 x 3 = 12 (multiplication)]] '''Arithmetic''' has to deal with elementary/basic levels of math, such as division, multiplication, subtraction, and addition. Basically, just working with numbers. This SHOULD be a level familiar with you. If you are not familiar with arithmetic math/rules, then PLEASE review through Arithmetic, as you won't survive even the 1st step of Algebra. Trust me, the basics are THAT important. ===Fractions=== [[File:Cake quarters.svg|thumb|left|A Cake with fractions]] '''Fractions''' (from Latin ''fractus'', "broken") are parts of a whole. On the left side in the image of the cake, there is only <math>3/4</math>'s of the cake showing, the other <math>1/4</math> has been eaten/taken away. The number, ''3'', in <math>3/4</math>, is what is known as a '''numerator''' (Numerator: Number at the top, tells us of how much of the number is being talked about/being used). The number, ''4'', in <math>3/4</math>, is what is known as a '''denominator''' (Denominator: Number showing the all time total). ; ;Simplest form/reduced form A reduced form of a fraction is a fraction that cannot be divided by any number other than 1, and the denominator is greater than 1. So <math>2/4</math> is NOT in simplest form, since we can divide 2 and 4, by 2... which results in the following number: <math>1/2</math>. Though, not every fraction can be divided by 2, there are fractions, such as: <math>5/35</math>, <math>7/21</math>, and <math>30/5</math>. The two first fractions are not divisible by 2, and <math>30/5</math> can not be divided by 2 on both sides, but only on <math>30</math>. It's important to simplify as if you were in a test, your teacher will mark your problems as incorrect if you didn't simplify your fractions. Keep in mind that simplifying a fraction into its simplest/reduced form doesn't change its value, both the original (unsimplified) fraction and its reduced form represent the same exact value/quantity. So, <math>\tfrac{2}{4}</math> and <math>\tfrac{1}{2}</math> represents the same quantity, a half! Here, we will present a few fractions for you to simplify. ====Sample problems for ''simplifying fractions'' (use ''/'' as the fraction line)==== <quiz display=simple points="1/1"> { |type="{}"} <math>\tfrac{6}{8}=</math>{ 3/4_7 } { |type="{}"} <math>\tfrac{4}{60}=</math>{ 1/15_7 } { |type="{}"} <math>\tfrac{30}{90}=</math>{ 1/3_7 } { |type="{}"} <math>\tfrac{8}{18}=</math>{ 4/9_7 } { |type="{}"} <math>\tfrac{9}{72}=</math>{ 1/8_7 } { |type="{}"} <math>\tfrac{64}{46}=</math>{ 32/23|1 9/23_7 } { |type="{}"} <math>\tfrac{206}{340}=</math>{ 103/170_7 } </quiz> ===== Adding or Subtracting Fractions ===== [[File:Fractionsworkalgebra.PNG|thumb|right|What we just worked on, summarized]] To simply add or subtract fractions, make sure the denominators of the fractions you are adding or subtracting are the same. If they are not, find the least common denominator (LCD). For example, if you want to add <math>\tfrac{4}{2}</math> and <math>\tfrac{4}{6}</math>, you first have to multiply the 2 in <math>\tfrac{4}{2}</math> by '''3''', which equals '''6'''... BUT you cannot just multiply 2 only, you also have to multiply 4 by 3, since that's what you did to 2, the denominator. If you change the denominator, you have to change the numerator. ('''This step is crucial as it allows you to preserve the same value of the fraction''' but with just a different representation) Alright, we got that out of the way, so once we have <math>\tfrac{12}{6}</math> + <math>\tfrac{4}{6}</math>, we can simply add. So <math>12</math> + <math>4</math> = <math>16</math>, but don't add the denominators, they stay the same. So the answer is <math>\tfrac{16}{6}</math>, and then we simplify down to <math>\tfrac{8}{3}</math> dividing by 2 on both the numerator and denominator. But... did you notice something? <math>\tfrac{8}{3}</math>? That doesn't seem right, does it? The denominator is smaller than the numerator. When you have a fraction like this, you have to convert it to a '''mixed fraction''' (skip to [[Speak_Math_Now!/Week_1:_Introduction_To_Algebra#Improper_Fraction_--.3E_Mixed_Fraction|section 2.1.1.4]]). ===== Multiplying Fractions ===== To multiply fractions, its easiest to first simplify your fraction to simplest terms. Once you have done that, you can simply multiply the numerators and the denominators. And obviously, simplify your final product, if you can. So, we have <math>\tfrac{6}{8}</math> and <math>\tfrac{2}{6}</math>. You could multiply the numerators and denominators straight away and simplify at the end if you are comfortable, but to make it easier and clearer, we should simplify the fractions first. We simplify 6 and 8 by dividing both by 2, we also divide 2 and 6 by 2. So the fractions are now <math>\tfrac{3}{4}</math> and <math>\tfrac{1}{3}</math>. You simply multiply those two fractions by multiplying the numerator by the numerator, and doing the same for the denominators. After completing this process, you will get a solution (in fraction form). <math>\tfrac{3}{4}</math> × <math>\tfrac{1}{3}</math> <math>=</math> <math>\tfrac{3}{12}</math>. <math>\tfrac{3}{12}</math> is not going to be our final product, though, since we can simplify the fraction by dividing the fraction by 3, which results in <math>\tfrac{1}{4}</math>. ===== Dividing Fractions ===== There is an interesting twist when it comes to dividing fractions. You have to turn the fraction you want to divide by (second fraction) upside-down, also known as "Keep, Change, Flip" where you keep the first fraction the same, change the operation to multiplication, and replace the second fractions numerator with the denominator and the denominator with the numerator. Not only that, you have to turn the division symbol (÷) into a multiplication symbol (× or •). After that, you use your skills you learned in multiplying a fraction, and you multiply both of the fractions. Simplify if you need to. So, <math>\tfrac{6}{8}</math> ÷ <math>\tfrac{7}{12}</math>. Change the division symbol to a multiplication symbol, and turn the fraction you want to divide by upside-down (the upside-down fraction is known as a '''reciprocal'''). So <math>\tfrac{6}{8}</math> × (or •) <math>\tfrac{12}{7}</math>. Multiply the numerators and denominators. The answer is <math>\tfrac{72}{56}</math>, simplified down to <math>\tfrac{9}{7}</math>. ===== Improper Fraction --> Mixed Fraction ===== Divide the numerator by the denominator. The '''quotient''' (result of the division taking place/number above the division line) will be the whole number of the mixed fraction, while the numerator will be the remainder. The denominator remains unchanged, so don't change the denominator at all! {{notice|If you would like to take the quiz on Fractions, please go to '''[[Speak Math Now!/Week 1: Introduction To Algebra/Fractions Quiz]]'''}} See also: https://www.tes.com/lessons/bJieZ4sFPJbSTw/fractions-4-mixed-numbers-and-improper-fractions ===Decimals=== Ever wondered how to write 8<math>\tfrac{47}{100}</math> as a decimal? Well, you've got the answer: 8.47! How did we get that answer? Let's look at a few more and maybe you'll see the pattern: # 6<math>\tfrac{98}{100}</math> = 6.98 # 2<math>\tfrac{56}{100}</math> = 2.56 # 9<math>\tfrac{27}{100}</math> = 9.27 # 5<math>\tfrac{83}{100}</math> = 5.83 You see? We simply put the mixed number in front of the dot, and with the numerator, we slap that behind the dot! Throw out the 100, it's not important when building your decimal. Decimals are all about place value, the value of a number in a specific place in a number. So, when we have <math>6.72</math>, the <math>6</math> is in the Ones place. Now, let's throw <math>9</math> in the tens place, which is 10 times bigger than the Ones place: <math>96.72</math>. But... that's doesn't seem enough, does it? Let's throw in a <math>6, 2, 8</math> and a <math>3</math> in there! And now, we have: <math>628,396.72</math>. Woah! That's a pretty big number, but we can easily break this number down to it's place value. Let's do it! So, our number, <math>628,396.72</math>, is the number we need to break down. Let's start from the decimal point, and move left: * The number <math>6</math> is in the Ones place. '''x10''' * The number <math>9</math> is in the Tens place. '''x10''' * The number <math>3</math> is in the Hundreds place. '''x10''' * The number <math>8</math> is in the Thousands place. '''x10''' * The number <math>2</math> is in the Ten thousands place. '''x10''' * The number <math>6</math> is in the Hundred Thousands place. Now we have broken up the numbers left of the decimal--What about the numbers on the ''right''? Let's throw in a <math>5, 2, 4</math> and a <math>7</math>. Now, we have <math>628,396.725,247</math>. Let's break this number up like we did above. So, our number, <math>628,396.725,247</math>, is the number we need to break down. This time, we need to start on the decimal point, and move ''right'': * The number <math>7</math> is in the Tenths place. '''x-10''' * The number <math>2</math> is in the Hundredths place. '''x-10''' * The number <math>5</math> is in the Thousandths place. '''x-10''' * The number <math>2</math> is in the Ten Thousandths place. '''x-10''' * The number <math>4</math> is in the Hundred Thousandths place. '''x-10''' * The number <math>7</math> is in the Millionths place. We have just now gone over the importance of Place Value in the Decimal World. Now, we will go into how to work with decimals, in the Decimal World! See also: http://www.shmoop.com/fractions-decimals/place-value-naming-decimals.html ==== Adding/Subtracting Decimals ==== To add decimals, in addition column-style, put the decimals in its place with the decimals lined up. Then simply add on. So, for <math>1.5</math> + <math>2.5</math> we'd line up the decimal points. But, if we had a problem like <math>1.15</math> + <math>2.0</math>, we'd add a <math>0</math> after the <math>0</math> that is behind the decimal. Adding a zero to a place in a decimal means "no value". So <math>10</math> basically means no ones, and <math>100</math>, means no ones or hundreds. Same things goes for subtracting as well folks. =====Sample problems for ''adding/subtracting decimals''===== <quiz display="simple" points="1/1"> { |type="{}"} 6.8 - 2.5 = { 4.3_6 } { |type="{}"} 3.4 + 5.6 = { 9_6 } { |type="{}"} 9 + 4.50 = { 13.5_6 } { |type="{}"} 41.89 + 25.00 = { 66.89_6 } { |type="{}"} 9.01 + 3.089 = { 12.099_6 } { |type="{}"} 10.90 + 11.1 = { 22_6 } { |type="{}"} 9.5 + 3.44 = { 12.94_6 } { |type="{}" coef="2.5"} 9.00 x 2.00 = { 18_6 } </quiz> ==== Multiplying Decimals ==== [[File:9.82x5.73 multiplication image.svg|thumb|A visual representation of the multiplication example]] Multiplying decimals isn't as hard as it really seems to be. So, we have <math>9.83</math> × <math>5.73</math>. For most people, column multiplication is a lot easier than side-by-side multiplication. That being mentioned, let us column these numbers: <math>9.83</math><br>× <math>5.73</math> ------- Now that we have our problem, we should simply ignore the decimal points and just multiply as usual, so you should get this answer once you are done with that (remember to add a zero (and grow with zeros in each line) to each and every line of addition): <math>9.83</math><br> × <math>5.73</math> ------- <math>2949</math> <br> <math>+</math> <math>68810</math> <br> <math>+</math> <math>491500 </math> ------- With the simple usage of addition, we should get: <math>9.83</math><br> × <math>5.73</math> ------- <math>2949</math><br> <math>+</math> <math>68810</math><br> <math>+ </math> <math>491500</math> ------- <math>563259</math> Now, we need to bring back our handy dandy decimal point, but where? In <math>9.83</math> and <math>5.73</math>, there are FOUR numbers in these 2 numbers overall that are behind the decimal point (in each number, there are two numbers behind the decimal points). So, we have <math>9.83</math> and <math>5.73</math>. Now, that totals up to four numbers overall behind the decimal point. So in <math>563259</math>, we need to move the decimal point four times (beginning from the right). So watch as follows: <math>563259.</math><br> <math>56325.9</math><br> <math>5632.59</math><br> <math>563.259</math><br> <math>56.3259</math> That simple. Now, review your work, your whole work should look like this: <math>9.83</math><br> × <math>5.73</math> ------- <math>2949</math><br> <math>+</math><math>68810</math><br> <math>+</math><math>491500</math> ------- <math>56.3259</math> ==== Dividing Decimals ==== ;Dividing a decimal by a whole number If you want to divide a decimal by a whole number, you should divide the 2 numbers, omitting the decimal point. After you are done dividing, add the decimal point to the '''quotient''' (final product/answer at the top of the long division symbol). The decimal should be right above the decimal point in the '''dividend''' (number in the box/number that is being divided). It's quite easy and simple, as long as you know how to do long division and if you are still familiar with long division. Hey, this seems ''too'' easy--Let's figure out how to divide a decimal by a decimal! ;Dividing a decimal by a decimal The trick to dividing a decimal by a decimal is to shift the decimal point as many times as it gets to a whole number, so follow along: <math>69.45</math> ÷ <math>5.78</math>. Now, we simply move the decimal point as many times as we need to make the number we are going to use to divide 69.45 a whole number, so watch as followed:<br> <math>69.45</math> ÷ <math>5.78</math> →<br> <math>694.5</math> ÷ <math>57.8</math> →<br> <math>6945</math>. ÷ <math>578</math>. Now that we have finally got our dividend a whole number (and now our first number that we are going to divide), we can go ahead and divide normally (using long division). In the end, <math>69.45</math> divided by <math>5.78</math> should get you <math>12.0155709</math>! A pretty simple one we could go is <math>6.4</math> ÷ <math>0.4</math>, here, we simply move our dots like so:<br> <math>6.4</math> ÷ <math>0.4</math><br> <math>64</math> ÷ <math>04.</math><br> <math>64</math> ÷ <math>4</math><br> Then, we can simply divide, heck... we don't even need to do long division! The answer should pop in your head, which is <math>16</math>. {{notice|If you would like to take the quiz on Decimals, please go to '''[[Speak Math Now!/Week 1: Introduction To Algebra/Decimals Quiz]]'''}} ===Percentages=== A good definition of "percent" is a fraction in which the denominator is the number <math>100</math>. For example, the numbers <math>59%</math>, <math>63%</math>, <math>91%</math>, and <math>85%</math>, are the same as just saying <math>\tfrac{59}{100}</math>, <math>\tfrac{63}{100}</math>, <math>\tfrac{91}{100}</math>, and <math>\tfrac{85}{100}</math>. You could also say 59 out of 100 parts, 63 out of 100 parts, 91 out of 100 parts, and 85 out of 100 parts. ====Converting Percentages==== Now that we got the basis of percentages and how they operate, we should look into changing percentages. ===== Percentage → Decimal ===== Let's look in turning a percentage into a decimal point first. It's very simple. Let's say you have <math>\tfrac{9}{100}</math>, which, in percentage form, is <math>9%</math>. So, we have 9%. Now, we want to change it to a decimal (I don't know, think of a reason). We simply convert the percentage symbol into a decimal point, so like this: <math>9.</math>. Now, we have <math>9.</math>, so then we move the decimal number two places to the left, like so: <math>9.</math> → <math>.9</math> → <math>.09</math>. So now, we have <math>0.09</math>. We added the 2 zeros in because there is no value in the tenths place, and because <math>.09</math> does not look quite right. Looks a bit off. ===== Samples problems for ''converting percentages to decimals'' ===== <quiz display="simple" points="1/1"> { |type="{}"} 59% = { 0.59_5 } { |type="{}"} 63% = { 0.63_5 } { |type="{}"} 91% = { 0.91_5 } { |type="{}"} 85% = { 0.85_5 } { |type="{}"} 9% = { 0.09_5 } { |type="{}"} 9834% = { 98.34_5 } { |type="{}"} 20% = { 0.2_5 } { |type="{}"} 4% = { 0.04_5 } { |type="{}"} 7.6% = { 0.076_5 } { |type="{}"} 6% = { 0.06_5 } </quiz> ===== Decimal → Percentage ===== Now to convert a decimal into percentage we essentially do the complete opposite. We have <math>98.34</math>. We need this to be a percentage (easier to read). Move the decimal point two places to the right. So, watch: <math>98.34</math> → <math>983.4</math> → <math>9834.</math> --Now, we have <math>9834.</math>, but the decimal point, since it's now a percentage, should not be there, but instead, a percentage should talk the decimal point's place. Now, we have our final result of <math>9834%</math>. ==== Finding percent of a number ==== [[File:Universität Bonn.jpg|thumb|right|Would this be the fictional university these students were trying to get accepted to?]] So, 95 students applied to a university (the fictional [[User:Atcovi/Mustafa Einhoonansebadoi University|Mustafa Einhoonansebadoi University]], for example), and only 20% of the students made it. 20%? What? With this in mind, we want to find <math>20%</math> of <math>95</math>. We take the percentage, <math>20%</math>, and divide it by <math>100</math>. So we get <math>20/100</math> = <math>.2</math>. Then, we multiply <math>.2</math> by <math>95</math>, in which we get <math>19</math>. So <math>20%</math> of <math>95</math> is <math>19</math>. If that didn't make sense .2 x 95 is 2 x 95 = 190 but because we multipled 2 by ten in the tenths place to the ones place we -10 off of the number to reverse what we just did. Therefore, only 19 students out of 95 students made it into the fictional Mustafa Einhoonansebadoi University. {{BookCat}} 100cjoqo24wjxgqwp3x9d3428po578u 0 330717 2819254 2819244 2026-07-24T12:32:39Z MathXplore 2888076 added [[Category:Nutrition]] using [[Help:Gadget-HotCat|HotCat]] 2819254 wikitext text/x-wiki See also [[OpenStax]] == <big>OpenStax</big> Nutrition for Nurses == == Summary == ''Nutrition for Nurses'' is structured to support the flexible integration of nutrition content across both system-based and nursing competency-based curricula. It can be used whether nutrition is taught as a standalone course or part of another nursing course. The table of contents for ''Nutrition for Nurses'' presents content in 20 chapters, organized into 9 thematic units. The text emphasizes evidence-based practice and holistic assessment to facilitate the integration of nutritional awareness for pre-licensure nursing students in the provision of client-centered care. ''Nutrition for Nurses'' helps students develop sound clinical judgment as well as a deep understanding of the impact of nutrition on body systems across the lifespan. Written and reviewed by highly experienced faculty, ''Nutrition for Nurses'' includes a detailed narrative, extensive features and learning resources, and ample student support. The presentation utilizes concepts promoting the development of clinical judgment by building upon the systematic model developed by the National Council of State Boards of Nursing (NCSBN). * [https://openstax.org/details/books/nutrition OpenStax Nutrition for Nurses] (original content). Available as pdf or web view. * [https://audileo.com/audiobooks/openstax/nutrition-for-nurses/ OpenStax Nutrition for Nurses audiobook]. Available as audio textbook. [[Category:Nutrition]] 3rdg5mdh8tuha1ox8vzbv51r1ouqigs 0 330718 2819246 2026-07-24T12:01:21Z Andy?yes 3006471 Created page with "See also [[OpenStax]] == <big>OpenStax</big> <big>Clinical Nursing Skills</big> == == Summary == ''Clinical Nursing Skills'' is designed to equip nursing students with the practical knowledge and hands-on skills necessary to provide comprehensive patient care. The material emphasizes the application of clinical judgment in a variety of settings, ensuring that students are prepared to deliver high-quality care across different patient populations and clinical scenarios...." 2819246 wikitext text/x-wiki See also [[OpenStax]] == <big>OpenStax</big> <big>Clinical Nursing Skills</big> == == Summary == ''Clinical Nursing Skills'' is designed to equip nursing students with the practical knowledge and hands-on skills necessary to provide comprehensive patient care. The material emphasizes the application of clinical judgment in a variety of settings, ensuring that students are prepared to deliver high-quality care across different patient populations and clinical scenarios. The content utilizes concepts promoting the development of clinical judgment by building upon the systematic model developed by the National Council of State Boards of Nursing (NCSBN). ''Clinical Nursing Skills'' provides detailed instructions on basic procedures such as hygiene, mobility, vital signs assessment, medication administration, and wound care. It also guides students through more complex skills, including intravenous therapy, catheterization, tracheostomy care, and emergency interventions. By integrating the Clinical Judgment Measurement Model, the material helps students recognize, analyze, prioritize, create, act, and evaluate outcomes in various clinical situations, fostering critical thinking and clinical decision making. By studying ''Clinical Nursing Skills'', students will gain the confidence and competence needed to perform essential nursing tasks, make informed clinical decisions, and provide compassionate, patient-centered care, which will prepare students for success in their clinical rotations and future professional practice. * OpenStax ''Clinical Nursing Skills'' (original content). Available as pdf or web view. * OpenStax ''Clinical Nursing Skills'' audiobook Available as audio textbook. 44t9w3ota2wuowl9dqmvjzasu0ibuo6 2819247 2819246 2026-07-24T12:01:59Z Andy?yes 3006471 /* Summary */ 2819247 wikitext text/x-wiki See also [[OpenStax]] == <big>OpenStax</big> <big>Clinical Nursing Skills</big> == == Summary == ''Clinical Nursing Skills'' is designed to equip nursing students with the practical knowledge and hands-on skills necessary to provide comprehensive patient care. The material emphasizes the application of clinical judgment in a variety of settings, ensuring that students are prepared to deliver high-quality care across different patient populations and clinical scenarios. The content utilizes concepts promoting the development of clinical judgment by building upon the systematic model developed by the National Council of State Boards of Nursing (NCSBN). ''Clinical Nursing Skills'' provides detailed instructions on basic procedures such as hygiene, mobility, vital signs assessment, medication administration, and wound care. It also guides students through more complex skills, including intravenous therapy, catheterization, tracheostomy care, and emergency interventions. By integrating the Clinical Judgment Measurement Model, the material helps students recognize, analyze, prioritize, create, act, and evaluate outcomes in various clinical situations, fostering critical thinking and clinical decision making. By studying ''Clinical Nursing Skills'', students will gain the confidence and competence needed to perform essential nursing tasks, make informed clinical decisions, and provide compassionate, patient-centered care, which will prepare students for success in their clinical rotations and future professional practice. * [https://openstax.org/details/books/clinical-nursing-skills OpenStax ''Clinical Nursing Skills''] (original content). Available as pdf or web view. * [https://audileo.com/audiobooks/openstax/clinical-nursing-skills/ OpenStax ''Clinical Nursing Skills'' audiobook] Available as audio textbook. 0xmwawlqmz06vnbesd5vl1ujya4eon2 2819253 2819247 2026-07-24T12:32:34Z MathXplore 2888076 added [[Category:Nursing]] using [[Help:Gadget-HotCat|HotCat]] 2819253 wikitext text/x-wiki See also [[OpenStax]] == <big>OpenStax</big> <big>Clinical Nursing Skills</big> == == Summary == ''Clinical Nursing Skills'' is designed to equip nursing students with the practical knowledge and hands-on skills necessary to provide comprehensive patient care. The material emphasizes the application of clinical judgment in a variety of settings, ensuring that students are prepared to deliver high-quality care across different patient populations and clinical scenarios. The content utilizes concepts promoting the development of clinical judgment by building upon the systematic model developed by the National Council of State Boards of Nursing (NCSBN). ''Clinical Nursing Skills'' provides detailed instructions on basic procedures such as hygiene, mobility, vital signs assessment, medication administration, and wound care. It also guides students through more complex skills, including intravenous therapy, catheterization, tracheostomy care, and emergency interventions. By integrating the Clinical Judgment Measurement Model, the material helps students recognize, analyze, prioritize, create, act, and evaluate outcomes in various clinical situations, fostering critical thinking and clinical decision making. By studying ''Clinical Nursing Skills'', students will gain the confidence and competence needed to perform essential nursing tasks, make informed clinical decisions, and provide compassionate, patient-centered care, which will prepare students for success in their clinical rotations and future professional practice. * [https://openstax.org/details/books/clinical-nursing-skills OpenStax ''Clinical Nursing Skills''] (original content). Available as pdf or web view. * [https://audileo.com/audiobooks/openstax/clinical-nursing-skills/ OpenStax ''Clinical Nursing Skills'' audiobook] Available as audio textbook. [[Category:Nursing]] drcvvn2knhkd9g0tx0wmfhcrwvyi0d6 2 330719 2819257 2026-07-24T12:34:34Z Tet-Math6 3103001 I have introduced the Nodal Current Splitter. 2819257 wikitext text/x-wiki = <big>This is a way to solve Unbalanced Bridge Circuits using only Ohms Law instead of Kirchhoff's more difficult mesh current method.</big> = <big>This is not a usual current divider which separates 2 PARALLEL currents. This is a Nodal Current Splitter. It separates 2 SERIES currents just like a voltage divider separates 2 series voltages. It does that by using the difference between the 2 series currents & tells you what that difference is at the same time.</big> <big>To see how nodal current splitters can solve problems using Ohms Law instead of Kirchhoff's method consider the simple T-FRAME circuit shown below,</big> ivaoekco9nm7wvuzk3nu20c2xkwpd0j 2819258 2819257 2026-07-24T12:37:10Z Tet-Math6 3103001 Better Wording 2819258 wikitext text/x-wiki = <big>This is a way to solve Unbalanced Bridge Circuits using only Ohms Law instead of Kirchhoff's more difficult mesh current method.</big> = <big>This is not a typical current divider which separates 2 PARALLEL currents. This is a Nodal Current Splitter. It separates 2 SERIES currents just like a voltage divider separates 2 series voltages. It does that by using the difference between the 2 series currents & tells you what that difference is at the same time.</big> <big>To see how nodal current splitters can solve problems using Ohms Law instead of Kirchhoff's method consider the simple T-FRAME circuit shown below.</big> cantrsqpq26za1ub2s2jbgeyga4ojbs 2819260 2819258 2026-07-24T12:43:43Z Tet-Math6 3103001 2819260 wikitext text/x-wiki = <big>This is a way to solve Unbalanced Bridge Circuits using only <u>Ohms Law</u> instead of Kirchhoff's more difficult mesh current method.</big> = <big>This is not a typical current divider which separates 2 PARALLEL currents. This is a Nodal Current Splitter. It separates 2 SERIES currents just like a voltage divider separates 2 series voltages. It does that by using the difference between the 2 series currents & tells you what that difference is at the same time.</big> <big>To see how nodal current splitters can solve problems using <u>Ohms Law</u> instead of Kirchhoff's method consider the simple T-FRAME circuit shown below.</big> l7kdc0rzzexf16oyimsole9m4ls8yik 2819280 2819260 2026-07-24T15:10:51Z Tet-Math6 3103001 Beginning to Add Circuit Diagrams & Equations 2819280 wikitext text/x-wiki = <big>This is a way to solve Unbalanced Bridge Circuits using only <u>Ohms Law</u> instead of Kirchhoff's more difficult mesh current method.</big> = === <big>This is not about a typical current divider which separates 2 PARALLEL currents. This is a Nodal Current Splitter. It separates 2 SERIES currents just like a voltage divider separates 2 series voltages. It does that by using the difference between the 2 series currents & tells you what that difference is at the same time.</big> === === <big>To see how nodal current splitters can solve problems using Ohms Law instead of Kirchhoff's method consider the simple T-FRAME circuit shown below.</big> === === <big><math>I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === === <big>These 2 equations are much more than just adaptions of Ohms Law. They are actually powerful problem solving formulas that have been hiding right in front of our eyes ever since Kirchhoff's Mesh equations showed up. They are not simple Current Dividers ether. These 2 are Common Mode Nodal Current Splitters. These 2 formulas split the '''Common mode Nodal Current''' '''I₃''' into 2 '''Discrete Nodal Currents''' '''I₁ & I₂'''. The other type of Nodal Current Splitter could be called Discrete mode Nodal Current Splitters. Wait until you see how Discrete mode Nodal Current Splitters can shred unbalanced Wheatstone Bridge problems. And do it using only 1 Nodal Current instead of 3 Kirchhoff's Mesh equations or the usually required 5 nodal current equations.</big> === === <big><math>I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === === <big><math>I_3=I_1+I_2\quad therefore\quad I_3=\frac{\ V_1-I_3R_3\ }{R_1}+\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === === <big><math>I_3=\frac{\ V_1R_2+V_2R_1-I_3R_2R_3-I_3R_1R_3\ }{R_1R_2}</math></big> === === <big><math>I_3R_1R_2+I_3R_1R_3+I_3R_2R_3=\ V_1R_2+V_2R_1</math></big> === === <big><math>I_3=\frac{V_1R_2+V_2R_1}{\ R_1R_2+R_3\bigl(R_1+R_2\bigr)\ }</math></big> === === <big>Then calculate '''I₁''' & '''I₂''' us</big><big>ing the original 2 Common mode Nodal Current Splitters.</big> === === <big><math>I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === 16mp6k2o2ur14dep6zv0vmw0wmxt6iz 2819281 2819280 2026-07-24T15:15:00Z Tet-Math6 3103001 Tidy Monster 2819281 wikitext text/x-wiki = <big>There is a way to solve Unbalanced Bridge Circuits using only Ohms Law instead of using Kirchhoff's more difficult mesh current method.</big> = === <big>This is not about a typical current divider which separates 2 PARALLEL currents. This is a Nodal Current Splitter. It separates 2 SERIES currents just like a voltage divider separates 2 series voltages. It does that by using the difference between the 2 series currents & tells you what that difference is at the same time.</big> === === <big>To see how nodal current splitters can solve problems using Ohms Law instead of Kirchhoff's method consider the simple T-FRAME circuit shown below.</big> === === <big><math>I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === === <big>These 2 equations are much more than just adaptions of Ohms Law. They are actually powerful problem solving formulas that have been hiding right in front of our eyes ever since Kirchhoff's Mesh equations showed up. They are not simple Current Dividers ether. These 2 are Common Mode Nodal Current Splitters. These 2 formulas split the '''Common mode Nodal Current''' '''I₃''' into 2 '''Discrete Nodal Currents''' '''I₁ & I₂'''. The other type of Nodal Current Splitter could be called Discrete mode Nodal Current Splitters. Wait until you see how Discrete mode Nodal Current Splitters can shred unbalanced Wheatstone Bridge problems. And do it using only 1 Nodal Current instead of 3 Kirchhoff's Mesh equations or the usually required 5 nodal current equations.</big> === === <big><math>I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === === <big><math>I_3=I_1+I_2\quad therefore\quad I_3=\frac{\ V_1-I_3R_3\ }{R_1}+\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === === <big><math>I_3=\frac{\ V_1R_2+V_2R_1-I_3R_2R_3-I_3R_1R_3\ }{R_1R_2}</math></big> === === <big><math>I_3R_1R_2+I_3R_1R_3+I_3R_2R_3=\ V_1R_2+V_2R_1</math></big> === === <big><math>I_3=\frac{V_1R_2+V_2R_1}{\ R_1R_2+R_3\bigl(R_1+R_2\bigr)\ }</math></big> === === <big>Then calculate '''I₁''' & '''I₂''' us</big><big>ing the original 2 Common mode Nodal Current Splitters.</big> === === <big><math>I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === dyc9hog67mj5109ya4t46pe4dj9r11e 2819282 2819281 2026-07-24T15:39:10Z Tet-Math6 3103001 Tidy Monster Again 2819282 wikitext text/x-wiki = <big>There is a way to solve Unbalanced Bridge Circuits using only Ohms Law instead of using Kirchhoff's more difficult mesh current method.</big> = === <big>This is not about a typical current divider which separates 2 PARALLEL currents. This is a Nodal Current Splitter. It separates 2 SERIES currents just like a voltage divider separates 2 series voltages. It does that by using the difference between the 2 series currents & tells you what that difference is at the same time.</big> === === <big>To see how nodal current splitters can solve problems using Ohms Law instead of Kirchhoff's more complicated method consider the simple T-FRAME circuit shown below.</big> === === <big><math>I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === === <big>These 2 equations are much more than just adaptions of Ohms Law. They are actually powerful problem solving formulas that have been hiding right in front of our eyes ever since Kirchhoff's Mesh equations showed up. They are not simple Current Dividers ether. These 2 are Common Mode Nodal Current Splitters. These 2 formulas split the '''Common mode Nodal Current''' '''I₃''' into 2 '''Discrete Nodal Currents''' '''I₁ & I₂'''. The other type of Nodal Current Splitter could be called Discrete mode Nodal Current Splitters. Wait until you see how Discrete mode Nodal Current Splitters can shred unbalanced Wheatstone Bridge problems. And do it using only 1 Nodal Current instead of 3 Kirchhoff's Mesh equations or the usually required 5 nodal current equations.</big> === === <big><math>I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === === <big><math>I_3=I_1+I_2\quad therefore\quad I_3=\frac{\ V_1-I_3R_3\ }{R_1}+\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === === <big><math>I_3=\frac{\ V_1R_2+V_2R_1-I_3R_2R_3-I_3R_1R_3\ }{R_1R_2}</math></big> === === <big><math>I_3R_1R_2+I_3R_1R_3+I_3R_2R_3=\ V_1R_2+V_2R_1</math></big> === === <big><math>I_3=\frac{V_1R_2+V_2R_1}{\ R_1R_2+R_3\bigl(R_1+R_2\bigr)\ }</math></big> === === <big>Then calculate '''I₁''' & '''I₂''' us</big><big>ing the original 2 Common mode Nodal Current Splitters.</big> === === <big><math>I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === su3qv19dfigh6d226oceovelqte0tmq 2819312 2819282 2026-07-24T19:09:52Z Tet-Math6 3103001 Added Vital Wiring Diagram 2819312 wikitext text/x-wiki = <big>There is a way to solve Unbalanced Bridge Circuits using only Ohms Law instead of using Kirchhoff's more difficult mesh current method.</big> = === <big>This is not about a typical current divider which separates 2 PARALLEL currents. This is a Nodal Current Splitter. It separates 2 SERIES currents just like a voltage divider separates 2 series voltages. It does that by using the difference between the 2 series currents & tells you what that difference is at the same time.</big> === === <big>To see how nodal current splitters can solve problems using Ohms Law instead of Kirchhoff's more complicated method consider the simple T-FRAME circuit shown below.</big> === === <big><math>I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === === <big>These 2 equations are much more than just adaptions of Ohms Law. They are actually powerful problem solving formulas that have been hiding right in front of our eyes ever since Kirchhoff's Mesh equations showed up. They are not simple Current Dividers ether. These 2 are Common Mode Nodal Current Splitters. These 2 formulas split the '''Common mode Nodal Current''' '''I₃''' into 2 '''Discrete Nodal Currents''' '''I₁ & I₂'''. The other type of Nodal Current Splitter could be called Discrete mode Nodal Current Splitters. Wait until you see how Discrete mode Nodal Current Splitters can shred unbalanced Wheatstone Bridge problems. And do it using only 1 Nodal Current instead of 3 Kirchhoff's Mesh equations or the usually required 5 nodal current equations.</big> === === <big><math>I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === [[File:Cali Cat.jpg|thumb|609x609px]] === <big><math>I_3=I_1+I_2\quad therefore\quad I_3=\frac{\ V_1-I_3R_3\ }{R_1}+\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === === <big><math>I_3=\frac{\ V_1R_2+V_2R_1-I_3R_2R_3-I_3R_1R_3\ }{R_1R_2}</math></big> === === <big><math>I_3R_1R_2+I_3R_1R_3+I_3R_2R_3=\ V_1R_2+V_2R_1</math></big> === === <big><math>I_3=\frac{V_1R_2+V_2R_1}{\ R_1R_2+R_3\bigl(R_1+R_2\bigr)\ }</math></big> === === <big>Then calculate '''I₁''' & '''I₂''' us</big><big>ing the original 2 Common mode Nodal Current Splitters.</big> === === <big><math>I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === a1vzve7ly88z3fo81vohxv5cd0ad1lq 2819314 2819312 2026-07-24T19:35:38Z Tet-Math6 3103001 Worked on the T-Frame circuit section 2819314 wikitext text/x-wiki = <big>There is a way to solve Unbalanced Bridge Circuits by using only Ohms Law instead of using Kirchhoff's more difficult mesh current method.</big> = === <big>This is not about a typical current divider which separates 2 PARALLEL currents. This is a Nodal Current Splitter. It separates 2 SERIES currents just like a voltage divider separates 2 series voltages. It does that by using the difference between the 2 series currents & tells you what that difference is at the same time.</big> === === <big>To see how nodal current splitters can solve problems using Ohms Law instead of Kirchhoff's more complicated method consider the simple T-FRAME circuit shown below</big> === === <big>Where:</big> === === <big><math>I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === === <big>These 2 equations are much more than just adaptions of Ohms Law. They are actually powerful problem solving formulas that have been hiding right in front of our eyes ever since Kirchhoff's Mesh equations showed up. They are not simple Current Dividers ether. These 2 are Common Mode Nodal Current Splitters. These 2 formulas split the '''Common mode Nodal Current''' '''I₃''' into 2 '''Discrete Nodal Currents''' '''I₁ & I₂'''. The other type of Nodal Current Splitter could be called Discrete mode Nodal Current Splitters. Wait until you see how Discrete mode Nodal Current Splitters can shred unbalanced Wheatstone Bridge problems. And do it using only 1 Nodal Current instead of 3 Kirchhoff's Mesh equations or the usually required 5 nodal current equations.</big> === = <big>The T-Frame Circuit</big> = === <big>You can derive the formula for I3 using Ohms Law thus bypassing Kirchhoff.</big> === === <big><math>I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === [[File:Cali Cat.jpg|thumb|609x609px|<big>Cali Cat</big>]] === <big><math>I_3=I_1+I_2\quad therefore\quad I_3=\frac{\ V_1-I_3R_3\ }{R_1}+\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === === <big><math>I_3=\frac{\ V_1R_2+V_2R_1-I_3R_2R_3-I_3R_1R_3\ }{R_1R_2}</math></big> === === <big><math>I_3R_1R_2+I_3R_1R_3+I_3R_2R_3=\ V_1R_2+V_2R_1</math></big> === === <big><math>I_3=\frac{V_1R_2+V_2R_1}{\ R_1R_2+R_3\bigl(R_1+R_2\bigr)\ }</math></big> === === <big>Then calculate currents '''I₁''' & '''I₂''' us</big><big>ing the original 2 Common mode Nodal Current Splitters</big> <big>you</big> <big>use</big><big>d</big> <big>to d</big><big>erived I₃ from in the 1st place.</big> === === <big><math>I_1=\frac{\ V_1-I_3R_3\ }{R_1}\qquad\ I_2=\frac{\ V_2-I_3R_3\ }{R_2}</math></big> === = <big>The H-Frame Circuit</big> = d1mrrf8hl5epfi25zsf859ib83pk0hn 2 330720 2819259 2026-07-24T12:42:30Z Tet-Math6 3103001 Started my Sandbox 2819259 wikitext text/x-wiki = <big>This is a way to solve Unbalanced Bridge Circuits using only <u>Ohms Law</u> instead of Kirchhoff's more difficult mesh current method.</big> = editedit source <big>This is not a typical current divider which separates 2 PARALLEL currents. This is a Nodal Current Splitter. It separates 2 SERIES currents just like a voltage divider separates 2 series voltages. It does that by using the difference between the 2 series currents & tells you what that difference is at the same time.</big> <big>To see how nodal current splitters can solve problems using <u>Ohms Law</u> instead of Kirchhoff's method consider the simple T-FRAME circuit shown below</big><big>.</big> tisno80esnj75a6osugrcurnyetevgf 6 330721 2819272 2026-07-24T13:52:46Z Young1lim 21186 {{Information |Description=C04.SA0: Address and Dereference Operators (20260724 - 20260723) |Source={{own|Young1lim}} |Date=2026-07-24 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2819272 wikitext text/x-wiki == Summary == {{Information |Description=C04.SA0: Address and Dereference Operators (20260724 - 20260723) |Source={{own|Young1lim}} |Date=2026-07-24 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} dxm2nf2vp4vn8cbhu7uksyg7n6aupzd 6 330722 2819273 2026-07-24T13:55:37Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2B Single Level (20260724 - 20260723) |Source={{own|Young1lim}} |Date=2026-07-24 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2819273 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2B Single Level (20260724 - 20260723) |Source={{own|Young1lim}} |Date=2026-07-24 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 452ngfir4ra2iyzvuj1loq4fncs89w8 6 330723 2819274 2026-07-24T13:56:23Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2C Multi-Level (20260724 - 20260723) |Source={{own|Young1lim}} |Date=2026-07-24 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2819274 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2C Multi-Level (20260724 - 20260723) |Source={{own|Young1lim}} |Date=2026-07-24 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 626sx19koosqfbt50yrkn3j2uppc4dp 6 330724 2819276 2026-07-24T14:08:19Z Young1lim 21186 {{Information |Description=Laurent.5: Permutation 6C (20260723 - 20260722) |Source={{own|Young1lim}} |Date=2026-07-24 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2819276 wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Permutation 6C (20260723 - 20260722) |Source={{own|Young1lim}} |Date=2026-07-24 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 2tg4t8zel6vql8ko61f5k5aa3wgneb0 6 330725 2819278 2026-07-24T14:09:11Z Young1lim 21186 {{Information |Description=Laurent.5: Permutation 6C (20260724 - 20260723) |Source={{own|Young1lim}} |Date=2026-07-24 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2819278 wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Permutation 6C (20260724 - 20260723) |Source={{own|Young1lim}} |Date=2026-07-24 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} l4w9n6rajed811x38v9080dhnkh0j62 6 330726 2819294 2026-07-24T16:46:36Z Young1lim 21186 {{Information |Description=Data.1A: Data Object (202600721 - 20260720) |Source={{own|Young1lim}} |Date=2026-07-25 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2819294 wikitext text/x-wiki == Summary == {{Information |Description=Data.1A: Data Object (202600721 - 20260720) |Source={{own|Young1lim}} |Date=2026-07-25 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 1wpzm5yo26xg2uvg1bp62qpbgy87qit 2 330727 2819301 2026-07-24T17:50:27Z Asmaadaghrir 3103065 /* */ 2819301 wikitext text/x-wiki '''Lab Technician (15+ yrs exp.) | Tech & Innovation Content Creator | Sharing Insights & Ideas''' k1xz8iytctlg2wly4ordxzbwm5n43le 0 330728 2819304 2026-07-24T18:12:56Z Howie2024 2995240 Insert Spectral Analysis of Matrix Dilation Operators 2819304 wikitext text/x-wiki = Probability Dilation Theory / Spectral Analysis of Matrix Dilation Operators = This page is a subpage of [[Probability Dilation Theory]]. == Overview == This page investigates the spectral properties of Matrix Dilation Operators introduced as a proposed generalization of Probability Dilation Theory (PDT). The objective is to examine how eigenvalues, eigenvectors, and positive matrix theory may characterize the long-term behaviour of repeated matrix dilation. This page is exploratory and extends the mathematical framework of PDT without modifying the core diagonal formulation. == Background == The generalized matrix dilation operator is <math> T_M(P) = \frac{MP}{\|MP\|_1}, </math> where * <math>P</math> is a probability vector, * <math>M</math> is a non-negative matrix, * normalization preserves total probability. Repeated iteration produces <math> P_{n+1} = T_M(P_n). </math> The resulting nonlinear iteration motivates the study of spectral properties of the underlying matrix. == Eigenvalues and Eigenvectors == Consider <math> Mv=\lambda v, </math> where * <math>\lambda</math> is an eigenvalue, * <math>v</math> is the corresponding eigenvector. ::contentReference[oaicite:0]{index=0} The dominant eigenvalue often governs the asymptotic behaviour of repeated matrix multiplication. Because Matrix PDT includes normalization after each iteration, the resulting dynamics differ from ordinary linear iteration while still retaining connections to classical spectral theory. == Perron–Frobenius Theory == When <math> M </math> is a positive matrix, the Perron–Frobenius theorem establishes that * a largest positive eigenvalue exists, * the dominant eigenvector has strictly positive components, * repeated multiplication tends toward the dominant eigendirection under suitable conditions. These classical results motivate the investigation of long-term behaviour in Matrix PDT. Whether analogous convergence results hold after repeated normalization remains an open mathematical question. == Fixed Points == A fixed point satisfies <math> T_M(P)=P. </math> Equivalently, <math> MP = \alpha P, </math> for some positive scalar <math> \alpha. </math> Thus every fixed point of Matrix PDT corresponds to a normalized positive eigenvector of the dilation matrix. This relationship provides one of the principal motivations for studying spectral properties. == Stability == Suppose <math> P^* </math> is a fixed point. Questions of interest include * local stability, * global stability, * convergence rates, * sensitivity to perturbations, * dependence upon the spectrum of <math> M. </math> These remain topics for future investigation. == Entropy Evolution == Matrix coupling introduces interactions between probability states that may alter entropy differently from diagonal PDT. Open questions include * monotonicity of entropy, * entropy production, * equilibrium entropy, * dependence upon spectral radius, * relation between dominant eigenvectors and entropy-maximizing states. == Continuous-Time Perspective == If <math> M=I+\Delta t\,A, </math> with <math> \Delta t\rightarrow0, </math> one may formally investigate continuous-time limits of Matrix PDT. Possible limiting equations include nonlinear probability evolution equations on the probability simplex. Rigorous derivation of these limits remains an open problem. == Discussion == Spectral analysis provides one possible mathematical framework for understanding Matrix Dilation Operators. Connections with positive matrices, eigenvalue theory, and Perron–Frobenius theory suggest that long-term behaviour may be characterized by classical spectral methods together with nonlinear normalization. Whether these ideas yield new mathematical results within PDT remains an open area of research. == Open Questions == Future investigations include * characterization of fixed points, * uniqueness of equilibrium distributions, * spectral conditions for convergence, * Lyapunov stability, * entropy and spectral radius, * continuous-time limits, * operator-theoretic formulations, * infinite-dimensional extensions. == See also == * [[Probability Dilation Theory]] * [[Probability Dilation Theory/Matrix Dilation Operators]] * [[Probability Dilation Theory/Convergence and Fixed Points]] * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows]] * [[Probability Dilation Theory/Dilation Vector Field]] * [[Probability Dilation Theory/Euler Methods and Continuous-Time PDT]] * [[Probability Dilation Theory/Measure-Theoretic Foundations]] == References == * O. Perron (1907), ''Zur Theorie der Matrices''. * G. Frobenius (1912), ''Über Matrizen aus positiven Elementen''. * R. A. Horn and C. R. Johnson, ''Matrix Analysis''. * Howard Richardson, ''Probability Dilation Theory'', Wikiversity. mojag4kgv9u4d8kbhog5o9b5f8i153t 2819306 2819304 2026-07-24T18:13:33Z Howie2024 2995240 /* Probability Dilation Theory / Spectral Analysis of Matrix Dilation Operators */ 2819306 wikitext text/x-wiki This page is a subpage of [[Probability Dilation Theory]]. == Overview == This page investigates the spectral properties of Matrix Dilation Operators introduced as a proposed generalization of Probability Dilation Theory (PDT). The objective is to examine how eigenvalues, eigenvectors, and positive matrix theory may characterize the long-term behaviour of repeated matrix dilation. This page is exploratory and extends the mathematical framework of PDT without modifying the core diagonal formulation. == Background == The generalized matrix dilation operator is <math> T_M(P) = \frac{MP}{\|MP\|_1}, </math> where * <math>P</math> is a probability vector, * <math>M</math> is a non-negative matrix, * normalization preserves total probability. Repeated iteration produces <math> P_{n+1} = T_M(P_n). </math> The resulting nonlinear iteration motivates the study of spectral properties of the underlying matrix. == Eigenvalues and Eigenvectors == Consider <math> Mv=\lambda v, </math> where * <math>\lambda</math> is an eigenvalue, * <math>v</math> is the corresponding eigenvector. ::contentReference[oaicite:0]{index=0} The dominant eigenvalue often governs the asymptotic behaviour of repeated matrix multiplication. Because Matrix PDT includes normalization after each iteration, the resulting dynamics differ from ordinary linear iteration while still retaining connections to classical spectral theory. == Perron–Frobenius Theory == When <math> M </math> is a positive matrix, the Perron–Frobenius theorem establishes that * a largest positive eigenvalue exists, * the dominant eigenvector has strictly positive components, * repeated multiplication tends toward the dominant eigendirection under suitable conditions. These classical results motivate the investigation of long-term behaviour in Matrix PDT. Whether analogous convergence results hold after repeated normalization remains an open mathematical question. == Fixed Points == A fixed point satisfies <math> T_M(P)=P. </math> Equivalently, <math> MP = \alpha P, </math> for some positive scalar <math> \alpha. </math> Thus every fixed point of Matrix PDT corresponds to a normalized positive eigenvector of the dilation matrix. This relationship provides one of the principal motivations for studying spectral properties. == Stability == Suppose <math> P^* </math> is a fixed point. Questions of interest include * local stability, * global stability, * convergence rates, * sensitivity to perturbations, * dependence upon the spectrum of <math> M. </math> These remain topics for future investigation. == Entropy Evolution == Matrix coupling introduces interactions between probability states that may alter entropy differently from diagonal PDT. Open questions include * monotonicity of entropy, * entropy production, * equilibrium entropy, * dependence upon spectral radius, * relation between dominant eigenvectors and entropy-maximizing states. == Continuous-Time Perspective == If <math> M=I+\Delta t\,A, </math> with <math> \Delta t\rightarrow0, </math> one may formally investigate continuous-time limits of Matrix PDT. Possible limiting equations include nonlinear probability evolution equations on the probability simplex. Rigorous derivation of these limits remains an open problem. == Discussion == Spectral analysis provides one possible mathematical framework for understanding Matrix Dilation Operators. Connections with positive matrices, eigenvalue theory, and Perron–Frobenius theory suggest that long-term behaviour may be characterized by classical spectral methods together with nonlinear normalization. Whether these ideas yield new mathematical results within PDT remains an open area of research. == Open Questions == Future investigations include * characterization of fixed points, * uniqueness of equilibrium distributions, * spectral conditions for convergence, * Lyapunov stability, * entropy and spectral radius, * continuous-time limits, * operator-theoretic formulations, * infinite-dimensional extensions. == See also == * [[Probability Dilation Theory]] * [[Probability Dilation Theory/Matrix Dilation Operators]] * [[Probability Dilation Theory/Convergence and Fixed Points]] * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows]] * [[Probability Dilation Theory/Dilation Vector Field]] * [[Probability Dilation Theory/Euler Methods and Continuous-Time PDT]] * [[Probability Dilation Theory/Measure-Theoretic Foundations]] == References == * O. Perron (1907), ''Zur Theorie der Matrices''. * G. Frobenius (1912), ''Über Matrizen aus positiven Elementen''. * R. A. Horn and C. R. Johnson, ''Matrix Analysis''. * Howard Richardson, ''Probability Dilation Theory'', Wikiversity. a2cfnclhyt0i8e0c92iu3pp5jer3mif 2819307 2819306 2026-07-24T18:14:37Z Howie2024 2995240 /* Eigenvalues and Eigenvectors */ 2819307 wikitext text/x-wiki This page is a subpage of [[Probability Dilation Theory]]. == Overview == This page investigates the spectral properties of Matrix Dilation Operators introduced as a proposed generalization of Probability Dilation Theory (PDT). The objective is to examine how eigenvalues, eigenvectors, and positive matrix theory may characterize the long-term behaviour of repeated matrix dilation. This page is exploratory and extends the mathematical framework of PDT without modifying the core diagonal formulation. == Background == The generalized matrix dilation operator is <math> T_M(P) = \frac{MP}{\|MP\|_1}, </math> where * <math>P</math> is a probability vector, * <math>M</math> is a non-negative matrix, * normalization preserves total probability. Repeated iteration produces <math> P_{n+1} = T_M(P_n). </math> The resulting nonlinear iteration motivates the study of spectral properties of the underlying matrix. == Eigenvalues and Eigenvectors == Consider <math> Mv=\lambda v, </math> where * <math>\lambda</math> is an eigenvalue, * <math>v</math> is the corresponding eigenvector. The dominant eigenvalue often governs the asymptotic behaviour of repeated matrix multiplication. Because Matrix PDT includes normalization after each iteration, the resulting dynamics differ from ordinary linear iteration while still retaining connections to classical spectral theory. == Perron–Frobenius Theory == When <math> M </math> is a positive matrix, the Perron–Frobenius theorem establishes that * a largest positive eigenvalue exists, * the dominant eigenvector has strictly positive components, * repeated multiplication tends toward the dominant eigendirection under suitable conditions. These classical results motivate the investigation of long-term behaviour in Matrix PDT. Whether analogous convergence results hold after repeated normalization remains an open mathematical question. == Fixed Points == A fixed point satisfies <math> T_M(P)=P. </math> Equivalently, <math> MP = \alpha P, </math> for some positive scalar <math> \alpha. </math> Thus every fixed point of Matrix PDT corresponds to a normalized positive eigenvector of the dilation matrix. This relationship provides one of the principal motivations for studying spectral properties. == Stability == Suppose <math> P^* </math> is a fixed point. Questions of interest include * local stability, * global stability, * convergence rates, * sensitivity to perturbations, * dependence upon the spectrum of <math> M. </math> These remain topics for future investigation. == Entropy Evolution == Matrix coupling introduces interactions between probability states that may alter entropy differently from diagonal PDT. Open questions include * monotonicity of entropy, * entropy production, * equilibrium entropy, * dependence upon spectral radius, * relation between dominant eigenvectors and entropy-maximizing states. == Continuous-Time Perspective == If <math> M=I+\Delta t\,A, </math> with <math> \Delta t\rightarrow0, </math> one may formally investigate continuous-time limits of Matrix PDT. Possible limiting equations include nonlinear probability evolution equations on the probability simplex. Rigorous derivation of these limits remains an open problem. == Discussion == Spectral analysis provides one possible mathematical framework for understanding Matrix Dilation Operators. Connections with positive matrices, eigenvalue theory, and Perron–Frobenius theory suggest that long-term behaviour may be characterized by classical spectral methods together with nonlinear normalization. Whether these ideas yield new mathematical results within PDT remains an open area of research. == Open Questions == Future investigations include * characterization of fixed points, * uniqueness of equilibrium distributions, * spectral conditions for convergence, * Lyapunov stability, * entropy and spectral radius, * continuous-time limits, * operator-theoretic formulations, * infinite-dimensional extensions. == See also == * [[Probability Dilation Theory]] * [[Probability Dilation Theory/Matrix Dilation Operators]] * [[Probability Dilation Theory/Convergence and Fixed Points]] * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows]] * [[Probability Dilation Theory/Dilation Vector Field]] * [[Probability Dilation Theory/Euler Methods and Continuous-Time PDT]] * [[Probability Dilation Theory/Measure-Theoretic Foundations]] == References == * O. Perron (1907), ''Zur Theorie der Matrices''. * G. Frobenius (1912), ''Über Matrizen aus positiven Elementen''. * R. A. Horn and C. R. Johnson, ''Matrix Analysis''. * Howard Richardson, ''Probability Dilation Theory'', Wikiversity. lf793awn28dso24me5kzmmyj0wgggpe 2 330730 2819329 2026-07-25T01:38:06Z Evan Mercer 3071189 Its my sandbox do i need to explain 2819329 wikitext text/x-wiki = Algebra 1: course design plan = ''A design document for rebuilding the whole course. Intended to live at [[Algebra 1/Course design]] or on the course talk page.'' == The problem with the current course == The existing units were written at different times by different people and it shows. Specific issues: * '''No consistent unit skeleton.''' Unit 1 is chatty and unstructured, Unit 4 is a bare definition dump, Unit 7 opens with a proper introduction and objectives. A learner cannot build a routine. * '''Order is wrong in places.''' Functions (Unit 4) arrives before graphing (Unit 5) and before slope (Unit 6). Students need to plot many concrete relationships before the abstraction "function" means anything. Slope belongs with linear functions, not in a unit of its own after graphing. * '''Almost no assessment.''' There is one quiz, on a subpage, at the end of Unit 2. Wikiversity has the quiz extension built in and it is barely used. * '''Errors of fact.''' Unit 2 lists "per" under both multiplication and division, and treats "of" as a general multiplication keyword. * '''Notation stops at Algebra 1 level and stays there.''' Nothing prepares a learner for how the same ideas are written by mathematicians. This is the gap the notation thread below is designed to close. == Design principle 1: one skeleton, every unit == Every unit page uses the same seven sections in the same order. Predictability is worth more than variety here, because most Wikiversity learners are self-directed and need to know where things are. # '''A problem you cannot solve yet.''' Open with a concrete question the unit will answer. Do not resolve it until the worked example section. # '''Core ideas.''' Concepts, each with a worked example immediately after it. Not a list of definitions followed by a list of examples. # '''Check yourself.''' A short embedded <code><nowiki><quiz display=simple></nowiki></code> of two to four questions after each major idea, not just at the end. # '''Writing it properly.''' The notation upgrade. See principle 3. # '''The mistakes that cost marks.''' Named, numbered, specific. # '''Practice''' with answers on the page. # '''Unit quiz''' on a subpage, graded, with feedback on every option. == Design principle 2: fix the sequence == Proposed order. Bold entries are structural changes from the current course. {| class="wikitable" !# !Unit !Why here |- |1 |Foundations: number sets, operations, order of operations, properties |Must establish <math>\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}</math> early or the notation thread has nothing to stand on |- |2 |Translating English into algebra |Cannot model anything without it |- |3 |Linear equations in one variable |The workhorse |- |4 |'''Ratio, proportion, percent, units''' |New. The most common real use of algebra, and currently missing entirely |- |5 |'''The coordinate plane and relations''' |Moved before functions. Concrete plotting first |- |6 |Functions |Moved after graphing. Abstraction needs examples underneath it |- |7 |Linear functions: slope, intercepts, forms |'''Merges current Units 5 and 6.''' Slope is not a separate topic from linear graphing |- |8 |Modelling with linear functions; lines of best fit |Where units 4 and 7 pay off |- |9 |Inequalities and interval notation |Currently Unit 7 |- |10 |Systems of linear equations | |- |11 |Exponents and exponential growth | |- |12 |Polynomials and their arithmetic | |- |13 |Factoring | |- |14 |Quadratic functions and their graphs | |- |15 |Solving quadratics; the discriminant | |- |16 |Radicals, rational exponents, Pythagoras | |- |17 |Rational expressions and domain restrictions | |- |18 |'''Sequences and series''' |New. The natural home for <math>\Sigma</math> |- |19 |'''Data, averages and spread''' |New. The natural home for <math>\bar{x}</math> and <math>\sigma</math> |} Each unit page should carry a small prerequisite line at the top, because Wikiversity learners arrive from search engines in the middle of a course rather than at the start. == Design principle 3: the notation thread == This is the spine of the rebuild and the thing that makes the course different from every other free algebra course. '''The idea.''' Unit 2 teaches English into symbols. The notation thread runs the same skill in the opposite direction, symbols into English, at a steadily rising level of sophistication. Every unit ends with a short section, always titled '''Writing it properly''', which re-expresses that unit's content in the notation a mathematician would actually use, and teaches the learner to read it aloud. '''Why it works.''' The content stays at Algebra 1 difficulty. Only the notation gets more grown up. A student meeting <math>\{x \in \mathbb{R} : 2x + 1 = 7\}</math> in Unit 3 is not learning harder algebra, they are learning that the answer to an equation is a ''set'', which they already understood informally. By the time <math>\sum_{i=1}^{n} x_i</math> arrives in Unit 18 it is the twelfth new symbol, not the first, and the learner has a habit of decoding rather than panicking. '''Rules for the thread.''' * Never introduce a symbol without its spoken form. <math>x \mapsto 2x</math> is read "x maps to two x", and the page must say so. * Never introduce a symbol without the informal version it replaces, side by side. * Never introduce more than three new symbols in one unit. * Every symbol goes into the running [[Algebra 1/Symbol reference|symbol reference]] page, which the learner can keep open in a tab. '''The ladder.''' {| class="wikitable" !Unit !Symbols introduced !Spoken as |- |1 |<math>\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}</math>; <math>\in</math>; <math>\notin</math>; <math>\neq</math>; <math>\approx</math>; <math>\vert x \vert</math> |"is an element of"; "the absolute value of x" |- |2 |<math>:=</math>; formal grouping |"is defined to be" |- |3 |<math>\{x : \ldots\}</math>; <math>\iff</math>; <math>\varnothing</math> |"the set of x such that"; "if and only if"; "the empty set" |- |4 |<math>\propto</math>; <math>a : b</math> |"is proportional to"; "a to b" |- |5 |<math>(x, y)</math>; <math>\mathbb{R}^2</math>; <math>A \times B</math> |"R two"; "A cross B" |- |6 |<math>f : A \to B</math>; <math>x \mapsto f(x)</math>; <math>\operatorname{dom} f</math> |"f from A to B"; "x maps to f of x" |- |7 |<math>\Delta y / \Delta x</math>; <math>x_1, y_1</math> |"delta y over delta x"; "x sub one" |- |8 |<math>[a, b)</math>; <math>\cup</math>; <math>\cap</math>; <math>\leq</math>; <math>\geq</math> |"union"; "intersection" |- |9 |system brace; <math>\wedge</math> |"and" |- |10 |<math>x^{-n}</math>; scientific notation | |- |11 |<math>a_n x^n + \cdots + a_0</math>; first sight of <math>\sum</math> |"a sub n"; "the sum of" |- |12 |<math>\prod</math> lightly |"the product of" |- |13 |<math>\pm</math>; <math>\Delta</math> as discriminant; <math>x_1, x_2</math> as roots |"plus or minus" |- |14 |<math>\sqrt[n]{x} = x^{1/n}</math> |"the nth root of x" |- |15 |domain restriction <math>x \neq a</math> | |- |16 |<math>a_n</math>; recursive vs closed form | |- |18 |<math>\sum_{i=1}^{n} a_i</math> in full, with index and bounds |"the sum, from i equals one to n, of a sub i" |- |19 |<math>\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i</math>; <math>\sigma</math> |"x bar"; "sigma" |} Sigma appears twice on purpose. Unit 11 shows it as shorthand a learner can simply read. Unit 18 teaches them to write it themselves. == Design principle 4: quizzes everywhere == Two kinds, used differently. '''Inline self-checks.''' Two to four questions, placed directly after the idea they test, using <code><nowiki><quiz display=simple></nowiki></code> to suppress the points table. Purpose is immediate feedback, not grading. Every wrong option gets a feedback line that names the specific misconception, not just "incorrect". '''Unit quizzes.''' On a subpage, ten to fifteen questions, full <code><nowiki><quiz></nowiki></code> tag with the points table, feedback on every option, and <code>coef</code> weighting so the questions that test the unit's central idea count for more. === Editorial rules for quiz markup === Learned the hard way, and worth putting on the style page: * Every question must be separated by a '''blank line''' or the extension merges them. * Inside quiz tags, a line beginning with <code>+</code> or <code>-</code> is read as an answer. Never begin a prose line with a minus sign inside a quiz. Write "negative 5" or reword. * Symbols must start at the very beginning of the line. A single leading space breaks the question. * Text fields need spaces inside the braces. <code>{ x-5 }</code> works, <code>{x-5}</code> does not. * Text field answers are plain text and cannot contain <code><nowiki><math></nowiki></code>. Always list spacing variants with pipes: <code>{ x-5|x - 5|x−5 _6 }</code>. * Add <code>(i)</code> for case insensitivity on any answer containing letters. * Use <code>_n</code> to size the box to roughly the answer length plus two, or learners guess the answer's length from the box. * Numerical answers accept ranges (<code>91-95</code>) and tolerances (<code>100 5%</code>). * Do not put wiki tables inside quiz tags. The pipes collide. * You cannot test a quiz from the edit preview. The submit button is greyed out. Test in a sandbox subpage first. == Design principle 5: errors are content == Every unit closes with a numbered list titled '''The mistakes that cost the most marks'''. Not generic advice. Specific, observed, high-frequency errors, each with the wrong answer shown next to the right one. This is the section learners will actually come back to before a test, so it is worth writing carefully. == Suggested build order == Rewriting nineteen units at once will stall. Suggested order of work: # '''Unit 2''', as the template. It is the shortest and most self-contained, so use it to settle the skeleton, the tone and the quiz conventions. ''(Done.)'' # '''Unit 1''', which currently needs the most work and must be fixed before the notation thread can start. # '''The symbol reference page''', so later units have somewhere to link. # '''Unit 3''', the highest-traffic page in any algebra course. # Everything else in order. == Unit 1: what needs fixing == Unit 1 is the weakest page and it is load bearing. Problems: * The etymology of "al-jabr" and the history of al-Khwarizmi are interesting but arrive before the learner knows what a variable is. Move them to a sidebar. * The empty box notation is introduced and then abandoned. It is a good device and should be carried further, because "the box is now called x" is exactly the right first intuition. * The balance rule is stated once and never justified. It deserves a proper treatment with a physical balance image, because it is the single rule the next fifteen units depend on. * Number sets are never introduced. Without <math>\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}</math> the learner has no way to answer "what kind of thing is x allowed to be", which becomes urgent in Unit 6 (domains), Unit 15 (restrictions) and Unit 16 (roots). * Properties of operations (commutative, associative, distributive) are never named. Distributivity in particular has to be solid before Unit 12. * Arithmetic warm ups on fractions, decimals and percents are the right instinct, but should be a linked prerequisite check with a quiz, not body text in the opening unit. Proposed Unit 1 contents: what algebra is and what it buys you; the box and the variable; number sets and <math>\in</math>; the four operations and their properties; order of operations; the balance rule with justification; absolute value; a diagnostic quiz on prerequisite arithmetic. r1b0sw4exvk8jn9yb5orla2kxzha1tu 2 330731 2819330 2026-07-25T01:42:51Z Evan Mercer 3071189 Created page with "= Algebra 1: course design plan = ''A design document for rebuilding the whole course. Intended to live at [[Algebra 1/Course design]] or on the course talk page.'' == The problem with the current course == The existing units were written at different times by different people and it shows. Specific issues: * '''No consistent unit skeleton.''' Unit 1 is chatty and unstructured, Unit 4 is a bare definition dump, Unit 7 opens with a proper introduction and objectives...." 2819330 wikitext text/x-wiki = Algebra 1: course design plan = ''A design document for rebuilding the whole course. Intended to live at [[Algebra 1/Course design]] or on the course talk page.'' == The problem with the current course == The existing units were written at different times by different people and it shows. Specific issues: * '''No consistent unit skeleton.''' Unit 1 is chatty and unstructured, Unit 4 is a bare definition dump, Unit 7 opens with a proper introduction and objectives. A learner cannot build a routine. * '''Order is wrong in places.''' Functions (Unit 4) arrives before graphing (Unit 5) and before slope (Unit 6). Students need to plot many concrete relationships before the abstraction "function" means anything. Slope belongs with linear functions, not in a unit of its own after graphing. * '''Almost no assessment.''' There is one quiz, on a subpage, at the end of Unit 2. Wikiversity has the quiz extension built in and it is barely used. * '''Errors of fact.''' Unit 2 lists "per" under both multiplication and division, and treats "of" as a general multiplication keyword. * '''Notation stops at Algebra 1 level and stays there.''' Nothing prepares a learner for how the same ideas are written by mathematicians. This is the gap the notation thread below is designed to close. == Design principle 1: one skeleton, every unit == Every unit page uses the same seven sections in the same order. Predictability is worth more than variety here, because most Wikiversity learners are self-directed and need to know where things are. # '''A problem you cannot solve yet.''' Open with a concrete question the unit will answer. Do not resolve it until the worked example section. # '''Core ideas.''' Concepts, each with a worked example immediately after it. Not a list of definitions followed by a list of examples. # '''Check yourself.''' A short embedded <code><nowiki><quiz display=simple></nowiki></code> of two to four questions after each major idea, not just at the end. # '''Writing it properly.''' The notation upgrade. See principle 3. # '''The mistakes that cost marks.''' Named, numbered, specific. # '''Practice''' with answers on the page. # '''Unit quiz''' on a subpage, graded, with feedback on every option. == Design principle 2: fix the sequence == Proposed order. Bold entries are structural changes from the current course. {| class="wikitable" ! # !! Unit !! Why here |- | 1 || Foundations: number sets, operations, order of operations, properties || Must establish <math>\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}</math> early or the notation thread has nothing to stand on |- | 2 || Translating English into algebra || Cannot model anything without it |- | 3 || Linear equations in one variable || The workhorse |- | 4 || '''Ratio, proportion, percent, units''' || New. The most common real use of algebra, and currently missing entirely |- | 5 || '''The coordinate plane and relations''' || Moved before functions. Concrete plotting first |- | 6 || Functions || Moved after graphing. Abstraction needs examples underneath it |- | 7 || Linear functions: slope, intercepts, forms || '''Merges current Units 5 and 6.''' Slope is not a separate topic from linear graphing |- | 8 || Modelling with linear functions; lines of best fit || Where units 4 and 7 pay off |- | 9 || Inequalities and interval notation || Currently Unit 7 |- | 10 || Systems of linear equations || |- | 11 || Exponents and exponential growth || |- | 12 || Polynomials and their arithmetic || |- | 13 || Factoring || |- | 14 || Quadratic functions and their graphs || |- | 15 || Solving quadratics; the discriminant || |- | 16 || Radicals, rational exponents, Pythagoras || |- | 17 || Rational expressions and domain restrictions || |- | 18 || '''Sequences and series''' || New. The natural home for <math>\Sigma</math> |- | 19 || '''Data, averages and spread''' || New. The natural home for <math>\bar{x}</math> and <math>\sigma</math> |} Each unit page should carry a small prerequisite line at the top, because Wikiversity learners arrive from search engines in the middle of a course rather than at the start. == Design principle 3: the notation thread == This is the spine of the rebuild and the thing that makes the course different from every other free algebra course. '''The idea.''' Unit 2 teaches English into symbols. The notation thread runs the same skill in the opposite direction, symbols into English, at a steadily rising level of sophistication. Every unit ends with a short section, always titled '''Writing it properly''', which re-expresses that unit's content in the notation a mathematician would actually use, and teaches the learner to read it aloud. '''Why it works.''' The content stays at Algebra 1 difficulty. Only the notation gets more grown up. A student meeting <math>\{x \in \mathbb{R} : 2x + 1 = 7\}</math> in Unit 3 is not learning harder algebra, they are learning that the answer to an equation is a ''set'', which they already understood informally. By the time <math>\sum_{i=1}^{n} x_i</math> arrives in Unit 18 it is the twelfth new symbol, not the first, and the learner has a habit of decoding rather than panicking. '''Rules for the thread.''' * Never introduce a symbol without its spoken form. <math>x \mapsto 2x</math> is read "x maps to two x", and the page must say so. * Never introduce a symbol without the informal version it replaces, side by side. * Never introduce more than three new symbols in one unit. * Every symbol goes into the running [[Algebra 1/Symbol reference|symbol reference]] page, which the learner can keep open in a tab. '''The ladder.''' {| class="wikitable" ! Unit !! Symbols introduced !! Spoken as |- | 1 || <math>\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}</math>; <math>\in</math>; <math>\notin</math>; <math>\neq</math>; <math>\approx</math>; <math>\vert x \vert</math> || "is an element of"; "the absolute value of x" |- | 2 || <math>:=</math>; formal grouping || "is defined to be" |- | 3 || <math>\{x : \ldots\}</math>; <math>\iff</math>; <math>\varnothing</math> || "the set of x such that"; "if and only if"; "the empty set" |- | 4 || <math>\propto</math>; <math>a : b</math> || "is proportional to"; "a to b" |- | 5 || <math>(x, y)</math>; <math>\mathbb{R}^2</math>; <math>A \times B</math> || "R two"; "A cross B" |- | 6 || <math>f : A \to B</math>; <math>x \mapsto f(x)</math>; <math>\operatorname{dom} f</math> || "f from A to B"; "x maps to f of x" |- | 7 || <math>\Delta y / \Delta x</math>; <math>x_1, y_1</math> || "delta y over delta x"; "x sub one" |- | 8 || <math>[a, b)</math>; <math>\cup</math>; <math>\cap</math>; <math>\leq</math>; <math>\geq</math> || "union"; "intersection" |- | 9 || system brace; <math>\wedge</math> || "and" |- | 10 || <math>x^{-n}</math>; scientific notation || |- | 11 || <math>a_n x^n + \cdots + a_0</math>; first sight of <math>\sum</math> || "a sub n"; "the sum of" |- | 12 || <math>\prod</math> lightly || "the product of" |- | 13 || <math>\pm</math>; <math>\Delta</math> as discriminant; <math>x_1, x_2</math> as roots || "plus or minus" |- | 14 || <math>\sqrt[n]{x} = x^{1/n}</math> || "the nth root of x" |- | 15 || domain restriction <math>x \neq a</math> || |- | 16 || <math>a_n</math>; recursive vs closed form || |- | 18 || <math>\sum_{i=1}^{n} a_i</math> in full, with index and bounds || "the sum, from i equals one to n, of a sub i" |- | 19 || <math>\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i</math>; <math>\sigma</math> || "x bar"; "sigma" |} Sigma appears twice on purpose. Unit 11 shows it as shorthand a learner can simply read. Unit 18 teaches them to write it themselves. == Design principle 4: quizzes everywhere == Two kinds, used differently. '''Inline self-checks.''' Two to four questions, placed directly after the idea they test, using <code><nowiki><quiz display=simple></nowiki></code> to suppress the points table. Purpose is immediate feedback, not grading. Every wrong option gets a feedback line that names the specific misconception, not just "incorrect". '''Unit quizzes.''' On a subpage, ten to fifteen questions, full <code><nowiki><quiz></nowiki></code> tag with the points table, feedback on every option, and <code>coef</code> weighting so the questions that test the unit's central idea count for more. === Editorial rules for quiz markup === Learned the hard way, and worth putting on the style page: * Every question must be separated by a '''blank line''' or the extension merges them. * Inside quiz tags, a line beginning with <code>+</code> or <code>-</code> is read as an answer. Never begin a prose line with a minus sign inside a quiz. Write "negative 5" or reword. * Symbols must start at the very beginning of the line. A single leading space breaks the question. * Text fields need spaces inside the braces. <code><nowiki>{ x-5 }</nowiki></code> works, <code><nowiki>{x-5}</nowiki></code> does not. * Text field answers are plain text and cannot contain <code><nowiki><math></nowiki></code>. Always list spacing variants with pipes: <code><nowiki>{ x-5|x - 5|x−5 _6 }</nowiki></code>. * Add <code>(i)</code> for case insensitivity on any answer containing letters. * Use <code>_n</code> to size the box to roughly the answer length plus two, or learners guess the answer's length from the box. * Numerical answers accept ranges (<code>91-95</code>) and tolerances (<code>100 5%</code>). * Do not put wiki tables inside quiz tags. The pipes collide. * You cannot test a quiz from the edit preview. The submit button is greyed out. Test in a sandbox subpage first. == Design principle 5: errors are content == Every unit closes with a numbered list titled '''The mistakes that cost the most marks'''. Not generic advice. Specific, observed, high-frequency errors, each with the wrong answer shown next to the right one. This is the section learners will actually come back to before a test, so it is worth writing carefully. == Suggested build order == Rewriting nineteen units at once will stall. Suggested order of work: # '''Unit 2''', as the template. It is the shortest and most self-contained, so use it to settle the skeleton, the tone and the quiz conventions. ''(Done.)'' # '''Unit 1''', which currently needs the most work and must be fixed before the notation thread can start. # '''The symbol reference page''', so later units have somewhere to link. # '''Unit 3''', the highest-traffic page in any algebra course. # Everything else in order. == Unit 1: what needs fixing == Unit 1 is the weakest page and it is load bearing. Problems: * The etymology of "al-jabr" and the history of al-Khwarizmi are interesting but arrive before the learner knows what a variable is. Move them to a sidebar. * The empty box notation is introduced and then abandoned. It is a good device and should be carried further, because "the box is now called x" is exactly the right first intuition. * The balance rule is stated once and never justified. It deserves a proper treatment with a physical balance image, because it is the single rule the next fifteen units depend on. * Number sets are never introduced. Without <math>\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}</math> the learner has no way to answer "what kind of thing is x allowed to be", which becomes urgent in Unit 6 (domains), Unit 15 (restrictions) and Unit 16 (roots). * Properties of operations (commutative, associative, distributive) are never named. Distributivity in particular has to be solid before Unit 12. * Arithmetic warm ups on fractions, decimals and percents are the right instinct, but should be a linked prerequisite check with a quiz, not body text in the opening unit. Proposed Unit 1 contents: what algebra is and what it buys you; the box and the variable; number sets and <math>\in</math>; the four operations and their properties; order of operations; the balance rule with justification; absolute value; a diagnostic quiz on prerequisite arithmetic. 6v74909tu35ncsi9i15eekmqeat4tux 2819331 2819330 2026-07-25T01:43:42Z Evan Mercer 3071189 2819331 wikitext text/x-wiki ''Prerequisite: [[Algebra 1/Unit 1: Introduction To Algebra|Unit 1]]. You should know what a variable is and that an equation stays true if you do the same thing to both sides.'' = Unit 2: Translating English into algebra = == A problem you cannot solve yet == <blockquote>A taxi charges a $3 flat fee plus $2 per mile. Your ride cost $17. How far did you go?</blockquote> You already have the tools to solve this. What you do not have yet is a way to get from that ''sentence'' to something you can solve. That conversion is the whole of this unit. It is not busywork. It is where most students actually lose marks, not in the algebra that follows. == The one big idea == '''Word order does not always match symbol order.''' Most people assume English reads left to right into maths left to right. Sometimes it does: : "8 decreased by 3" becomes <math>8 - 3</math> Sometimes it flatly does not: : "8 less than 3" becomes <math>3 - 8</math> The numbers appear in the same order in both sentences, yet the correct expressions are opposites. Translate word by word and you will get the second one wrong every time. So the first question to ask about any phrase is not "which operation is this?" but: '''Does this phrase keep the order, or flip it?''' For addition and multiplication a flip does not change the answer, so you can be sloppy and survive. For subtraction and division a flip changes everything. That is why subtraction and division are where the damage happens. == The flip list == Memorise these six. Everything else keeps the order. {| class="wikitable" ! Phrase !! Example !! Translation |- | '''A more than B''' || 5 more than ''x'' || <math>x + 5</math> |- | '''A added to B''' || 5 added to ''x'' || <math>x + 5</math> |- | '''A less than B''' || 5 less than ''x'' || <math>x - 5</math> |- | '''A fewer than B''' || 5 fewer than ''x'' || <math>x - 5</math> |- | '''A subtracted from B''' || 5 subtracted from ''x'' || <math>x - 5</math> |- | '''A divided into B''' || 5 divided into ''x'' || <math>\frac{x}{5}</math> |} The pattern behind all six: the phrase names the ''thing being done'' before it names the ''thing it is done to''. "5 more than ''x''" is a statement about ''x''. The ''x'' is the subject and the 5 is the adjustment, so ''x'' goes first. A trick that works. Rewrite the phrase using the words "starting from". "5 less than ''x''" becomes "starting from ''x'', go down 5", which is <math>x - 5</math>. If you cannot naturally say "starting from", you are probably not looking at a flip phrase. === Check yourself === <quiz display=simple> {Translate: 5 less than <math>x</math> |type="()"} + <math>x - 5</math> || Correct. "Less than" is a flip phrase, so <math>x</math> is the starting amount. - <math>5 - x</math> || This is the trap. The words appear in this order, but "less than" flips them. Start from <math>x</math>. - <math>x + 5</math> || Right order, wrong operation. "Less" means subtract. - <math>5x</math> || That is multiplication. Nothing here signals a product. {Translate: 5 decreased by <math>x</math> |type="()"} + <math>5 - x</math> || Correct. "Decreased by" keeps the order, so the 5 is the starting amount. - <math>x - 5</math> || "Decreased by" is not on the flip list. Compare it with "5 less than <math>x</math>", which does flip. - <math>x + 5</math> || Wrong operation. - <math>\frac{5}{x}</math> || Nothing here signals division. {Translate: 7 subtracted from <math>k</math> |type="()"} + <math>k - 7</math> || Correct. Whatever follows the word "from" is the starting amount. - <math>7 - k</math> || The word "from" points at <math>k</math>, so <math>k</math> goes first. - <math>7k</math> || Wrong operation. - <math>\frac{k}{7}</math> || Wrong operation. {Translate: 3 divided into <math>n</math> |type="()"} + <math>\frac{n}{3}</math> || Correct. Think of long division: 3 goes into <math>n</math>, so 3 is the divisor and sits underneath. - <math>\frac{3}{n}</math> || This would be "3 divided '''by''' <math>n</math>". The two phrases are opposites. - <math>3n</math> || Wrong operation. - <math>n - 3</math> || Wrong operation. </quiz> == Operation by operation == === Addition === '''Order keeping:''' plus, increased by, the sum of ___ and ___, the total of ___ and ___ '''Flipping:''' more than, added to {| class="wikitable" ! English !! Algebra |- | ''x'' plus 7 || <math>x + 7</math> |- | the sum of 6 and ''z'' || <math>6 + z</math> |- | ''x'' increased by 7 || <math>x + 7</math> |- | 7 more than ''x'' || <math>x + 7</math> |} Addition is commutative, so <math>x + 7</math> and <math>7 + x</math> are equal and both are correct. Write it the way your teacher models it, but do not lose sleep over it. === Subtraction === '''Order keeping:''' minus, decreased by, the difference of ___ and ___, take away '''Flipping:''' less than, fewer than, subtracted from {| class="wikitable" ! English !! Algebra |- | 8 minus ''x'' || <math>8 - x</math> |- | 8 decreased by ''x'' || <math>8 - x</math> |- | the difference of 8 and ''x'' || <math>8 - x</math> |- | 8 less than ''x'' || <math>x - 8</math> |- | 8 subtracted from ''x'' || <math>x - 8</math> |- | subtract 8 from ''x'' || <math>x - 8</math> |} Subtraction is '''not''' commutative. <math>8 - x</math> and <math>x - 8</math> are different numbers; each is the negative of the other. Getting this backwards is the single most common error in Algebra 1. Note the last two rows. Both "subtract A from B" and "A subtracted from B" give <math>B - A</math>. The word '''from''' is your signal: whatever follows "from" is the starting amount, so it goes first. === Multiplication === '''Keywords:''' times, multiplied by, the product of ___ and ___, twice (times 2), triple (times 3), each {| class="wikitable" ! English !! Algebra |- | 4 times ''u'' || <math>4u</math> |- | the product of 4 and ''u'' || <math>4u</math> |- | twice a number ''n'' || <math>2n</math> |- | ''u'' multiplied by 4 || <math>4u</math> |} By convention, write the number before the letter: <math>4u</math>, not <math>u4</math>. ==== A note on the word "of" ==== "Of" means multiply '''only when it follows a fraction, a decimal or a percent.''' {| class="wikitable" ! English !! Algebra |- | half of ''x'' || <math>\tfrac{x}{2}</math> |- | one third of ''x'' || <math>\tfrac{x}{3}</math> |- | 20% of ''x'' || <math>0.20x</math> |} An earlier version of this page gave "6 of ''r''" as an example meaning <math>6r</math>. That is not English anybody speaks, and treating "of" as a general multiplication keyword will confuse you the moment you meet "the sum '''of'''", "the quotient '''of'''" and "the difference '''of'''", where the word means nothing of the sort. Keep the rule restricted to fractions and percents. === Division === '''Order keeping:''' divided by, the quotient of ___ and ___, the ratio of ___ to ___, per, out of '''Flipping:''' divided into {| class="wikitable" ! English !! Algebra |- | ''n'' divided by 3 || <math>\frac{n}{3}</math> |- | the quotient of ''z'' and 4 || <math>\frac{z}{4}</math> |- | the ratio of ''w'' to 6 || <math>\frac{w}{6}</math> |- | miles per hour || <math>\frac{\text{miles}}{\text{hours}}</math> |- | 3 divided into ''n'' || <math>\frac{n}{3}</math> |} '''"Per" is division, always.''' An earlier version of this page listed "per" as a multiplication keyword ''and'' as a division keyword, which is simply an error. "Per" means "for each one", which is a rate, which is a quotient. Miles per gallon is <math>\tfrac{\text{miles}}{\text{gallons}}</math>. Cost per person is <math>\tfrac{\text{cost}}{\text{people}}</math>. '''"Divided into" versus "divided by".''' Think about how you learned long division: "3 goes into 12 four times." The number doing the going into is the divisor, so it ends up underneath. "3 divided into 12" is <math>\tfrac{12}{3}</math>, and "12 divided by 3" is also <math>\tfrac{12}{3}</math>. Same value, opposite word order. '''A warning about "times greater".''' English is genuinely ambiguous here. "3 times greater than ''x''" is used by most people to mean <math>3x</math>, but read literally it means <math>x + 3x = 4x</math>. Careful writers avoid the phrase. If a question asks "how many times greater is ''j'' than ''m''?", it wants the ratio <math>\tfrac{j}{m}</math>. === Check yourself === Type your answer into each box. Write products without a multiplication sign, so two times ''n'' is <code>2n</code>. Write division with a slash, so ''n'' over 3 is <code>n/3</code>. <quiz display=simple> {&nbsp; |type="{}"} 9 more than a number ''n'' is { n+9|n + 9|9+n|9 + n _6 } {&nbsp; |type="{}"} The difference of ''n'' and 9 is { n-9|n - 9 _6 } {&nbsp; |type="{}"} The quotient of 20 and ''k'' is { 20/k _6 } {&nbsp; |type="{}"} Twice a number ''n'' is { 2n|2*n _4 } {&nbsp; |type="{}"} 20 divided into ''k'' is { k/20 _6 } </quiz> == Grouping and parentheses == This is the hardest part of the unit. Here is the actual rule: '''You need parentheses whenever an operation is applied to the ''result'' of another operation.''' Two signals to watch for. '''1. The word "quantity".''' This word exists in maths problems for exactly one reason: to tell you where the parentheses go. : "the quantity ''x'' plus 3, times 2" becomes <math>2(x + 3)</math> '''2. A "the ___ of A and B" phrase used as a single chunk.''' When "the sum of ''a'' and ''b''" is itself being multiplied, divided or subtracted, it has to be bundled. {| class="wikitable" ! English !! Correct !! Wrong |- | twice the sum of ''x'' and 5 || <math>2(x + 5)</math> || <math>2x + 5</math> |- | the sum of ''x'' and 5, doubled || <math>2(x + 5)</math> || <math>2x + 5</math> |- | twice ''x'', increased by 5 || <math>2x + 5</math> || <math>2(x + 5)</math> |- | the difference of ''a'' and ''b'', divided by 6 || <math>\frac{a - b}{6}</math> || <math>a - \frac{b}{6}</math> |- | 3 less than the product of 4 and ''n'' || <math>4n - 3</math> || |- | 3 less than 4, times ''n'' || <math>(4 - 3)n</math> || |} Compare rows 1 and 3. The same words rearranged, with a comma doing the work, give completely different expressions. Punctuation carries real mathematical meaning in these sentences, so read it. '''Fraction bars group automatically.''' Written as a stacked fraction, <math>\frac{a - b}{6}</math> needs no parentheses, because the bar already does the bundling. Written on one line as <code>(a-b)/6</code>, it does. === Check yourself === <quiz display=simple> {Translate: twice the sum of <math>x</math> and 5 |type="()"} + <math>2(x + 5)</math> || Correct. "The sum of <math>x</math> and 5" is a single chunk, and the doubling applies to all of it. - <math>2x + 5</math> || This doubles only the <math>x</math>. The phrase says the whole sum is doubled. - <math>2x + 10</math> || This is what <math>2(x+5)</math> ''expands'' to, but the question asked you to translate, not to expand. As a translation it hides the structure. - <math>x + 10</math> || Neither the operation nor the grouping matches. {Translate: twice <math>x</math>, increased by 5 |type="()"} + <math>2x + 5</math> || Correct. The comma separates the doubling from the increase, so only <math>x</math> is doubled. - <math>2(x + 5)</math> || Read the comma. It closes off "twice <math>x</math>" before the increase is mentioned. - <math>2(x) + 2(5)</math> || These brackets do nothing useful and suggest the 5 is being doubled. - <math>7x</math> || You cannot combine <math>2x</math> and 5; they are unlike terms. {Translate: the difference of <math>a</math> and <math>b</math>, divided by 6 |type="()"} + <math>\frac{a - b}{6}</math> || Correct. The whole difference goes on top. - <math>a - \frac{b}{6}</math> || This divides only <math>b</math> by 6. The phrase divides the whole difference. - <math>\frac{6}{a - b}</math> || "Divided by" keeps the order, so the difference is on top. - <math>\frac{a}{6} - b</math> || Neither the grouping nor the order matches. </quiz> == Expressions and equations == An '''expression''' has no equals sign: <math>x + 5</math>. It is a value. You can simplify it but you cannot solve it. An '''equation''' has an equals sign: <math>x + 5 = 12</math>. It is a claim. You can solve it. The word that creates an equation is '''is''', along with ''equals, is equal to, results in, gives, yields, was'' and ''will be''. {| class="wikitable" ! English !! Algebra |- | 5 more than a number || <math>n + 5</math> (expression) |- | 5 more than a number is 12 || <math>n + 5 = 12</math> (equation) |- | the sum of a number and its double is 21 || <math>n + 2n = 21</math> |} Comparison words work the same way and return in [[Algebra 1/Unit 7: Inequalities|Unit 7]]: ''is greater than'' becomes <math>></math>, ''is less than'' becomes <math><</math>, ''is at least'' becomes <math>\geq</math>, ''is at most'' becomes <math>\leq</math>. Watch this collision carefully: * "5 less than <math>x</math>" is <math>x - 5</math>, an expression built on subtraction * "5 '''is''' less than <math>x</math>" is <math>5 < x</math>, an inequality One word changes everything. === Check yourself === <quiz display=simple> {Which of these is an equation? |type="()"} + <math>3n - 4 = 11</math> || Correct. It contains an equals sign, so it makes a claim you can test and solve. - <math>3n - 4</math> || No equals sign, so this is an expression. There is nothing to solve. - <math>3n</math> || An expression. - <math>n</math> || A single term, which is also an expression. {Translate: 5 is less than <math>x</math> |type="()"} + <math>5 < x</math> || Correct. The word "is" turns this into a comparison rather than a subtraction. - <math>x - 5</math> || This is "5 less than <math>x</math>", without the word "is". That one word changes the whole meaning. - <math>5 - x</math> || Not a comparison, and the wrong order for a subtraction too. - <math>5 > x</math> || The inequality points the wrong way. {Translate: the sum of a number and its double is 21 |type="()"} + <math>n + 2n = 21</math> || Correct. "Its double" refers back to the same number, so it must be the same letter. - <math>n + 2m = 21</math> || A second letter would mean a second, unrelated number. "Its double" refers to the first one. - <math>n + 2n</math> || The word "is" makes this an equation, so it needs the equals sign and the 21. - <math>2n = 21</math> || This drops the first <math>n</math> from the sum. </quiz> == Writing it properly == ''This section appears in every unit. It shows the notation a mathematician would use for what you just learned, so that by the end of the course the symbols are familiar rather than frightening. The maths does not get harder here. Only the handwriting does.'' === The definition symbol === In the worked example below, the first step is "let <math>m</math> be the number of miles driven". Written formally, that is: : <math>m := \text{the number of miles driven}</math> The symbol <math>:=</math> is read '''"is defined to be"'''. It matters because it is doing something different from an ordinary equals sign. <math>3 + 2m = 17</math> is a ''claim'' that might be true or false. <math>m := \text{miles}</math> is a ''decision'' you are making about what a letter will mean. Claims can be false. Definitions cannot. Some writers use <math>\equiv</math> or just <math>=</math> with the word "let" in front. All three are common. What matters is that you write the definition down somewhere, every time. === Reading an expression from the outside in === You have been building expressions from the inside out. Mathematicians usually ''read'' them from the outside in, naming the last operation first: {| class="wikitable" ! Expression !! Read from the outside in |- | <math>2(x + 3)</math> || "a doubling, of a sum" |- | <math>2x + 3</math> || "a sum, of a doubling and 3" |- | <math>\frac{a - b}{6}</math> || "a quotient, whose numerator is a difference" |} This is worth practising because it is exactly the skill of this unit running backwards, and because the outermost operation is the one you will undo ''first'' when you start solving equations in [[Algebra 1/Unit 3: Linear and Literal Equations, and how to solve them|Unit 3]]. === Coming later === Once you can name what kind of number <math>x</math> is allowed to be, you will be able to write the answer to an equation as a '''set''' rather than a sentence, like <math>\{x \in \mathbb{R} : 3 + 2x = 17\}</math>. That arrives in Unit 3. Add <math>:=</math> to your [[Algebra 1/Symbol reference|symbol reference]] page now. == The worked example == <blockquote>A taxi charges a $3 flat fee plus $2 per mile. Your ride cost $17. How far did you go?</blockquote> '''Step 1. Name the unknown.''' Let <math>m := \text{the number of miles driven}</math>. Write it down. Do not keep it in your head. '''Step 2. Translate piece by piece.''' * "$2 per mile" becomes <math>2m</math>, a rate multiplied by a quantity * "a $3 flat fee plus $2 per mile" becomes <math>3 + 2m</math> * "your ride cost $17" uses ''cost'' as an "is" word, giving <math>3 + 2m = 17</math> '''Step 3. Solve''' using the balance rule from Unit 1. : <math>3 + 2m = 17</math> : <math>2m = 14 \qquad \text{(subtract 3 from both sides)}</math> : <math>m = 7 \qquad \text{(divide both sides by 2)}</math> '''Step 4. Answer the question that was asked.''' Not "<math>m = 7</math>" but "the ride was 7 miles". Then check: <math>3 + 2(7) = 3 + 14 = 17</math>. Correct. == Practice == # 9 more than a number ''n'' # 9 less than a number ''n'' # a number ''n'' less than 9 # the difference of ''n'' and 9 # 12 subtracted from ''y'' # subtract ''y'' from 12 # twice the sum of ''x'' and 5 # twice ''x'', plus 5 # the quotient of 20 and ''k'' # 20 divided into ''k'' # one third of the sum of ''a'' and ''b'' # 5 less than three times a number # the sum of a number and 4, divided by 2 # a number is 8 more than twice another number # the ratio of ''p'' to ''q'', decreased by 7 # 40% of a number is 30 # the product of 6 and the quantity ''x'' minus 2 # a number decreased by the quotient of that number and 5 === Answers === # <math>n + 9</math> # <math>n - 9</math> # <math>9 - n</math> (compare with 2: here 9 is the starting point) # <math>n - 9</math> # <math>y - 12</math> (the word "from" points at <math>y</math>) # <math>12 - y</math> (the word "from" points at 12) # <math>2(x + 5)</math> # <math>2x + 5</math> # <math>\frac{20}{k}</math> # <math>\frac{k}{20}</math> # <math>\frac{a + b}{3}</math> # <math>3n - 5</math> # <math>\frac{n + 4}{2}</math> # <math>x = 2y + 8</math> # <math>\frac{p}{q} - 7</math> # <math>0.40n = 30</math> # <math>6(x - 2)</math> # <math>n - \frac{n}{5}</math> If you missed 3, 6 or 10, go back to the flip list. Those three test exactly one thing. == The mistakes that cost the most marks == # '''Translating "less than" from left to right.''' "5 less than <math>x</math>" is <math>x - 5</math>, never <math>5 - x</math>. # '''Dropping parentheses after "the sum of", "the difference of" or "the quantity".''' "Twice the sum of <math>x</math> and 3" is <math>2(x + 3)</math>, not <math>2x + 3</math>. # '''Treating "per" as multiplication.''' It is division, without exception. # '''Never naming the variable.''' Write the definition line before anything else. Half of all word problem errors are really bookkeeping errors. # '''Solving for <math>x</math> and stopping.''' The question asked for a distance, a price or an age. Answer ''that''. == Unit quiz == ''Editors: this quiz can stay here or be cut to [[Algebra 1/Unit 2: The Translation of Algebra/Translation of Algebra Quiz]]. The <code>coef</code> weightings give double marks to the flip phrases, which are the central idea of the unit.'' <quiz> {'''1.''' Translate: 12 subtracted from <math>y</math> |type="()" coef="2"} + <math>y - 12</math> || Correct. The word "from" points at <math>y</math>, so <math>y</math> is the starting amount. - <math>12 - y</math> || A flip phrase. Compare "subtract <math>y</math> '''from''' 12", which does give this answer. - <math>12y</math> || Wrong operation. - <math>\frac{y}{12}</math> || Wrong operation. {'''2.''' Translate: <math>w</math> decreased by 4 |type="()"} + <math>w - 4</math> || Correct. "Decreased by" keeps the order. - <math>4 - w</math> || "Decreased by" is not a flip phrase. - <math>w + 4</math> || "Decreased" means subtract. - <math>4w</math> || Wrong operation. {'''3.''' A car travels <math>d</math> miles in <math>t</math> hours. Its speed in miles per hour is: |type="()" coef="2"} + <math>\frac{d}{t}</math> || Correct. "Per" always means division, and it keeps the order given. - <math>dt</math> || "Per" is never multiplication. It describes a rate, which is a quotient. - <math>\frac{t}{d}</math> || Right operation, upside down. Miles per hour puts miles on top. - <math>d - t</math> || Wrong operation. {'''4.''' Translate: three times the quantity <math>n</math> minus 8 |type="()" coef="2"} + <math>3(n - 8)</math> || Correct. The word "quantity" is there precisely to tell you where the brackets go. - <math>3n - 8</math> || This would be "three times <math>n</math>, minus 8". The word "quantity" changes it. - <math>3n - 24</math> || This is the expanded form of the right answer, but as a translation it hides the structure the sentence described. - <math>n - 24</math> || The 3 has to multiply something. {'''5.''' Which of the following are '''equations'''? |type="[]"} + <math>2x = 10</math> || Yes. It has an equals sign. + <math>y < 3x + 1</math> || Careful. This is strictly an ''inequality'' rather than an equation, but like an equation it makes a claim you can test. Both are accepted here. - <math>4a + 7</math> || No equals sign, so this is an expression. - <math>\frac{p}{q} - 7</math> || An expression. {'''6.''' Translate: 6 divided into <math>x</math> |type="()" coef="2"} + <math>\frac{x}{6}</math> || Correct. The number doing the "going into" is the divisor and sits underneath. - <math>\frac{6}{x}</math> || This is "6 divided '''by''' <math>x</math>". "Into" and "by" are opposites. - <math>6x</math> || Wrong operation. - <math>x - 6</math> || Wrong operation. {'''7.''' Which phrase translates to <math>x + 9</math>? |type="[]"} + 9 more than <math>x</math> || Yes, a flip phrase, so <math>x</math> comes first. + <math>x</math> increased by 9 || Yes, and this one keeps the order. + the sum of <math>x</math> and 9 || Yes. Addition is commutative, so this is the same value. - 9 less than <math>x</math> || That is <math>x - 9</math>. {'''8.''' Translate: the ratio of <math>a</math> to <math>b</math>, increased by 5 |type="()"} + <math>\frac{a}{b} + 5</math> || Correct. "Ratio of A to B" keeps the order, and the comma closes it off before the increase. - <math>\frac{a}{b + 5}</math> || The 5 is added to the whole ratio, not to <math>b</math>. - <math>\frac{a + 5}{b}</math> || The 5 is added to the whole ratio, not to <math>a</math>. - <math>\frac{b}{a} + 5</math> || The ratio is upside down. {'''9.''' A number is 4 less than half of another number. Which equation says this? |type="()" coef="2"} + <math>x = \tfrac{y}{2} - 4</math> || Correct. "Half of" is a fraction followed by "of", so it is division, and "4 less than" flips. - <math>x = 4 - \tfrac{y}{2}</math> || "Less than" flips, so the half is the starting amount. - <math>x = \tfrac{y - 4}{2}</math> || This halves the difference. The sentence halves first, then subtracts. - <math>x = 2y - 4</math> || "Half of" is division by 2, not multiplication by 2. {'''10.''' What does the symbol <math>:=</math> mean? |type="()"} + is defined to be || Correct. It records a decision about what a letter will stand for, rather than making a claim that could be false. - is approximately equal to || That is <math>\approx</math>. - is not equal to || That is <math>\neq</math>. - is less than or equal to || That is <math>\leq</math>. {&nbsp; |type="{}"} '''11.''' A gym charges a $20 joining fee plus $15 per month. Let <math>m</math> be the number of months. Write an expression for the total cost: { 20+15m|20 + 15m|15m+20|15m + 20 _10 } {&nbsp; |type="{}"} '''12.''' Using your answer to question 11, write an equation saying the total cost is $95. Type only the right hand side after the equals sign: { 95 _4 } </quiz> == What comes next == [[Algebra 1/Unit 3: Linear and Literal Equations, and how to solve them|Unit 3]] takes the equations you can now build and teaches systematic methods for solving them. Everything from here onwards assumes you can read a sentence and write the equation hiding inside it, so this unit is worth overlearning. ---- [[Algebra 1]] &middot; [[Algebra 1/Unit 3: Linear and Literal Equations, and how to solve them|Unit 3: Linear and Literal Equations]] &rarr; 723cdatrkvikmihqu1jdczsjv6nxy85 3 330733 2819346 2026-07-25T09:19:56Z MathXplore 2888076 vandalism1 ([[m:User:ZbVl/VD|Vandoom]]) 2819346 wikitext text/x-wiki == 2026-07-25 == [[File:Information.svg|25px|alt=Information icon]] Hello, I’m letting you know that one or more of your recent contributions have been reverted because they did not appear constructive. 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Thank you.<!-- Glow-vandalism1 @ 1784971226797.4s --><nowiki></nowiki> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 09:20, 25 July 2026 (UTC) szfoqi9a4ryiv2p4j5f8m7ch0v8jce5