User:JWSchmidt

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Welcome to the Wikiversity user page of John Schmidt. If you need to contact me in a hurry, the best bet is to send me an [[:en:Special:Emailuser/JWSchmidt|email]] or leave a note on my [[User talk:JWSchmidt|talk page]]. Hopefully in the future we will all be able to instantly link into [[w:iChat|video chat]] discussion groups. Maybe some day Wikiversity will exist as an [[w:Immersion (virtual reality)|immersive]] virtual reality environment. See [[Education in virtual worlds]].

'''Multiple accounts'''. I have several Wikiversity user accounts such as [[User:JWSchmidt|User:JWS]]. Most of them are not used very much. &quot;JWS&quot; and &quot;JWSchmidt&quot; are the two main accounts. Others include [[User:JS]] (was renamed to [[User:JWSchmidt|User:JS~enwikiversity]], [[User:JWSurf]], [[User:JohnWSchmidt]], [[User:clueless]] (was renamed to [[User:JWSchmidt|User:Clueless~enwikiversity]]), [[User:JWSchmidt|User:testnocookies]], [[User:JWSchmidt|User:Trout of Doubt]], [[User:Beetlebaum]]. These accounts have been made for several reasons: to &quot;claim&quot; account names similar to my own real name, to test how the MediaWiki software &quot;treats&quot; new accounts or for special learning activities. 

'''The Rules'''. Feel free to edit this page, but since it is my user page, I usually get the last word.

==About me==
I'm interested in exploring [[wiki]] technology as a tool for supporting online learning. I enjoy using Wikiversity as playground for [[Portal:Education/Wikiversity model#The Wikiversity model|learning]]. At the [[betawikiversity:User:JWSchmidt|multi-lingual Wikiversity hub]] I have a special interest in the development of [[betawikiversity:Wikiversity talk:Research guidelines/En#Talk before change|the Wikiversity research guidelines]].{{Scholarly ethics}}

===My Wikiversity projects and activities===
[[User:JWSchmidt/2006#Learning projects of interest|Go here]] for the list of Wikiversity editing projects that was on the earlier version of this page. An alternative list is provided below.
{{User:JWSchmidt/Blog/Navigation}}
====My Wikiversity blog====
[[User:JWSchmidt/Blog|My Wikiversity blog]] is where I sometimes write about topics I am trying to understand. The blog originated on this page and is really just an extension of my user page. Everyone is welcome to edit my user pages and add comments/questions. However, I always get the option of having the last edit on my user page!

Back at the [[User:JWSchmidt/Blog/Start|start of the blog]] I spent a significant amount of time tracking the software problems of the Wikiversity website. 

One of the most unique entries in the blog is an [[User:JWSchmidt/Portal|essay]] about wiki and virtual realities.

Some of the blog entries are rather mundane accounts of mundane editing tasks such as the January 2007 effort to make [[User:JWSchmidt/Blog/2 February 2007|portal pages]] with easy-to-edit box-like sections including a section for featured content.

Sometimes I blog about issues at the level of the entire [[User:JWSchmidt/Blog/9 May 2007|Wikimedia Foundation]]. Although I have spent most of my &quot;wiki time&quot; at Wikiversity since it launched, I still think of myself as a &quot;Wikimedian&quot;.

====Wiki Scholar====
[[Image:Logo Movie.gif|thumb|left|135px|[[Portal:Wiki Scholar/About the movie|About this movie]].]]
In addition to personal learning blogs, I think Wikiversity should have some kind of &quot;community blog&quot;. My original idea for this was more like a &quot;magazine&quot; (see: [[Portal:Wiki Scholar]]). Another alternative is to use the [[Wiki Journal]] format. It might be better to just use a commercial blogging service that can feed into [[m:Planet Wikimedia|Planet Wikimedia]].

{{Free learning}}

====School of Free Learning====
The [[School:Free Learning|School of Free Learning]] is dedicated to the idea that learners should be encouraged to explore the topics that are of greatest personal interest. Like all [[Wikiversity:Schools|Wikiversity schools]], the School of Free Learning is a [[Wikiversity:Content development|content development project]] where Wikiversity participants can collaborate to create, develop and organize [[learning resource]]s.

====Science====
As a science geek, much of my wiki editing falls under the broad subject of [[Portal:Science|science]]. I'm particularly interested in [[Portal:Life Sciences|biology]]. While the term &quot;biology&quot; is new, interest in living organisms is integral to human nature. In Western culture, interesting [[Aristotle/On the Soul: discussion group|writing about biology]] goes right back to the point where alphabetic writing reached the Western world. From Aristotle to Francis Crick, naturalists and scientists have been fascinated by the invisible forces that animate living organisms and generate our conscious experiences. The struggle against invisibility was a major theme in the development of biology. Life is nearly [[Age of the Earth|as old as the Earth]] and as evolved on Earth, life is made possible by special molecular processes and even most [[Cell biology|cells]] are too small to be clearly seen. A major remaining challenge for biology is to work out the details of how the specialized molecular and cellular processes of the brain produce memory and [[Consciousness studies|consciousness]].{{Science}}

In an age of amazing advances in [[School:Medicine|medicine]] and discovery of our biological origins, an interesting cultural phenomenon is that some [[Portal:Religious studies|religions]] continue to deny our biological origins. Commercial forces acting through politics probably warp the reasoned application of scientific knowledge more than any other force. Tobacco companies deny that smoking causes cancer, oil companies deny the costs of pollution and danger of human-induced climate change, drug manufacturers pressure government agencies to approve useless and harmful drugs.....entire mis-information industries exist to produce junk science and confuse public understanding of scientific issues. We need good science education and [[Science Journalism|science journalism]] to keep politicians from being able to legislate contrary to scientific reality. The State of Kansas continues to teeter, torn between a political faction that views [[Science as religion/Darwinism|Darwinism as religion]] and another faction that views creationism as [[pseudoscience]].
{{Basic sciences}}

====Science Fiction====
[[Image:Lightanimation.gif|thumb|right|320px]]
History of science is often taught as a story of discovery after discovery after discovery. It is tempting to ignore all the confusion and errors of the past. If scientific results are always tentative and open to questioning, how do we know which current ideas will later be shown to be incomplete or wrong? Sometimes the hardest part of science is escaping from what we think we know and finding ways to open our thinking to new ideas. [[Exploring science through fiction|Science fiction]] provides a playground for thinking about new ideas and exploring the boundary between the known and the imagined. [[Protoscience|New sciences]] can form and the [[Science Fiction Challenge|boundary of science]] always seems to be shifting and being re-defined. Can we use new tools like wiki to keep up with the pace of change in science? Can [[How to use wiki technology as a free learner#Collaborative video creation|new media]] like collaborative science fiction writing help keep our thinking free enough to keep pace with science-driven technological and cultural changes?

====Wiki Studies====
I hope Wikiversity can become a good source of [[learning resource]]s for wiki in general, how to participate in Wikimedia wiki projects, and how to be a good online participant in the Open Culture movement.
*[[betawikiversity:Wikiversity:Research ethics/En#Research on wiki communities|Research on wikis]]
*[[Action research]]
*[[Introduction to Wikiversity]]
*[[MediaWiki/Introduction|Topic:MediaWiki]]
*[[Web 2.0]]
*[[Portal:Wiki|Wiki Portal]]
{{Editing help}}
{{Web 2.0}}
{{WORK&amp;PLAY}}

==Useful links==
*[http://en.wikiversity.org/w/index.php?title=Special:AllMessages&amp;ot=php system messages]

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== [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] ==

I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&amp;#126;2026-28640-56|&amp;#126;2026-28640-56]] ([[User talk:&amp;#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC)

:What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:35, 12 May 2026 (UTC)
: Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC)
::Sounds good -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:13, 15 May 2026 (UTC)
:::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC)

== 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:14, 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 &quot;practicing journalism&quot; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 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&lt;math&gt;\not=&lt;/math&gt; 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 &quot;Media &amp; Democracy&quot; 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 &quot;rehost Wikinews here&quot; 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 &quot;active editions&quot;?
:* 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 &quot;active editions&quot;?
::*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 &lt;em&gt;about&lt;/em&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 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|&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;''' Blue Rasberry '''&lt;/span&gt;]][[User talk:Bluerasberry|&lt;span style=&quot;cursor:help&quot;&gt;&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;(talk)&lt;/span&gt;&lt;/span&gt;]] 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 &quot;regular&quot; 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|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 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 &lt;code&gt;autopatrolled&lt;/code&gt; (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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:42, 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:16, 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: &quot;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.&quot;
:* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects?
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:13, 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:55, 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)

== Proposed user group and/or possible policy changes ==

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 &lt;code&gt;delete&lt;/code&gt; [but not &lt;code&gt;undelete&lt;/code&gt; or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and &lt;code&gt;protect&lt;/code&gt;), 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, ''&quot;II ...then you will be approved as a probationary custodian for a period of at least four weeks&quot;''.
:::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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:39, 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:06, 9 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)

== 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:24, 9 June 2026 (UTC)

== Image not displaying ==

Can anyone work out why this image isn't displaying?&lt;br&gt;
[[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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)

== wifi ==

hn [[Special:Contributions/&amp;#126;2026-34594-51|&amp;#126;2026-34594-51]] ([[User talk:&amp;#126;2026-34594-51|talk]]) 15:08, 13 June 2026 (UTC)</text>
      <sha1>m4azw8f54cncq9n72e5zxzb8u24xo2q</sha1>
    </revision>
    <revision>
      <id>2815497</id>
      <parentid>2815496</parentid>
      <timestamp>2026-06-13T15:55:33Z</timestamp>
      <contributor>
        <username>Juandev</username>
        <id>2651</id>
      </contributor>
      <comment>/* Towards an Ethics policy */ Reply</comment>
      <origin>2815497</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="38297" sha1="7ux2p5r6kwqrf3ofkvczgyvn8x9gq20" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
&lt;!-- MESSAGES GO BELOW --&gt;

== [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] ==

I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&amp;#126;2026-28640-56|&amp;#126;2026-28640-56]] ([[User talk:&amp;#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC)

:What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:35, 12 May 2026 (UTC)
: Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC)
::Sounds good -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:13, 15 May 2026 (UTC)
:::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC)

== 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:14, 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 &quot;practicing journalism&quot; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 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&lt;math&gt;\not=&lt;/math&gt; 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 &quot;Media &amp; Democracy&quot; 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 &quot;rehost Wikinews here&quot; 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 &quot;active editions&quot;?
:* 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 &quot;active editions&quot;?
::*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 &lt;em&gt;about&lt;/em&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 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|&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;''' Blue Rasberry '''&lt;/span&gt;]][[User talk:Bluerasberry|&lt;span style=&quot;cursor:help&quot;&gt;&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;(talk)&lt;/span&gt;&lt;/span&gt;]] 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 &quot;regular&quot; 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|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 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 &lt;code&gt;autopatrolled&lt;/code&gt; (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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:42, 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:16, 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: &quot;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.&quot;
:* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects?
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:13, 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:55, 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)

== Proposed user group and/or possible policy changes ==

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 &lt;code&gt;delete&lt;/code&gt; [but not &lt;code&gt;undelete&lt;/code&gt; or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and &lt;code&gt;protect&lt;/code&gt;), 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, ''&quot;II ...then you will be approved as a probationary custodian for a period of at least four weeks&quot;''.
:::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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:39, 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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)

== 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:24, 9 June 2026 (UTC)

== Image not displaying ==

Can anyone work out why this image isn't displaying?&lt;br&gt;
[[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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)

== wifi ==

hn [[Special:Contributions/&amp;#126;2026-34594-51|&amp;#126;2026-34594-51]] ([[User talk:&amp;#126;2026-34594-51|talk]]) 15:08, 13 June 2026 (UTC)</text>
      <sha1>7ux2p5r6kwqrf3ofkvczgyvn8x9gq20</sha1>
    </revision>
    <revision>
      <id>2815498</id>
      <parentid>2815497</parentid>
      <timestamp>2026-06-13T15:56:06Z</timestamp>
      <contributor>
        <username>Juandev</username>
        <id>2651</id>
      </contributor>
      <comment>/* Towards an Ethics policy */ Reply</comment>
      <origin>2815498</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="38467" sha1="11kirn3z1ghjobwesh2es8qa8cdbk85" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
&lt;!-- MESSAGES GO BELOW --&gt;

== [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] ==

I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&amp;#126;2026-28640-56|&amp;#126;2026-28640-56]] ([[User talk:&amp;#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC)

:What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:35, 12 May 2026 (UTC)
: Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC)
::Sounds good -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:13, 15 May 2026 (UTC)
:::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC)

== 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:14, 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 &quot;practicing journalism&quot; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 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&lt;math&gt;\not=&lt;/math&gt; 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 &quot;Media &amp; Democracy&quot; 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 &quot;rehost Wikinews here&quot; 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 &quot;active editions&quot;?
:* 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 &quot;active editions&quot;?
::*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 &lt;em&gt;about&lt;/em&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 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|&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;''' Blue Rasberry '''&lt;/span&gt;]][[User talk:Bluerasberry|&lt;span style=&quot;cursor:help&quot;&gt;&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;(talk)&lt;/span&gt;&lt;/span&gt;]] 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 &quot;regular&quot; 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|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 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 &lt;code&gt;autopatrolled&lt;/code&gt; (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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:42, 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:16, 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: &quot;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.&quot;
:* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects?
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:13, 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:55, 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)

== Proposed user group and/or possible policy changes ==

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 &lt;code&gt;delete&lt;/code&gt; [but not &lt;code&gt;undelete&lt;/code&gt; or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and &lt;code&gt;protect&lt;/code&gt;), 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, ''&quot;II ...then you will be approved as a probationary custodian for a period of at least four weeks&quot;''.
:::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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:39, 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:24, 9 June 2026 (UTC)

== Image not displaying ==

Can anyone work out why this image isn't displaying?&lt;br&gt;
[[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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)

== wifi ==

hn [[Special:Contributions/&amp;#126;2026-34594-51|&amp;#126;2026-34594-51]] ([[User talk:&amp;#126;2026-34594-51|talk]]) 15:08, 13 June 2026 (UTC)</text>
      <sha1>11kirn3z1ghjobwesh2es8qa8cdbk85</sha1>
    </revision>
    <revision>
      <id>2815590</id>
      <parentid>2815498</parentid>
      <timestamp>2026-06-13T22:50:02Z</timestamp>
      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <minor />
      <comment>Undid revision [[Special:Diff/2815496|2815496]] by [[Special:Contributions/~2026-34594-51|~2026-34594-51]] ([[User talk:~2026-34594-51|talk]])</comment>
      <origin>2815590</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="38318" sha1="7k94xk0m081n06p6sbjkk0y2raba7o4" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
&lt;!-- MESSAGES GO BELOW --&gt;

== [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] ==

I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&amp;#126;2026-28640-56|&amp;#126;2026-28640-56]] ([[User talk:&amp;#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC)

:What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:35, 12 May 2026 (UTC)
: Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC)
::Sounds good -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:13, 15 May 2026 (UTC)
:::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC)

== 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:14, 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 &quot;practicing journalism&quot; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 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&lt;math&gt;\not=&lt;/math&gt; 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 &quot;Media &amp; Democracy&quot; 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 &quot;rehost Wikinews here&quot; 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 &quot;active editions&quot;?
:* 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 &quot;active editions&quot;?
::*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 &lt;em&gt;about&lt;/em&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 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|&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;''' Blue Rasberry '''&lt;/span&gt;]][[User talk:Bluerasberry|&lt;span style=&quot;cursor:help&quot;&gt;&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;(talk)&lt;/span&gt;&lt;/span&gt;]] 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 &quot;regular&quot; 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|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 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 &lt;code&gt;autopatrolled&lt;/code&gt; (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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:42, 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:16, 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: &quot;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.&quot;
:* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects?
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:13, 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:55, 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)

== Proposed user group and/or possible policy changes ==

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 &lt;code&gt;delete&lt;/code&gt; [but not &lt;code&gt;undelete&lt;/code&gt; or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and &lt;code&gt;protect&lt;/code&gt;), 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, ''&quot;II ...then you will be approved as a probationary custodian for a period of at least four weeks&quot;''.
:::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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:39, 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:24, 9 June 2026 (UTC)

== Image not displaying ==

Can anyone work out why this image isn't displaying?&lt;br&gt;
[[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 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 (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 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)</text>
      <sha1>7k94xk0m081n06p6sbjkk0y2raba7o4</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Sandbox</title>
    <ns>4</ns>
    <id>1558</id>
    <revision>
      <id>2815540</id>
      <parentid>2815014</parentid>
      <timestamp>2026-06-13T18:27:25Z</timestamp>
      <contributor>
        <username>Morsecodeworld</username>
        <id>3093122</id>
      </contributor>
      <origin>2815540</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7560" sha1="pahluc4y39umfzw6e36ov9fanbqyhkq" xml:space="preserve">==Welcome==
{{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]],  Sandbox!'''|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Morsecodeworld|me personally]] if you would like some [[Help:Contents|help]].

Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple.

We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies.

To find your way around, check out:
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Wikiversity:Introduction|Introduction to Wikiversity]]
* [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]]
* [[Wikiversity:Browse|Browse]] or visit an educational level portal:&lt;br&gt;[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]]
* [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;

* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]]
* Learn [[Help:How to write an educational resource|how to write an educational resource]]
* Find out about [[Wikiversity:Research|research]] activities
* Give [[Wikiversity:Feedback|feedback]] about your observations
* Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]].

See you around Wikiversity! --[[User:Morsecodeworld|Morsecodeworld]] ([[User talk:Morsecodeworld|discuss]] • [[Special:Contributions/Morsecodeworld|contribs]]) 18:27, 13 June 2026 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}

= Learning Morse Code =

'''Learning Morse Code''' is a free educational course designed to teach the fundamentals of International Morse Code. Learners will study the history of Morse code, understand its structure, learn letters and numbers, practice encoding and decoding messages, and use modern tools to improve their skills.

Morse code remains an important part of communication history and is still used in amateur radio, emergency signaling, navigation training, education, and recreational activities.

== Learning Objectives ==

By completing this course, learners should be able to:

* Understand the origins and development of Morse code.
* Recognize Morse code timing rules.
* Memorize the Morse code alphabet and numbers.
* Encode plain text into Morse code.
* Decode Morse code into readable text.
* Practice Morse communication using audio and visual methods.
* Utilize modern Morse code learning tools.

== Prerequisites ==

No prior knowledge is required.

This course is suitable for:

* Beginners
* Students
* Amateur radio enthusiasts
* Educators
* Hobbyists interested in communication systems

== Course Outline ==

=== Module 1: Introduction to Morse Code ===

==== Topics ====

* History of the electric telegraph
* Development of Morse code
* Samuel Morse and Alfred Vail
* International Morse Code
* Modern uses of Morse code

==== Learning Activities ====

* Read the Morse code overview article.
* Identify examples of Morse code in modern communication.

==== Reading ====

* [[w:Morse code|Morse code]]

----

=== Module 2: Morse Code Timing ===

==== Topics ====

* Dots (dits)
* Dashes (dahs)
* Character spacing
* Word spacing
* Transmission speed

==== Example ====

The letter A:

&lt;code&gt;.-&lt;/code&gt;

The letter B:

&lt;code&gt;-...&lt;/code&gt;

==== Exercise ====

Practice tapping or speaking the rhythm of:

* A
* B
* C
* SOS

----

=== Module 3: Morse Code Alphabet ===

==== Morse Alphabet ====

{| class=&quot;wikitable&quot;
! Letter !! Morse Code
|-
| A || .-
|-
| B || -...
|-
| C || -.-.
|-
| D || -..
|-
| E || .
|-
| F || ..-.
|-
| G || --.
|-
| H || ....
|-
| I || ..
|-
| J || .---
|-
| K || -.-
|-
| L || .-..
|-
| M || --
|-
| N || -.
|-
| O || ---
|-
| P || .--.
|-
| Q || --.-
|-
| R || .-.
|-
| S || ...
|-
| T || -
|-
| U || ..-
|-
| V || ...-
|-
| W || .--
|-
| X || -..-
|-
| Y || -.--
|-
| Z || --..
|}

==== Exercise ====

Translate the following words into Morse code:

* RADIO
* HELLO
* LEARN
* MORSE

----

=== Module 4: Numbers and Symbols ===

==== Numbers ====

{| class=&quot;wikitable&quot;
! Number !! Morse Code
|-
| 0 || -----
|-
| 1 || .----
|-
| 2 || ..---
|-
| 3 || ...--
|-
| 4 || ....-
|-
| 5 || .....
|-
| 6 || -....
|-
| 7 || --...
|-
| 8 || ---..
|-
| 9 || ----.
|}

==== Common Punctuation ====

{| class=&quot;wikitable&quot;
! Symbol !! Morse Code
|-
| Period (.) || .-.-.-
|-
| Comma (,) || --..--
|-
| Question Mark (?) || ..--..
|-
| Slash (/) || -..-.
|-
| Hyphen (-) || -....-
|}

==== Exercise ====

Encode:

* 2025
* HELLO?
* TEST/123

----

=== Module 5: Encoding and Decoding Messages ===

==== Encoding Example ====

Text:

&lt;code&gt;HELLO&lt;/code&gt;

Morse:

&lt;code&gt;.... . .-.. .-.. ---&lt;/code&gt;

==== Decoding Example ====

Morse:

&lt;code&gt;... --- ...&lt;/code&gt;

Decoded:

&lt;code&gt;SOS&lt;/code&gt;

==== Exercise ====

Decode the following:

&lt;code&gt;.-- .. -.- .. ...- . .-. ... .. - -.--&lt;/code&gt;

----

=== Module 6: Audio and Visual Morse Code ===

==== Topics ====

* Listening to Morse code tones
* Flashlight signaling
* Light-based communication
* Sound recognition training

==== Activities ====

* Listen to Morse code recordings.
* Practice identifying letters by sound.
* Send a message using a flashlight.

----

=== Module 7: Modern Morse Code Tools ===

The following educational tools may be used for additional practice:

* [https://morsecode.live/ Morse Code Translator]
* [https://morsecode.live/machine/ Morse Code Machine]
* [https://morsecode.live/morse-code-chart/ Morse Code Chart]
* [https://morsecode.live/image-to-morse/ Image to Morse Code]
* [https://morsecode.live/audio-to-morse/ Audio to Morse Code]
* [https://morsecode.live/light/ Morse Code Light Translator]

==== Exercise ====

Use one of the tools above to:

# Encode a sentence.
# Decode a sentence.
# Compare the result with manual translation.

----

=== Final Assessment ===

Complete the following tasks:

# Encode your full name into Morse code.
# Encode a sentence containing at least 10 words.
# Decode a Morse message provided by another learner.
# Successfully identify five Morse code letters by sound.

== Additional Resources ==

=== Wikipedia ===

* [[w:Morse code|Morse code]]
* [[w:Samuel Morse|Samuel Morse]]
* [[w:Telegraphy|Telegraphy]]
* [[w:Amateur radio|Amateur radio]]

=== External References ===

* [https://www.itu.int/ International Telecommunication Union]
* [https://www.arrl.org/ American Radio Relay League]
* [https://en.wikipedia.org/wiki/Morse_code Morse Code Reference]

== See Also ==

* [[w:Telegraphy|Telegraphy]]
* [[w:Signal lamp|Signal Lamp]]
* [[w:Radio communication|Radio Communication]]

== References ==

&lt;references /&gt;

[[Category:Communication]]
[[Category:Telecommunications]]
[[Category:Language learning]]
[[Category:Open educational resources]]</text>
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      <text bytes="5206" sha1="coi2ayasx85dzx2jzlo0avihjn7gh7m" xml:space="preserve">= Learning Morse Code =

'''Learning Morse Code''' is a free educational course designed to teach the fundamentals of International Morse Code. Learners will study the history of Morse code, understand its structure, learn letters and numbers, practice encoding and decoding messages, and use modern tools to improve their skills.

Morse code remains an important part of communication history and is still used in amateur radio, emergency signaling, navigation training, education, and recreational activities.

== Learning Objectives ==

By completing this course, learners should be able to:

* Understand the origins and development of Morse code.
* Recognize Morse code timing rules.
* Memorize the Morse code alphabet and numbers.
* Encode plain text into Morse code.
* Decode Morse code into readable text.
* Practice Morse communication using audio and visual methods.
* Utilize modern Morse code learning tools.

== Prerequisites ==

No prior knowledge is required.

This course is suitable for:

* Beginners
* Students
* Amateur radio enthusiasts
* Educators
* Hobbyists interested in communication systems

== Course Outline ==

=== Module 1: Introduction to Morse Code ===

==== Topics ====

* History of the electric telegraph
* Development of Morse code
* Samuel Morse and Alfred Vail
* International Morse Code
* Modern uses of Morse code

==== Learning Activities ====

* Read the Morse code overview article.
* Identify examples of Morse code in modern communication.

==== Reading ====

* [[w:Morse code|Morse code]]

----

=== Module 2: Morse Code Timing ===

==== Topics ====

* Dots (dits)
* Dashes (dahs)
* Character spacing
* Word spacing
* Transmission speed

==== Example ====

The letter A:

&lt;code&gt;.-&lt;/code&gt;

The letter B:

&lt;code&gt;-...&lt;/code&gt;

==== Exercise ====

Practice tapping or speaking the rhythm of:

* A
* B
* C
* SOS

----

=== Module 3: Morse Code Alphabet ===

==== Morse Alphabet ====

{| class=&quot;wikitable&quot;
! Letter !! Morse Code
|-
| A || .-
|-
| B || -...
|-
| C || -.-.
|-
| D || -..
|-
| E || .
|-
| F || ..-.
|-
| G || --.
|-
| H || ....
|-
| I || ..
|-
| J || .---
|-
| K || -.-
|-
| L || .-..
|-
| M || --
|-
| N || -.
|-
| O || ---
|-
| P || .--.
|-
| Q || --.-
|-
| R || .-.
|-
| S || ...
|-
| T || -
|-
| U || ..-
|-
| V || ...-
|-
| W || .--
|-
| X || -..-
|-
| Y || -.--
|-
| Z || --..
|}

==== Exercise ====

Translate the following words into Morse code:

* RADIO
* HELLO
* LEARN
* MORSE

----

=== Module 4: Numbers and Symbols ===

==== Numbers ====

{| class=&quot;wikitable&quot;
! Number !! Morse Code
|-
| 0 || -----
|-
| 1 || .----
|-
| 2 || ..---
|-
| 3 || ...--
|-
| 4 || ....-
|-
| 5 || .....
|-
| 6 || -....
|-
| 7 || --...
|-
| 8 || ---..
|-
| 9 || ----.
|}

==== Common Punctuation ====

{| class=&quot;wikitable&quot;
! Symbol !! Morse Code
|-
| Period (.) || .-.-.-
|-
| Comma (,) || --..--
|-
| Question Mark (?) || ..--..
|-
| Slash (/) || -..-.
|-
| Hyphen (-) || -....-
|}

==== Exercise ====

Encode:

* 2025
* HELLO?
* TEST/123

----

=== Module 5: Encoding and Decoding Messages ===

==== Encoding Example ====

Text:

&lt;code&gt;HELLO&lt;/code&gt;

Morse:

&lt;code&gt;.... . .-.. .-.. ---&lt;/code&gt;

==== Decoding Example ====

Morse:

&lt;code&gt;... --- ...&lt;/code&gt;

Decoded:

&lt;code&gt;SOS&lt;/code&gt;

==== Exercise ====

Decode the following:

&lt;code&gt;.-- .. -.- .. ...- . .-. ... .. - -.--&lt;/code&gt;

----

=== Module 6: Audio and Visual Morse Code ===

==== Topics ====

* Listening to Morse code tones
* Flashlight signaling
* Light-based communication
* Sound recognition training

==== Activities ====

* Listen to Morse code recordings.
* Practice identifying letters by sound.
* Send a message using a flashlight.

----

=== Module 7: Modern Morse Code Tools ===

The following educational tools may be used for additional practice:

* [https://morsecode.live/ Morse Code Translator]
* [https://morsecode.live/machine/ Morse Code Machine]
* [https://morsecode.live/morse-code-chart/ Morse Code Chart]
* [https://morsecode.live/image-to-morse/ Image to Morse Code]
* [https://morsecode.live/audio-to-morse/ Audio to Morse Code]
* [https://morsecode.live/light/ Morse Code Light Translator]

==== Exercise ====

Use one of the tools above to:

# Encode a sentence.
# Decode a sentence.
# Compare the result with manual translation.

----

=== Final Assessment ===

Complete the following tasks:

# Encode your full name into Morse code.
# Encode a sentence containing at least 10 words.
# Decode a Morse message provided by another learner.
# Successfully identify five Morse code letters by sound.

== Additional Resources ==

=== Wikipedia ===

* [[w:Morse code|Morse code]]
* [[w:Samuel Morse|Samuel Morse]]
* [[w:Telegraphy|Telegraphy]]
* [[w:Amateur radio|Amateur radio]]

=== External References ===

* [https://www.itu.int/ International Telecommunication Union]
* [https://www.arrl.org/ American Radio Relay League]
* [https://en.wikipedia.org/wiki/Morse_code Morse Code Reference]

== See Also ==

* [[w:Telegraphy|Telegraphy]]
* [[w:Signal lamp|Signal Lamp]]
* [[w:Radio communication|Radio Communication]]

== References ==

&lt;references /&gt;

[[Category:Communication]]
[[Category:Telecommunications]]
[[Category:Language learning]]
[[Category:Open educational resources]]</text>
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      <comment>Restored revision 2811686 by [[Special:Contributions/Codename Noreste|Codename Noreste]] ([[en:w:User:BrandonXLF/Restorer|Restorer]])</comment>
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        <id>3039865</id>
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      <comment>Remove template/category (via JWB)</comment>
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Interested participant from Wikipedia under the name [[w:User:Rayc|Rayc]]

Graduate student with knowledge in Electrical engineering with a hope to become a professor. I also am interested in math, genetic algorithms, and statistics.


http://www.wired.com/news/games/0,2101,65052,00.html
http://www.simba.us/misc/dsmc/dsmca.html
http://bbs.keyhole.com/ubb/postlist.php/Cat/0/Board/EducationEducators
[[Web Design/An Introduction to Programming with Javascript]]
http://ocwfinder.com/

http://www.acvt.com.au/research/videotrace/
==Places to contact==
*Signpost
*Wikiprojects
*Wikia
::

==Sandboxes==
*[[/Rand]]
*[[/ProposalQuiz]]
*[[/Unipages]]
*[[/School status]]
*[[/Learning Blog]]
*[[/Youtube]]

==Trash==
[[/Scrap page]]
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This is not an encyclopedia article. If you find this page on any site other than &lt;span style=&quot;white-space: nowrap&quot;&gt;&lt;span&gt;&amp;#87;&amp;#105;&lt;!-- Wikiversity --&gt;&amp;#107;&amp;#105;&lt;/span&gt;&lt;span&gt;&amp;#118;&amp;#101;&amp;#114;&amp;#115;&amp;#105;&amp;#116;&amp;#121;&lt;/span&gt;,&lt;/span&gt; you are viewing a mirror site. Be aware that the page may be outdated and that the user this page belongs to may have no personal affiliation with any site other than &lt;span style=&quot;white-space: nowrap&quot;&gt;&lt;span&gt;&amp;#87;&amp;#105;&lt;!-- Wikiversity --&gt;&amp;#107;&amp;#105;&lt;/span&gt;&lt;span&gt;&amp;#118;&amp;#101;&amp;#114;&amp;#115;&amp;#105;&amp;#116;&amp;#121;&lt;/span&gt;&lt;/span&gt; itself. The original page is located at &lt;span style=&quot;white-space: nowrap&quot;&gt;[http://en.wiki&lt;!----&gt;versity.org/wiki/{{FULLPAGENAMEE}} &lt;span&gt;http://en.wiki&lt;/span&gt;&lt;!----&gt;&lt;span&gt;versity.org/wi&lt;/span&gt;&lt;span&gt;ki/{{FULLPAGENAMEE}}&lt;/span&gt;].&lt;/span&gt;
|[[Image:Wikimedia Foundation RGB logo with text.svg|60px|none|Wiki&lt;!----&gt;media Foundation]]
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        <id>3039865</id>
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== [[IMHA Research Archives]] ==
I propose to '''move to userspace''', including the subpages. I struggle to understand how Wikiversity readers are supposed to benefit from the material here and in the subpages. In the log, there is e.g. '10 February 2019 Marshallsumter discuss contribs deleted page IMHA Research Archives (content was: &quot;{&lt;nowiki/&gt;{Delete|Author request}} Thanks! -&quot;)', so the page was deleted before, but not the subpages.

We could also delete all the material if we have strong enough suspicion too much of it is copyright violation. In any case, moving to user space improves the matter a little by moving the content away from Google search. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 13:38, 9 November 2025 (UTC)

:Looking at some sub-pages, they can be deleted e.g., because they only consist of broken links or are largely empty. I deleted a couple but haven't been through all to check. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:27, 10 November 2025 (UTC)

As an example, let me give the wikitext content of [[IMHA Research Archives/3. Scientific litterature search, storage and use]]:
&lt;pre&gt;
==[[/Medicina Maritima - the Spanish scientific maritime health journal/]]==

==[[/PubMed/]]==

==[[/Google and Google Scholar/]]==

==[[/Zotero/]]==

==[https://www.dropbox.com/sh/d91z7bcyelfvk42/AAAkIvjtBnnFMbiU9ZLOdVL9a/Andrioti_database%20sources0310.pptx?dl=0 Maritime health web portal ressources ]==
&lt;/pre&gt;
The wikilinks are red; the external link to dropbox says &quot;You don't have access&quot;. This was made in 2016. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:04, 11 November 2025 (UTC)

:I suggest delete -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:27, 12 November 2025 (UTC)
:: I think we should avoid deletion as much as possible, instead moving to user space (bar copyvio, ethics violation, etc.). This is a good general principle. It greatly improves auditability and makes it so much easier for anyone to request undeletion since they know what content they are requesting for undeletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:52, 12 November 2025 (UTC)
:::Do not recreate Wikiversity from the educational and research project to the personal blog. That will lead to the cancelation of it by WMF. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:44, 20 November 2025 (UTC)
:::: The English Wikiversity has a long tradition of moving problematic content to user space, as per evidence collected at [[User:Dan_Polansky/About Wikiversity#Moving pages to userspace]]. If Wikimedia Foundation finds this problematic, they can start a discussion in Colloquium and state their concerns. They do not need to make explicit threats at first; they can start a discussion and explain why it is problematic. They can even do it from an anonymous IP and provide a well-articulated reasoning. And anyone else can start a discussion in Colloquium to change this tradition. I do not see why we should not want to change that tradition based on well-articulated, compelling reasoning. I see no reason why Juandev should be making threats instead of them, on a per RFD basis. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:58, 21 November 2025 (UTC)
:::: If Juandev is ''sincere'' about deleting very-low-value items ''from user space'', he should perhaps demonstrate that by asking his pages like [[:cs:Uživatel:Juandev/Problémy/Kov/Repase dvířek elektroskříně]] to be deleted; otherwise, I register a ''glaring inconsistence''. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:43, 21 November 2025 (UTC)
::What was the original delate page about @[[User:Jtneill|Jtneill]]? I guess that would be crucial for the decission. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:48, 20 November 2025 (UTC)
:::@[[User:Juandev|Juandev]] the couple of pages I checked and deleted were much like @[[User:Dan Polansky|Dan Polansky]] posted above i.e., headings with empty sections and/or broken links but no substantive content. But I think each sub-page needs checking. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:59, 20 November 2025 (UTC)
::::So I'm saying that the main page usually determines what the other pages are for. But if I don't know the page because it's been deleted, or why was deleted (deletion based on the founder's request is probably not the rule), it's hard to judge. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 22:16, 20 November 2025 (UTC)
:::::I've pasted the original content of the root page: [[IMHA Research Archives#Original page]] (i.e., prior to the content being removed and deletion requested) to help understand the context for the sub-pages. In 2018, Saltrabook blanked the page, indicating that the content had been moved elsewhere, and requested page deletion. Marshallsumter then deleted the main page but not the sub-pages. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:58, 21 November 2025 (UTC)
::::::I see, so if those subpages are usefull I would keept them, if not I would delete them. I dont see a point of providing free hosting to sombody, by moving many pages to their user space. The question is if we want to host (i.e. to have in the main ns) lists of links elsewhere. I have no opinion on that. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:11, 22 November 2025 (UTC)

: Let me clarify that while many of the subpages are like the example above, [[IMHA Research Archives/Scientific litterature search, storage and use/Zotero]] is different:
:: &quot;A continuous critical and evidence based learning is a core issue in clinical practice, research, teaching, publication and prevention activities. The Zotero Program is just one of many scientific literature management programs, that should be used for these purposes. Of course one can live without such a database but it helps a lot and can save a lot of time that could be used for more interesting issues. Not only that, but it helps to create better publications and knowledge. Without this program it can be very time consuming to publish a scientific article with the requested style for the references. Further in daily practice when you want to collect and cite a few references for a specific evidence in a clinical colloquium and discussion, this program is excellent. Therefore we strongly recommend that all maritime health persons learn how to use this excellent tool in their daily maritime health practice of all different types. There are good online courses for self-instruction on how to use Zotero. For example this one: Zotero fast online course But in order to increase IMHAR´s collective scientific strength in the use of EBM we would like to give training sessions in every possible opportunity, IMHA Symposia, seminars and other types of meetings. The database is useful for personal purposes but especially also for collaborative aims. At the IMHAR meeting in Paris Oct 7th 2016 we will give an introduction to the program by showing how it can be used in the daily practice and discuss strength and weaknesses compared to other similar databases.&quot;
: Even longer is e.g. [[IMHA Research Archives/Scientific litterature search, storage and use/Medicina Maritima - the Spanish scientific maritime health journal]].
: However, that does not mean these should be salvaged. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:53, 21 November 2025 (UTC)
:{{ping|Saltrabook}} I'm wondering if you can respond here to help us decide about whether to delete the IMHA Research Archives sub-pages or perhaps move them to your user space? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:58, 17 May 2026 (UTC)
: [[Special:Diff/2811248]] provides confirmation from Saltrabook to go ahead and delete these archives -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:56, 23 May 2026 (UTC)

== Undeletion request ==
{{archive top|'''Not done''' - discussion has been opened for over a month with no proper explanation as to why the page in question should be undeleted + author of the page failed to address any relevant arguments. I'd suggest reading [[Wikiversity:What is Wikiversity?]] and [[Wikiversity:Verifiability]] to find ways on how you can contribute to this site productively. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:49, 4 June 2026 (UTC)}}
It was deleted by an admin without discussion and with untrue rationale. If people take offense with the question that doesn't mean it's not a valid question and the page was good. Please undelete the Wikidebate page [https://web.archive.org/web/20250810030352/https://en.wikiversity.org/wiki/Is_it_likely_that_Earth_has_been_visited_by_aliens_millions_of_years_ago%3F Is it likely that Earth has been visited by aliens millions of years ago?]

There are lots of sources on the subject, the wikidebate is sourced very well compared to other wikidebates and wikiversity pages, and the page is educational, useful and of good quality. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 23:57, 10 April 2026 (UTC)

:Page: [[Is it likely that Earth has been visited by aliens millions of years ago?]]
:Ping: [[User:Atcovi|Atcovi]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:21, 11 April 2026 (UTC)
:There is no need for a discussion for straight garbage-level, pseudoscientific content.
:For '''Is it likely that Earth has been visited by aliens millions of years ago?''', the flaws for this page wouldn't even take someone more than a few minutes to assess:
:* Essentially, the &quot;pro&quot; arguments unproven claims being derived from irrelevant, established facts (basically: &quot;it is likely aliens have came because Earth has existed for so long [sources proving Earth's longevity]&quot;). These are not serious, scientifically-backed arguments - these are non sequiturs. It's as if I said Wikipedia has existed longer than my existence on Earth ([https://d1wqtxts1xzle7.cloudfront.net/74351725/eyJoIjogImZiODhmYzNkODU1N2UxMWExYzUyODJiYzgzZTRmZDM4OTBjODY5YWMzMjA3NDNmOWEyZTA0ZTU3ZGYwZjAyYTkiLCAidSI6ICJodHRwczovL3B1cmUuaHZhLm5sL3dz-libre.pdf?1636354596=&amp;response-content-disposition=inline%3B+filename%3DCritical_Point_of_View_A_Wikipedia_Reade.pdf&amp;Expires=1775872055&amp;Signature=GqbUZboYRvUYWi~aW40LT5eZSHrLuDL3o0-DxAH8vSvcJcGAuyByZWLF2oHTY6GlB72TqvZxpE-v9d4gvsA6myriYqO~QQQZgWxjT2JXjUWC-yiPcTF4l~lroJSi4dY0v9eKiBcU03l-aeUdrX8~UPfi0TfW0IhsmzH-VBR6X6FrzRpIqc6uM6n9YXfr5FRB3aCqqokU690af3n0Hguaub1Zgmh9qjYYqzBS0VOOHjKTTEQnDuadX3jl5CQeXYTaeCC3H0hMeVwHlratbrnuFEKC1aN0-5znCUoSzMEg21ECzGPTrSDM1W05dcK-u0ZTCeUGKAuC-2yRFL3sY46MIw__&amp;Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA#page=157 reputable source proving ''this'' fact]), therefore it's likely that my birth took place solely for the sake of me experiencing Wikipedia (0 backing). It makes no sense and no person with at least a high school-level of intelligence would take this seriously. 
:* What is worse is that the user is being misleading with their &quot;[the page is] sourced very well&quot; claim. The sources ''themselves'' don't even back up the claims. It's just used as proof for an established concept, where the user then uses this established concept to jump to an unsupported, laughable conclusion that is pulled out of thin air. It's utterly ridiculous to even consider such a page for mainspace since it clearly violates our [[Wikiversity:Verifiability]] policy. This is, once again, pseudoscientific content that has caused our website to reduce in quality over the last few years. 
:* Going source by source, we can see that:
:#[https://web.archive.org/web/20250918011642/https://timesofindia.indiatimes.com/blogs/thebigd/compress-earths-history-into-24-hrs-humans-came-at-1158-pm-yet-killed-70-of-wildlife/ ‘Compress Earth’s history into 24 hrs. Humans came at 11:58 pm, yet killed 70% of wildlife’] is literally just a blog post which doesn't even mention aliens or extraterrestrial life. It just talks about Earth's history in accordance with the 24-hour metric of time, and the author tries to use this article as a 'piece in the puzzle' of aliens &quot;possibly&quot; visiting Earth... which, once again, is unsupported and is not backed up anywhere in the article.
:#[https://web.archive.org/web/20250808053249/https://news.cornell.edu/stories/2023/11/jurassic-worlds-might-be-easier-spot-modern-earth The Cornell article does not even remotely support the idea that &quot;aliens visited Earth&quot;]. It mentions a ''chance'' of &quot;life there [a habitable exoplanet] might not be limited to microbes, but could include creatures as large and varied as the megalosauruses or microraptors that once roamed Earth.&quot;, but again, no justification to take this article as proof that &quot;aliens may have visited us!&quot;. There's no mention of aliens visiting Earth anywhere in the article. Once again this is only proving the background premise, but not the unsupported, nonsensical &quot;alien likelihood&quot; argument that the author of this garbage page is trying to push so desperately.
:#The Parker Solar Probe WP article does not even mention aliens either. It follows the same issues as the previous argument.
:And the other page this user complained about [https://en.wikiversity.org/wiki/User_talk:Atcovi#Deletion_of_educational_page_because_of_personal_opinion on my talk page] holds almost similar, maybe even more fatal mistakes, than this one. It has nothing to do with &quot;taking offense&quot;, this is just low-quality, garbage content. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:56, 11 April 2026 (UTC)
::Why do you think pro claims are required to be proven? It's possible to object to them and these are arguments, not contextualized to be statements of proven facts. And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future. Instead of insulting others' intelligence, maybe engage with the actual reasoning rather than censoring it away. And there are lots of sources, such as [https://interestingengineering.com/science/alien-civilizations-may-have-visited-earth-millions-of-years-ago-study-says Alien Civilizations May Have Visited Earth Millions of Years Ago, Study Says] etc etc. The sources are used for the arguments themselves individually. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:30, 11 April 2026 (UTC)
:::Because, once again, this is not a site that caters to rampant debating for the sake of &quot;we need to employ rationality and logic to solve the world's problems&quot;, we have policies that we need to fulfill. The claims made in the pro argument clearly do not meet [[Wikiversity:Verifiability]], since you cannot verify these arguments with the sources because they are not relevant. 
:::''&quot;And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future.&quot;'' The point being is that these arguments are not supported by the sources. Even the article you mention poses the idea as a hypothetical model. This is just you twisting the article to fit your unsupported narrative. I'll bring direct quotes for you to show why the linked article does not help you:
:::* ''One problem the researchers do make sure to point out is that '''they are working with only one data point: our own behaviors and capabilities for space exploration'''. “We tried to come up with a model that would involve the fewest assumptions about sociology that we could,” Carroll-Nellenback told Business Insider. '''We have no real way of knowing the motivations of an alien civilization'''.'' --&gt; proves that this is just speculation and no evidence-based arguments have been provided for the idea that aliens likely visited Earth.
:::And I'm not sure if you read my entire response, but I ''did'' engage with your &quot;actual reasoning&quot; and exposed its weaknesses and lack of adherence to Wikiversity policies. If we allowed content that was just filled with non sequiturs we would have content that fails Wikiversity's educational objectives and reduces the overall quality of this website, hence why such a harsh stance needs to be taken. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:50, 11 April 2026 (UTC)
::::Thanks for proving that the Wikimedia ecosystem is unfit to deliberate on controversial topics. The question is entirely valid and the content is far better sourced than nearly all Wikidebates and has no genuine flaws. The only possible issue with it as far as I can see is that now that Wikidebates has been paused people can't add objections if they do have sth specific to say about the topic that's not already included on that page which already had plenty of Cons and objections.
::::The page was more educational than most of Wikiversity and it was well-sourced – wikidebates was for arguments so people were invited to make arguments based on sourced things or outlined logic and the page met [[WV:V]] and most pages on Wikiversity aren't sourced as good. Doesn't look like people can see beyond their biases and personal views here but that's more evident in the marginalization and deletion of wikidebates and the low activity in that project than these selective deletions. A constructive thing to do would be to add reasoned Cons and objections not yet on the page and people had plenty of time to do that. There are and will be other sites for free constructive rational adversarial deliberation (not a big loss in that sense). [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 16:31, 22 April 2026 (UTC)
:::::Thank you for failing to address any of my arguments and going on an unrelated, nonsensical tangent that has nothing to do with the discussion. Once you start producing work that aligns with Wikiversity's content policies instead of typing up laughable, pseudoscientific garbage, then maybe your work can be accepted and not removed. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:59, 22 April 2026 (UTC)
::::::I suggest you stop ridiculing things and learn respectfully forming genuine points about the subject at hand. {{tq|the idea as a hypothetical model}} but please learn first about what arguments are and why they're not the same as a statement of objective proven fact. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 17:18, 22 April 2026 (UTC)
{{archive bottom}}

==Pages by Harold Foppele==
{{archive top|All deleted, user page blanked [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:33, 6 June 2026 (UTC)}}
[[User:Harold Foppele]] is locally blocked indefinitely and globally banned for sockpuppetry. There were also WMF and local community concerns expressed about copyright violation and AI (over)use. As a result, I think the Wikiversity pages created by this account warrant review with regard what should be deleted, what should be retained etc.:
* [[Completing the square]]
* [[Number of independent spatial modes in a spherical volume]]
* [[Quantum]]
** [[Quantum/Andrew N. Jordan]]
* [[Quantum A Matter Of Size]]
* [[Quantum A Spooky Action at a Distance]]
* [[Quantum: A Walk Through the Universe]]
* [[Quantum Computing Algorithms in the NISQ Era]]
* [[Quantum Formulas Collection]]
* [[Quantum harmonic oscillator]]
* [[Quantum Matter Elements and Particles]]
* [[Quantum mechanics]]
** [[Quantum mechanics/Timeline]]
* [[Quantum mechanics learning module]]
* [[Quantum mechanics measurements]]
* [[Quantum Noisy Qubits]]
* [[Quantum optics beam splitter experiments]]
* [[Quantum: The Secret of Cohesion: How Waves Hold Matter Together]]
* [[Quantum Ultra fast lasers]]
* [[Speed of sound experiments]]
* [[User:Harold Foppele]]
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:12, 17 May 2026 (UTC)

:'''Delete all''' Not worth keeping. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:27, 17 May 2026 (UTC)
{{Archive bottom}}

== [[Classical guitar pedagogy]] ==

According to the talk page, the author of this page intended to create this page for Wikipedia. At this moment in time (nearly 20 years later), the page is still riddled with red links and doesn't seem to fit Wikiversity's learning modules. Therefore, I propose that this page should be deleted. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:03, 19 May 2026 (UTC)

:'''Weak delete''' This at least has &lt;em&gt;something&lt;/em&gt; that someone could use, but agreed that it's not particularly useful and not likely to be developed. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:25, 20 May 2026 (UTC)
: '''Move''' to [[w:User:Grégory Leclair/Classical guitar pedagogy]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:18, 23 May 2026 (UTC)

== [[Film writing]] ==
{{archive top|Consensus to keep after vote changes. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:20, 31 May 2026 (UTC)}}
Undeveloped since 2007. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:05, 19 May 2026 (UTC)

:&lt;del&gt;'''Delete''' Nothing here. Great idea in principle, tho. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:25, 20 May 2026 (UTC)&lt;/del&gt;&lt;ins&gt;'''Keep''': It's now at least developed enough to be something. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:42, 31 May 2026 (UTC)&lt;/ins&gt;
: '''Keep''' as part of [[:Category:Film]] resources. I've tidied the page, so it looks less abandoned. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:57, 20 May 2026 (UTC)
:@[[User:Atcovi|Atcovi]] @[[User:Koavf|Koavf]] The page seems to have been tidied up. Do you want to reevaluate your votes? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:40, 31 May 2026 (UTC)
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==[[United States UFO files]]==
{{archive top|Deleted, but the author of the resource did not respond here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:16, 31 May 2026 (UTC)}}
Seems to be WP-like; material copied from [[w:United States UFO files]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:46, 21 May 2026 (UTC)

:'''Delete''', but why would a PROD template not suffice? My logic was that it is a newly created page (made just today), and isn't a big project/difficult page to deal with. Do we not deal with newly created pages that appear to not satisfy Wikiversity's objectives/mission with a PROD template? Wouldn't we best reserve RFDs for long-standing pages (like the two pages above this section being listed for deletion) or ''after'' the PROD template isn't enough to determine the fate of such pages (per [[Wikiversity:Deletion policy#Proposed deletion (prod)|here]]: &quot;Anyone still considering that the resource should be deleted [after the placement of the PROD template] may discuss deletion.&quot;)? A PROD template may also be useful in this case to alert the author that the page is not compatible with Wikiversity's learning objectives and communicates a concise opportunity to refine the page with the 90-day limit. Maybe even in this case, a speedy would've been enough (possibly fitting [[Wikiversity:Deletion policy#Criteria for speedy deletion|#12]]: &quot;No research objectives or discussion in history. Welcome users and resources when likely to be expanded shortly.&quot;).
:Interested to hear your thoughts as I want to make sure this is clear, as I've been cleaning up a lot of 'dead' pages around Wikiversity and find myself confused on whether to use PROD or RFD. Thanks, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:08, 21 May 2026 (UTC)
: Yes, could be speedy deleted. Otherwise, I don't know about the merits about leaving it around for 90 days, hence me bringing it to here. There is some comment in [[Wikiversity:Deletion policy]] about the specific deletion templates not being so important. More important I think is to flag for discussion. However, we could also improve the proposed policy to make the process clearer. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:20, 21 May 2026 (UTC)
: Ping {{u|User:Realcosmixyt}} for comment -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:54, 24 May 2026 (UTC)
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== [[Emergency Operation Centre GIS]] ==
{{archive top|Consensus to delete. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:39, 30 May 2026 (UTC)}}
Undeveloped for over a decade (only thing present is just an outline). —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:44, 22 May 2026 (UTC)

:*'''Delete'''
:―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:59, 22 May 2026 (UTC)
:* '''Delete'''. Insufficiently developed. Was moved from [[b:Emergency Operation Centre GIS]].
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:13, 23 May 2026 (UTC)
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==[[Mippedia]] ==
{{archive top|Consensus to delete, and the author of the template did not respond to Jtneill's comment. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:37, 30 May 2026 (UTC)}}
I propose the deletion of the page &quot;[[Mippedia]]&quot;, due to the subject not being backed by reputable sources. Pages with the same subject has been deleted multiple times on the Indonesian Wikipedia. The original writer of the page did it solely to promote his wiki site. [[User:ANNAFscience|ANNAFscience]] ([[User talk:ANNAFscience|discuss]] • [[Special:Contributions/ANNAFscience|contribs]]) 10:39, 23 May 2026 (UTC)
: {{ping|Sevent Me}} any comment? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:10, 23 May 2026 (UTC)
:'''Delete''' I don't know what the point of this is. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:26, 23 May 2026 (UTC)
: '''Delete'''. Advertising. Points to a non-English, copyright restricted website. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:58, 24 May 2026 (UTC)
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==[[Wikiphilosophers]]==
Moving from {{tl|prod}} by {{at|Atcovi}}: &quot;similar &quot;philosophy&quot;-related content has been removed in the past [issue of pseudoscience] + very little moderation (mirroring the issues of [[Wikidebates]]) + lacks educational value.&quot; The project has also been nominated for deletion on its talk page: [[Talk:Wikiphilosophers]]. There are many subpages: 
{{Special:PrefixIndex/Wikiphilosophers/}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:45, 24 May 2026 (UTC)

:'''Delete'''. Unfortunately, this project wasn't as successful as I had hoped. Kind regards, [[User:Perquirius|Perquirius]] ([[User talk:Perquirius|overleg]] • [[Special:Contributions/Perquirius|bijdragen]]) 14:29, 24 May 2026 (UTC)
::Don't forget to delete [[Template:Wikiphilosophers]], [[Template:Wikiphilosophers/doc]] and [[Template:Wikiphilosophers topics]] also. [[User:Perquirius|Perquirius]] ([[User talk:Perquirius|overleg]] • [[Special:Contributions/Perquirius|bijdragen]]) 14:30, 24 May 2026 (UTC)
:'''Delete''' Wikiversity should not be used as a host for failed sister projects. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 05:06, 6 June 2026 (UTC)

== [[Template:UserSkype]] ==
{{archive top|Deleted, template and category was removed from all pages. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:14, 14 June 2026 (UTC)}}
Service was discontinued over a year ago. I suggest deleting the Userbox and [[:Category:Users familiar with Skype]], as it can only confuse or mislead. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:17, 30 May 2026 (UTC)

:'''Delete''' per reasoning. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:40, 30 May 2026 (UTC)
: '''Delete''' -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:48, 31 May 2026 (UTC)
:Yup delete. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 16:59, 8 June 2026 (UTC)
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  <page>
    <title>User:Jade Knight</title>
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      <contributor>
        <username>PieWriter</username>
        <id>3039865</id>
      </contributor>
      <minor />
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      <text bytes="947" sha1="hnlo8bo6ma69c6sj003ldqjesbdweu7" xml:space="preserve">{{French mentor}}{{#babel:en|fr-3|es-2|jer-1|cy-1|zh-1|de-1|la-1}}

I'm Michael-Forest.  I [used to] live in the Western US, and I [also used to be] also quite active on [[w:User:Jade Knight|Wikipedia]].  I used to be [[Wikiversity:Custodianship|Custodian]] here at Wikiversity, but don't have much time for it any more.

I've been most active in the [[Topic:European History|Department of European History]], the [[Topic:French|French Department]] and in the [[Topic:English Language|English Language Division]], though I was generally active in the Schools of [[School:History|History]], [[School:Relosophy|Relosophy]] and [[School:Language and Literature|Language and Literature]].

*[[User:Jade Knight/tasks|Tasks]]
*[[/Tools/]]

If you really want to reach me, [[Special:Emailuser/Jade_Knight|email me]].

{{User knowledge-hist-3}}{{User knowledge-ling-3}}
{{User knowledge-lang-4}}{{User knowledge-food-2}}

[[fr:Utilisateur:C'valyi d'Jade]]</text>
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  <page>
    <title>Wikiversity:Bureaucratship</title>
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      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <comment>+</comment>
      <origin>2815556</origin>
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      <text bytes="5289" sha1="sta54oyha807l3zjfxu4my5vvcwuw98" xml:space="preserve">{{policy|WV:B|WV:BUR|WV:CRAT|WV:BCRAT}}
{{2FA-required|enforced=yes}}
[[File:Wikiversity Bureaucrat.svg|right|130px|link=]]

'''Bureaucrats''' are part of [[Wikiversity]]'s [[Wikiversity:Support staff|support staff]]. They can promote users to [[Wikiversity:Custodianship|custodian]] or bureaucrat status, and grant or revoke [[Wikiversity:Curatorship|curator]], [[Wikiversity:Bots|bot]], and [[Wikiversity:Interface administrators|interface administrator]] permissions.

The English Wikiversity currently has {{NUMINGROUP:bureaucrat}} bureaucrats ([[Special:ListUsers/bureaucrat|full list]]).

== A bureaucrat's role ==
First and foremost, bureaucrats must be well-trusted members of the community. They must have a deep understanding of [[Wikiversity:Mission|Wikiversity's mission]] and processes, and must be excellent judges of [[Wikiversity:Consensus|consensus]]. They must demonstrate through their extensive contributions to Wikiversity that they are not rash in decision-making, nor uncivil to others, even those whom they are in disagreement with. They must also have the ability and willingness to thoroughly explain decisions that they make, as well as to admit fault, where appropriate.

Bureaucrats ''do not'' have the right to use their status to appropriate any undue influence in community discussions - their participation in such activities is on a par with any other community member, insofar as is possible. Whatever influence they may have should be akin to that of any other community member, according to the weight of their opinions or their previous participation in the project.

Bureaucrats are expected to follow the same policies as [[Wikiversity:Curatorship|curators]] and [[Wikiversity:Custodianship|custodians]].

== A bureaucrat's duties ==
Bureaucrats can add users to the [[Wikiversity:Curatorship|curator]], [[Wikiversity:Custodianship|custodian]], bureaucrat, [[Wikiversity:Bots|bot]], and/or [[Wikiversity:Interface administrators|interface administrator]] groups (via [[Special:UserRights]] - [[Special:Log/rights]]). Bureaucrats act as the final interpreter of consensus and are charged with the responsibility of declaring at an appropriate time, whether a custodian, bureaucrat, bot, or interface administrator candidate is granted a user group change or otherwise. Bureaucrats should respect the Wikiversity community's decision on these particular matters. This management process is intended to streamline processing of requests for user group changes, and to minimize ambiguity introduced to the process when non-bureaucrats intervene. However, note that custodians can determine consensus and add user group permissions for curator requests.

Bureaucrats should be careful about making mistakes in adding user rights because only curator, bot, and interface administrator groups can be removed by bureaucrats. To remove custodian or bureaucrat rights requires a [[m:Stewards|steward]] to do so.

== How can I question a bureaucrat's decision? ==
You can ask on a bureaucrat's [[Help:User talk page|user talk page]], request [[Wikiversity:Custodian feedback|feedback]] from other custodians and bureaucrats, or start a [[Wikiversity:Community Review|community review]]. The order is important, it reflects the order in which you should attempt to resolve a problem.

== How are bureaucrats created? ==
Bureaucrats can be nominated at [[Wikiversity:Candidates for Bureaucratship]]. However, no mentor is required. Nominations should be [[Wikiversity:Announcements|announced]] ([[MediaWiki:Sitenotice|site wide]]), and kept open for a period of '''at least two weeks''' before being acted upon. There needs to be '''a very strong majority''' of users in support of the decision to add or remove a candidate from the ''bureaucrat'' group.

== How are bureaucrats removed? ==
There are three ways:
# A bureaucrat can request removal of their tools from [[m:Steward requests/Permissions|stewards]].
# Requests for bureaucrat removal by others (unless it's an emergency) should first go through the process of talk page discussion, [[Wikiversity:Custodian feedback|custodian feedback request]], and [[Wikiversity:Community Review|community review]]. The final act would be a [[m:Steward requests/Permissions|request to stewards]] to remove the ''bureaucrat'' group from a user.
# The maximum time period of inactivity &lt;u&gt;without community review&lt;/u&gt; for holders of advanced administrative rights is two years per the [[:meta:Category:Global policies|global policy]] described at [[meta:Admin activity review|Admin activity review]]. After that time, a [[meta:Steward requests/Permissions|steward will be asked to remove the rights]].

==See also==
;Wikiversity
* [[Wikiversity:CheckUser policy]]
* [[Wikiversity:Curatorship]]
* [[Wikiversity:Custodianship]]
* [[Special:ListUsers/bureaucrat|List of current Wikiversity bureaucrats]]
* [[Wikiversity:Colloquium/archives/December 2006#Bureaucrats]]

;Sister wikis
* [[w:Wikipedia:Bureaucrats|Wikipedia:Bureaucrats]]
* [[meta:Bureaucrat|Meta:Bureaucrat]]
* [[b:Wikibooks:Administrators|Wikibooks:Administrators]]
* [[wikinews:Wikinews:Administrators|Wikinews:Administrators]]

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

[[Category:Wikiversity bureaucratship| ]]
[[de:Wikiversity:Pedelle#Bürokraten]]</text>
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  <page>
    <title>User:Blast</title>
    <ns>2</ns>
    <id>22778</id>
    <revision>
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      <timestamp>2026-06-14T02:06:56Z</timestamp>
      <contributor>
        <username>PieWriter</username>
        <id>3039865</id>
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      <text bytes="521" sha1="oro6cjwjvb3nx6lqyk47ni7khzre1yx" xml:space="preserve">Please visit [[Cooking|Topic:Culinary Arts]], an effort to create a department on the artistic preparation of food.

If you have come here from OHP to collaborate on [[Cooking|Topic:Culinary Arts]], and need a tutorial on wikimarkup ''et all'', start [[User:Blast/tutorial1|here]].

{{#babel:en|fr-2|de-1}}
{{user knowledge-food-2}}

== On other related projects ==
* [[Wikipedia:User:Blast san|Wikipedia]]
* [[Wikinews:User:Blast|Wikinews]]
* [[User:Octane|Commons]]
* [http://www.wikia.com/wiki/User:Blast Wikia portal]</text>
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  <page>
    <title>User:Test</title>
    <ns>2</ns>
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      <contributor>
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        <id>3039865</id>
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      <text bytes="92" sha1="f2e7wdj5hejrbk9u1tx2l4xk3w47akd" xml:space="preserve">I'm a '''test''' user. Mess with me all you want!

[[Category:Fake users]]
[[Category:Help]]</text>
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  <page>
    <title>User talk:Da Punk '95~enwikiversity</title>
    <ns>3</ns>
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      <id>2815593</id>
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      <contributor>
        <username>PieWriter</username>
        <id>3039865</id>
      </contributor>
      <minor />
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      <text bytes="7718" sha1="3db6to6ltygjrtwt7vsn9h7sq1vpp6o" xml:space="preserve">{{clear}}
{| style=&quot;border-spacing:8px;margin:0px -8px&quot; width=&quot;100%&quot;
|class=&quot;MainPageBG&quot; style=&quot;width: 55%; border:1px solid #084080; background-color:#F5FFFA; vertical-align:top;color:#000000;font-size: 85%&quot;|
{| width=&quot;100%&quot; cellpadding=&quot;2&quot; cellspacing=&quot;5&quot; style=&quot;vertical-align:top; background-color:#F5FFFA&quot;
! &lt;div style=&quot;margin: 0; background-color:#CEF2E0; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #084080; text-align:left; color:#082840; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;&quot;&gt; '''Hello Punk Boi 8! [[Wikiversity:Welcome, newcomers|Welcome]] to [[Wikiversity:What is Wikiversity?|Wikiversity]]!''' If you decide that you need help, check out [[Wikiversity:Help desk]], ask the [[Wikiversity:Support staff|support staff]], or ask me on my talk page. Please remember to [[Wikiversity:Signature|sign your name]] on talk pages using four tildes (~&amp;#126;~~); this will automatically produce your name and the date. Below are some recommended guidelines to facilitate your involvement. {{{1|}}} Happy Editing! -- [[User:JWSchmidt|JWSchmidt]] 13:47, 5 March 2007 (UTC)&lt;/div&gt;
|}
{| style=&quot;border-spacing:8px;margin:0px -8px&quot; width=&quot;100%&quot;
|class=&quot;MainPageBG&quot; style=&quot;width: 55%; border:1px solid #FFFFFF; background-color:#F5FFFA; vertical-align:top&quot;|
{| width=&quot;100%&quot; cellpadding=&quot;2&quot; cellspacing=&quot;5&quot; style=&quot;vertical-align:top; background-color:#F5FFFA&quot;
! &lt;div style=&quot;margin: 0; background-color:#084080; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #CEF2E0; text-align:left; color:#FFC000; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;&quot;&gt;Getting Started&lt;/div&gt;
|-
|style=&quot;color:#000&quot;|
* [[Wikiversity:Introduction|Get an Introduction to Wikiversity]]
* [[Wikiversity:Guided tour|Take a guided tour]]
* [[Help:Editing|How to edit a page]]
* [[Wikiversity:Be bold|Be bold in editing]]
* [[Portal:Learning Projects|Learning Projects]]
* [[Wikiversity:What Wikiversity is not|What Wikiversity is not]]
|-
! &lt;div style=&quot;margin: 0; background:#084080; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #cef2e0; text-align:left; color:#FFC000; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;&quot;&gt;Getting your info out there&lt;/div&gt;
|-
| style=&quot;color:#000&quot;|
* [[Wikiversity:Cite sources|Cite your sources]]
* [[Wikiversity:Disclosures|Neutral Point of View]]
* [[Wikiversity:Verifiability|Verifiability]]
|-
! &lt;div style=&quot;margin: 0; background:#084080; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #cef2e0; text-align:left; color:#FFC000; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;&quot;&gt;Getting more Wikiversity rules&lt;/div&gt;
|-
| style=&quot;color:#000&quot;|
* [[Wikiversity:Policies|Policy Library]]
|-
|}
|class=&quot;MainPageBG&quot; style=&quot;width: 55%; border:1px solid #FFFFFF; background-color:#F5FFFA; vertical-align:top&quot;|
{| width=&quot;100%&quot; cellpadding=&quot;2&quot; cellspacing=&quot;5&quot; style=&quot;vertical-align:top; background-color:#F5FFFA&quot;
! &lt;div style=&quot;margin: 0; background-color:#084080; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #CEF2E0; text-align:left; color:#FFC000; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;&quot;&gt;Getting Help&lt;/div&gt;
|-
|style=&quot;color:#000&quot;|
* [[Wikiversity:Questions|Questions]]
* [[Wikiversity:Research|Research guidelines]]
* [[Wikiversity:Help desk|Help Desk]]
|-
! &lt;div style=&quot;margin: 0; background-color:#084080; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #cef2e0; text-align:left; color:#FFC000; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;&quot;&gt;Getting along&lt;/div&gt;
|-
|style=&quot;color:#000&quot;|
* [[Wikiversity:Civility|Civility]]
* [[Wikiversity:Signature|Sign your posts]]
* [[Wikiversity:Scholarly ethics|Scholarly ethics]]
|-
! &lt;div style=&quot;margin: 0; background-color:#084080; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #cef2e0; text-align:left; color:#FFC000; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;&quot;&gt;Getting technical&lt;/div&gt;
|-
|style=&quot;color:#000&quot;|
* [[Wikiversity:Colloquium|Colloquium]]
|-
! &lt;div style=&quot;margin: 0; background-color:#084080; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #cef2e0; text-align:left; color:#FFC000; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;&quot;&gt;Getting social&lt;/div&gt;
|-
|style=&quot;color:#000&quot;|
* [[Wikiversity:Chat|Live chat channel]]
|-
|}
|}
|}

== custodianship ==

&quot;Hi, I am seeing if you can be my Custionan Mentor.&quot; &lt;-- Quoting myself from [[Wikiversity:List of custodian mentors]]: &quot;right now, Wikiversity mainly needs people who can contribute to building Wikiversity content; for example, by working on learning projects. I think that custodian candidates should demonstrate (by good editing) their commitment to the Wikiversity project before expecting to become a custodian. Also, every custodian needs to have an active email address set in preferences.&quot; I think it is important that the Wikiversity community have a chance to get to know candidates for custodian ship. Good page editing is the best way to make yourself known to the community. It can also be useful to get to know potential custodian mentors by participating at the [[Wikiversity:Chat|#wikiversity-en IRC channel]]. --[[User:JWSchmidt|JWSchmidt]] 13:41, 23 March 2007 (UTC)

== [[Wikiversity:Request_custodian_action#Username_Change|Username Change]]==
see link please, ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] &lt;small&gt;uses the [[Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])&lt;/small&gt; 07:46, 6 July 2008 (UTC)

== Your account will be renamed ==

&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See [[m:Special:MyLanguage/Single User Login finalisation announcement|the announcement]] for more information.

Unfortunately, your account clashes with another account also called Da Punk '95. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Da Punk '95~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can [[Special:GlobalRenameRequest|pick out a different name]].

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours,&lt;br /&gt;[[m:User:Keegan (WMF)|Keegan Peterzell]]&lt;br /&gt;Community Liaison, Wikimedia Foundation
&lt;/div&gt; 23:33, 17 March 2015 (UTC)
&lt;!-- SUL finalisation notification --&gt;

== Renamed ==

&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr”&gt;
This account has been renamed as part of [[m:Special:MyLanguage/Single User Login finalisation announcement|single-user login finalisation]]. If you own this account you can [[{{#special:userlogin}}|log in using your previous username and password]] for more information. If you do not like this account's new name, you can choose your own using this form after logging in: [[{{#special:GlobalRenameRequest}}]]. -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]])
&lt;/div&gt; 06:05, 19 April 2015 (UTC)
&lt;!-- SUL post-rename notification --&gt;</text>
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    <title>Skill Builder: Invariant Tasks</title>
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      <comment>/* What to teach? How to teach it? */ Maybe, the best way to make a starting point of the lesson or a topic is by not directly used the long questions but in a way that make the student interested in the topic or should I say by telling a story that make them enjoy listening into it. Then, after you finish the story, ask them small questions about the story.</comment>
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      <text bytes="5354" sha1="0abulv4gimbgiqmdqdplwiyslgxjsw4" xml:space="preserve">[[Topic:Instructional Design]] &gt; [[Cognitive behaviors]] &gt; [[Invariant Tasks]] &gt; [[Skill Builder: Invariant Tasks]]

Source: [http://www.indiana.edu/~idtheory/methods/m2.html Invariant Tasks] by Charles M. Reigeluth. Used by Permission.

----

'''Skill Builder Instructions:''' This is your opportunity to practice the skills covered in this section of the lesson. As the Wikiversity is a collaborative learning space, you will contribute your original ideas to the work of those who have come before you by enhancing and adding to the possible responses within the case scenario laid out below. Are you saying to yourself, &quot;...but, isn't looking at the answers from a fellow learner the same as cheating?&quot; Not in this case. In fact, contributing your original ideas while participating with fellow learners and building upon their work is the intended goal of this Skill Builder exercise. The desired outcome is to build an ever evolving working document filled with not just one possible response to each question, but a host of instructional strategies that you can use on your next instructional design project. Have fun and get creative!

----

== What to teach? How to teach it? Maybe, the best way to make a starting point of the lesson or a topic is by not directly used the long questions but in a way that make the student interested in the topic or should I say by telling a story that make them enjoy listening into it. Then, after you finish the story, ask them small questions about the story. ==
Imagine a friend of yours, Jennifer, has just been hired to tutor a sixth grader, Sam. She's all excited, because it's her first tutoring job. However, she's very worried, too, because she has never tutored before. She has come to you for advice. You had just read somewhere that the most important concerns in any instruction are &quot;what to teach&quot; and &quot;how to teach it&quot;. Let's assume that Jennifer has already found out that Sam is supposed to learn the names of the first seven Presidents of the United States. So let's turn our attention to &quot;how to teach it&quot;. Based on the need to create strong links within memory, what would you say is the most important instructional strategy you could suggest to your friend?  '''Think about it, click &quot;edit&quot; for this section, and add your answer below:'''

*[[User:Phonebein|Phonebein]] 18:33, 29 March 2007 (UTC) I think the instructional strategy I would use would be to create a funny rhyme using some or all of the parts of the president's names.

== What are the recommended teacher actions? ==
What should Jennifer be doing while Sam is stating the names of the first seven Presidents of the United States? Should she just sit there or should she do something?   What do you think she should do? '''Think about it, click &quot;edit&quot; for this section, and add your answer below:'''


== What about when the learner get stuck? == 
Picture Jennifer asking Sam to name the Presidents. If he only knows who the first one is (George Washington), would you start by asking him to name all of them? That would be silly. So what other guideline should you give Jennifer? If this all seems obvious, it shows you have already picked up an intuitive understanding of some of the most basic principles of instruction from your observations or studies. Congratulations! But beware that we are quickly moving on to less obvious principles. '''Think about it, click &quot;edit&quot; for this section, and add your answer below:'''


== What about larger amounts of content? == 
Let's assume that Sam needs to learn the names of all the Presidents of the United States instead of just seven. Forty Presidents is a lot to learn. Would you present all 40 at once and then elicit practice on all 40 at once? '''Think of what you would recommend to make it easier for him to learn all those names, click &quot;edit&quot; for this section, and jot your answer below:'''


== How to prompt the learner? ==
Let's assume the names are difficult for Sam. What else can you recommend to help him remember the names in the first chunk? '''Keep in mind the need to strengthen those links, click &quot;edit&quot; for this section, and jot your answer below:'''


== What is more powerful and efficient than repetition to facilitate memorization? ==
However, aside from motivational strategies such as games to increase the learner's effort to memorize the information, there is another feature which can be even more powerful and efficient than repetition to facilitate memorization.  '''Try to think of what it is, click &quot;edit&quot;, and write your answer below:'''


== What about motivation? ==
It is hard to remember all those names, and Sam is bound to have trouble at first. What can Jennifer do to keep him from getting discouraged—to keep his attitude positive and his concentration high? High motivation translates into high effort, and that means quicker and better learning. What would you recommend to Jennifer?  '''Think about it, click &quot;edit&quot;, and add your answer below:'''

== Exercise Synthesis ==
To see possible responses to this Skill Builder Exercise (prepared by the original authors of this lesson), click here: [[Skill Builder: Invariant Tasks Synthesis]]. Note that these responses are purposely placed on a separate screen to provide general guidance should you become stuck, but NOT to imply a single correct response to the case scenario.</text>
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      <text bytes="5031" sha1="epmopwfnzgcmrfvdf7ykeomenuo6430" xml:space="preserve">[[Topic:Instructional Design]] &gt; [[Cognitive behaviors]] &gt; [[Invariant Tasks]] &gt; [[Skill Builder: Invariant Tasks]]

Source: [http://www.indiana.edu/~idtheory/methods/m2.html Invariant Tasks] by Charles M. Reigeluth. Used by Permission.

----

'''Skill Builder Instructions:''' This is your opportunity to practice the skills covered in this section of the lesson. As the Wikiversity is a collaborative learning space, you will contribute your original ideas to the work of those who have come before you by enhancing and adding to the possible responses within the case scenario laid out below. Are you saying to yourself, &quot;...but, isn't looking at the answers from a fellow learner the same as cheating?&quot; Not in this case. In fact, contributing your original ideas while participating with fellow learners and building upon their work is the intended goal of this Skill Builder exercise. The desired outcome is to build an ever evolving working document filled with not just one possible response to each question, but a host of instructional strategies that you can use on your next instructional design project. Have fun and get creative!

----

== What to teach? How to teach it? ==
Imagine a friend of yours, Jennifer, has just been hired to tutor a sixth grader, Sam. She's all excited, because it's her first tutoring job. However, she's very worried, too, because she has never tutored before. She has come to you for advice. You had just read somewhere that the most important concerns in any instruction are &quot;what to teach&quot; and &quot;how to teach it&quot;. Let's assume that Jennifer has already found out that Sam is supposed to learn the names of the first seven Presidents of the United States. So let's turn our attention to &quot;how to teach it&quot;. Based on the need to create strong links within memory, what would you say is the most important instructional strategy you could suggest to your friend?  '''Think about it, click &quot;edit&quot; for this section, and add your answer below:'''

*[[User:Phonebein|Phonebein]] 18:33, 29 March 2007 (UTC) I think the instructional strategy I would use would be to create a funny rhyme using some or all of the parts of the president's names.

== What are the recommended teacher actions? ==
What should Jennifer be doing while Sam is stating the names of the first seven Presidents of the United States? Should she just sit there or should she do something?   What do you think she should do? '''Think about it, click &quot;edit&quot; for this section, and add your answer below:'''


== What about when the learner get stuck? == 
Picture Jennifer asking Sam to name the Presidents. If he only knows who the first one is (George Washington), would you start by asking him to name all of them? That would be silly. So what other guideline should you give Jennifer? If this all seems obvious, it shows you have already picked up an intuitive understanding of some of the most basic principles of instruction from your observations or studies. Congratulations! But beware that we are quickly moving on to less obvious principles. '''Think about it, click &quot;edit&quot; for this section, and add your answer below:'''


== What about larger amounts of content? == 
Let's assume that Sam needs to learn the names of all the Presidents of the United States instead of just seven. Forty Presidents is a lot to learn. Would you present all 40 at once and then elicit practice on all 40 at once? '''Think of what you would recommend to make it easier for him to learn all those names, click &quot;edit&quot; for this section, and jot your answer below:'''


== How to prompt the learner? ==
Let's assume the names are difficult for Sam. What else can you recommend to help him remember the names in the first chunk? '''Keep in mind the need to strengthen those links, click &quot;edit&quot; for this section, and jot your answer below:'''


== What is more powerful and efficient than repetition to facilitate memorization? ==
However, aside from motivational strategies such as games to increase the learner's effort to memorize the information, there is another feature which can be even more powerful and efficient than repetition to facilitate memorization.  '''Try to think of what it is, click &quot;edit&quot;, and write your answer below:'''


== What about motivation? ==
It is hard to remember all those names, and Sam is bound to have trouble at first. What can Jennifer do to keep him from getting discouraged—to keep his attitude positive and his concentration high? High motivation translates into high effort, and that means quicker and better learning. What would you recommend to Jennifer?  '''Think about it, click &quot;edit&quot;, and add your answer below:'''

== Exercise Synthesis ==
To see possible responses to this Skill Builder Exercise (prepared by the original authors of this lesson), click here: [[Skill Builder: Invariant Tasks Synthesis]]. Note that these responses are purposely placed on a separate screen to provide general guidance should you become stuck, but NOT to imply a single correct response to the case scenario.</text>
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      <text bytes="307" sha1="q3s8ihbebdvr6xd6hykifszxihc5n74" xml:space="preserve">== Ghenson ==
&lt;b&gt;About Me:&lt;/b&gt; I am a web developer specializing in [http://www.php.net PHP] and SQL (both [http://www.mysql.com MySQL] and [http://www.postgresql.org PostgreSQL]), and I have been working in the education industry for the past three years.

[[User:Ghenson|Ghenson]] 12:34, 2 June 2007 (UTC)</text>
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      <text bytes="4507" sha1="ghyqrhvkoe4t3suqnwmsdejoiregk2t" xml:space="preserve">[[Image:Logo-cesarharada-blink.gif|left|thumb|150px|Logo de César Harada]]
&lt;div style=&quot;float:right;border:solid #ffd700 1px;margin:1px&quot;&gt;
{| cellspacing=&quot;0&quot; style=&quot;width:238px;background:#fffacd&quot;
! style=&quot;text-align:center;width:45px;height:45px;background:white;font-size:18pt; color:black&quot; | [[:Category:Wikipedians who use Skype| ]] [[Image:Crystal Clear app Internet Connection Tools.png|35px]]
| style=&quot;font-size:8pt;padding:4pt;line-height:1.25em&quot; | This user uses '''[[Skype]]'''. &lt;br /&gt;The username is '''{{{cesarminoruharada}}}'''.
|}
&lt;/div&gt;[[Category:Wikipedians who use Skype|&lt;noinclude&gt; &lt;/noinclude&gt;{{PAGENAME}}]]&lt;noinclude&gt;[[Category:Instant messaging user templates|Skype]]&lt;/noinclude&gt;
'''Cesar Harada''' (cesarharada) is now trying to start teaching in the [[Wikiversity:Main Page|Wikiversity]], in the [[Portal:Humanities|&lt;b&gt;School of Humanities&lt;/b&gt;]], [[School:Art and Design|&lt;b&gt;Arts and Design&lt;/b&gt;]], where he founded the [[New media arts|&lt;b&gt;New Media Arts Department&lt;/b&gt;]] [[w:June 14|14/6]]/[[w:2007|2007]].
He wishes that many life-long learners and explorers will join the project!  

Cesar Minoru Harada was born in [[w:Bayonne|Bayonne]] [[w:France|France]] in [[w:1983|1983]] from a french mother and a japonese father. He studied in [[w:France|France]] and [[w:Japan|Japan]], studied music in [[w:Saint-Malo|St Malo]] [[w:Cathedral|cathedral]]'s choir, [[w:glassblowing|glass-blow]] at the ADAC center [[w:Paris|Paris]] with master Pedro Veloso, [[w:Judo|judo]] with Henry Najos and Eric Pariset. He currently assists [[w:Tetsuo Harada|Tetsuo Harada]] sculptor, assisted [[w:William Kentridge|William Kentridge]] at la Poudriere for a co-realised animation film 2004. He graduated as section's major from &quot;école Boulle [http://fr.wikipedia.org/wiki/%C3%89cole_Boulle]&quot; from the arts applied section, studied graphic design in [[w:Central Saint Martins College of Art and Design|Central Saint Martins]] ([[w:London|London]] [[w:United Kingdom|UK]]) as an exchange student from the [[w:Grands établissements|Ecole Nationale Supérieure des Arts Décoratifs]] time student. He did his thesis about &quot;Troubles in Space Perception&quot;. He taught graphic design at the [[W:National Superior School of Architecture|École Nationale Supérieure d'Architecture de Versailles]] of [[w:Versailles|Versailles]] and did a Master II degree in [[New media arts|new media arts]] in [[w:University of Paris VIII: Vincennes - Saint-Denis|Paris 8]] with his essay &quot;Force Feedback&quot;. Between 2007 and 2009 he will be a student at the [[w:Royal College of Art|Royal College of Art]] in [[w:London|London]] with [[w:Tony Dunne|Tony Dunne]], [[w:Fiona Raby|Fiona Raby]] and [[w:Brendan Walker|Brendan Walker]]. 
&lt;br /&gt;
&lt;br /&gt;
Cesar Harada exhibited :
&lt;br /&gt;
'''2000''' &quot;formes&quot; (glass-blow sculptures) at &quot;la maison des ateliers&quot; Paris 75001,&lt;br /&gt;
'''2001''' &quot;pas cher&quot; (fashion acessory) at &quot;la manufacture des oeillets&quot; Ivry s/Seine,&lt;br /&gt;
'''2002''' &quot;INFO TRAFIC&quot; (choregraphic + video performance) at the french ministry of equipment la Défense, 2002 &quot;Happyjoy family&quot; (5 animation films) at the British Council Paris,&lt;br /&gt;
'''2003''' &quot;va chercher bonheur&quot; (multimédia) at the ENSAD Paris, &quot;pauvre de nous&quot; (installation) at the Humankind museum with midid6 collective Paris, &quot;QUEST&quot; in Islington water tunnel London, &quot;human experience&quot; in CSM London,&lt;br /&gt;
'''2004''' &quot;chaos magazine&quot;, &quot;MADRID 11-03-04&quot;, &quot;peace-war&quot;, &quot;la grotte&quot; in Kifisia art centrum and Polychronopoulos museum of Skironio Greece, took part in the &quot;pharaon&quot; action in the Paris tube Pyramide for C-album, &quot;le parc d'attraction&quot; in ENSAD Paris&lt;br /&gt;
'''2005''', exhibited &quot;FORMES&quot;, &quot;ligne-volume&quot;, &quot;Les champs de la Beauce 2005&quot; (animations films) at the Niigata prefectural museum of modern art Japan,&lt;br /&gt;
'''2006''' &quot;HARADA&quot; (book, installation, projection) at the Pompidou center's bookshop
&lt;br /&gt;
CesarMinoru Harada's films were screened in [[w:France|France]], [[w:United Kingdom|UK]], [[w:Italy|Italy]], [[w:Spain|Spain]], [[w:Greece|Greece]], [[w:Japan|Japan]], [[w:Russia|Russia]] ...

== External links ==
http://www.cesarharada.com [http://www.cesarharada.com]

== Some works ==
&lt;gallery&gt;
Image:Boat-cesarharada-available.jpg|''vela'', barque de filmage
Image:Human-experience cesarharada-csm.jpg|''human experience'', installation collaborative
Image:Arvo cesarharada.jpg | animation film on the music of [[w:Arvo Pärt|Arvo Pärt]]
&lt;/gallery&gt;


[[fr:Utilisateur:Cesarharada]]

[[Category:Students of new media arts]]</text>
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    <title>Quantum computing</title>
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{{center top}}&lt;big&gt;Welcome to '''Quantum Computing'''!&lt;/big&gt;{{center bottom}}

Quantum Computing is an important tool for code breakers. Hence the pioneers in this topic have often worked for organisations like the [[w:National Security Agency|National Security Agency]] in the USA, [[W:GCHQ|GCHQ]] in the [[W:UK|UK]]. These two organisations collaborate with [[W:Echelon|Echelon]] {{RightTOC}}

==Discussion questions==
* How might quantum computing impact computer science? 
* What applications does quantum computing have? 

==Learning paths and materials==
* [[School:Computer Science|School:Computer science]]
* [[Portal:Quantum computing]] - plan additional learning resources for quantum computing.
* [[Portal:Particle physics]]
* [[Quantum mechanics/Course|Study guide:Quantum mechanics I]]
* [[Making sense of quantum mechanics]]
* [[Quantum theory of the atom]]

==Readings==
===Wikipedia===
* [[w:Quantum computing|Quantum computing]]
* [[w:Quantum programming|Quantum programming]]
* [[w:Kane quantum computer|Kane quantum computer]]
* [[w:Quantum cognition|Quantum cognition]]
* [[w:Quantum gate|Quantum gate]]
* [[w:Timeline of quantum computing|Timeline of quantum computing]]
* [[w:Quantum complexity theory|Quantum complexity theory]]

==External links==
* June 2008 ''[http://www.telegraph.co.uk/earth/main.jhtml?xml=/earth/2008/07/01/scicomputer101.xml  Will the QC kill the PC?] ''

[[Category:Physics]]
[[Category:Computer science]]

{{BookCat}}</text>
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      <text bytes="11846" sha1="okjlnt0gp9e5hrrbt0js5n3y9jggp61" xml:space="preserve">&lt;div style=&quot;float:right;border:solid #ffd700 1px;margin:1px&quot;&gt;
{| cellspacing=&quot;0&quot; style=&quot;width:238px;background:#fffacd&quot;
! style=&quot;text-align:center;width:45px;height:45px;background:white;font-size:18pt; color:black&quot; |[[Image:Telefon, Nordisk familjebok.png|35px]]
| style=&quot;font-size:8pt;padding:4pt;line-height:1.25em&quot; | Member of the Wikiversity '''[[Skype]]'''. &lt;br /&gt;Skype username: '''jwsurf'''
|}
&lt;/div&gt;[http://en.wikiversity.org/w/index.php?title=User_talk:JWSchmidt&amp;action=edit&amp;section=new Leave a message for JWSchmidt] | [[User:JWSchmidt/Blog|my Wikiversity blog]]&lt;br style=&quot;clear:both;&quot;/&gt;
&lt;div style=&quot;background-color:#FFF594; padding:10px; border:1px solid black;&quot;&gt; &lt;big&gt;'''&quot;Openness and inclusiveness are ..... our radical means to our radical ends.&quot;'''&lt;/big&gt;
([[m:Translation requests/WQ/3/En|source]])&lt;/div&gt;
[[Image:Chrysalis5504.jpg|thumb|left|Wikiversity today.]]

[[w:Jimmy Wales|Jimmy Wales]] recently [http://wikimania2006.wikimedia.org/wiki/Opening_Plenary_%28transcript%29#Software_changes_for_stable_versions_.2852:30.29 said], &quot;I think we should run small experiments, tests, see what works, what doesn't, and be prepared to be flexible and change, and not be too locked into stone about how things should work.&quot;

I am interested in Wikiversity as a playground for explorers and active learners. I hope that the Wikiversity community can learn how to facilitate [[Wikiversity:Learning projects|learning projects]] that will allow Wikiversity participants to reach and discover their learning goals.
[[Image:Lepidoptera 001.jpg|thumb|right|250px|Wikiversity of the future.]]
I became interested in Wikiversity back in [http://meta.wikimedia.org/w/index.php?title=Talk:Wikiversity/Old&amp;diff=prev&amp;oldid=93004 2004]. After the 2005 community [http://meta.wikimedia.org/w/index.php?title=Wikiversity/Vote/en&amp;diff=prev&amp;oldid=208844 vote] on the original Wikiversity project proposal the Board of Trustees [[Wikiversity:Original proposal|asked for some changes]] to the proposal, I started [[m:User:Cormaggio/Wikiversity chat 04Nov2005|conspiring]] with folks like [[User:Cormaggio|Cormaggio]] to fix the Wikiversity proposal. I think the Board was correct to direct the Wikiversity community away from [[Wikiversity:Courses|conventional courses]]. We need to find new ways to use wiki technology to facilitate online learning. Towards that end, a new Wikiversity [[Wikiversity:The original learning model|model for e-learning]] was developed at the [[m:Wikiversity/Example|Wikimedia meta-wiki]] and included in the modified Wikiversity project proposal, as requested by the Board.

So the [[Wikiversity:Approved Wikiversity project proposal|modified Wikiversity project proposal]] was [[Wikiversity:History of Wikiversity#Wikimedia Foundation approval|finally approved]]. As an &quot;emergency&quot; measure, [[User:Cormaggio|Cormaggio]], acting as the first Wikiversity [[wikipedia:Wikipedia:Bureaucrats|bureaucrat]], made me a [[Wikiversity:Support staff|custodian]] on the English language Wikiversity. If you need the help of a custodian, use my [[User talk:JWSchmidt|talk page]], [http://en.wikiversity.org/wiki/Special:Emailuser/JWSchmidt send me an email], or go to the page that is used to [[Wikiversity:Request administrator action|Request custodian action]].

{{Template:Wikiversity}}
==Other Wiki Projects==
On 27 February 2003 I registered as [[w:User:JWSchmidt|JWSchmidt on the English language Wikipedia]]. In December of 2005 I became an [[w:Wikipedia:Requests for adminship/JWSchmidt|administrator at Wikipedia]]. There is more information about me on my [[wikipedia:User:JWSchmidt|Wikipedia user page]]. I am also a custodian at [[betawikiversity:User:JWSchmidt|Wikiversity beta]].

I remain intrigued and challenged by the way wiki projects first fragment then the fragments struggle to cooperate. I hope the Wikiversity community can support good editing for Wikibooks textbooks and Wikipedia encyclopedia articles.

==Learning projects of interest==
*[[Jargon project]] - promotes use of hyperlinks, gloassaries, invitations for people to ask questions.
*[[School:Free Learning|A. S. Neill School of Free Learning]] - students are assisted as they explore the topics that are of greatest personal interest.
*[[Wikiversity:Wiki as a tool for learning|Wiki learning project]] - explore how to use wikis in education
**[[Learning to learn a wiki way|Learn to learn a wiki way]]
*[[Human Genetic Uniqueness Project]] - What makes humans '''human'''?
**[[Molecular Paleontology Reading Group]]
*[[MediaWiki Project]] - develop new MediaWiki features for the Wikiversity community
*[[Wikiversity nonprofit corporation|Wikiversity NPC]] - $$$$$$$ for Wikiversity
*[[Cell biology|Cell Biology]] - Hub for Cell Biology-related learning activities
*[[Cell biology improvement drive]] - service project for Wikipedia
*[[Citing Sources]] - Fostering the Wikiversity culture of scholarly citation of sources
*[[Why 10 dimensions]] - learning about [[w:String theory|String theory]]
*[[Wikiversity the Movie]] - need to link to from relevant pages
*[[Wiki]] - learning how to use wiki technology to facilitate online learning.
*[[Science as religion|Science as Religion]] - explore the relationships between science and religion
*[[Creationism/Science teaching materials|Science teaching materials for creationism]] - What should Wikiversity do if a an editor wants to add learning materials to Wikiversity that aim to teach creationism as science?
*[[Unsolved problems in neuroscience]]
*[[Free Will as control]] [http://www.philosophytalk.org/pastShows/FreeWill.htm philosophytalk]
*[[Portal:Religious studies|Topic:New Age Religions]]
*[[Consilience]]
*[[Exploring science through fiction]]
*[[Wikihigh]]
*[[Stem cells|Stem Cells]]
*[[Virtual cell project]] [http://www.studiodaily.com/main/technique/tprojects/6850.html example]
*[[Introduction to medical microbiology|Introduction to Medical Microbiology]]
**[[Sexually transmitted diseases]]/[[Sexually transmitted diseases|Sexually Transmitted Diseases]]
**[[Infectious Disease and Public Health]]
*[[Wikiversity scholarship]] learning project for all Wikiversity participants
*[[GarageBand|Introduction to GarageBand]]
*[[Nobel Prize in Physiology or Medicine]]
====Divisions and Departments====
*[[Topic:Science research and the law|Topic:Science research and the Law]]
*[[Portal:Cell biology|Topic:Cell Biology]]
*[[Portal:Science|Topic:Basic sciences]]

{{Scholarly ethics}}

===Other===
{{Editing help}}
{{WikiversityActivities}}
*[[User:JWSchmidt/Blog/20 January 2007|Comments on courses]] at Wikiversity.
*[http://collaboration.wikia.com/wiki/Wikiversity History] and visions
*[http://chronicle.com/free/2005/12/2005121601t.htm Wikipedia, the Free Online Encyclopedia, Ponders a New Entity: Wikiversity] -  old news (pre-launch)
*[http://journal.fibreculture.org/issue3/issue3_scholz.html#21 It's New Media: But is it Art Education?] by Trebor Scholz (links to wikiversity.org)

==See also==
*[[Special:Version]]
*[[w:DjVu]] - allowed file type
*[[:Category:Uncategorized]] - for pages that need to be categorized...also; [[Special:Uncategorizedpages|Special:Uncategorized pages]]
*[[:Category:Requests for unblock]] - watch for requests
*[[:Category:Possible copyright violations]] - watch
*[[Wikiversity:Publishing original research]]
*[http://en.wikiversity.org/w/index.php?title=User:JWSchmidt/monobook.css&amp;curid=5371&amp;diff=43419&amp;oldid=25738 shaded discussion page code]
*[[:Template:Audio]] - inline link to audio
*[[Wikiversity:Frequently visited pages]]
*[[Wikiversity:Glossary]]
*[[User:JohnWSchmidt]] - for testing non-custodian user accounts
*[[Wikiversity:Introduction]] - 
*[[Wikiversity:Multiple points of view]]
*[[Wikiversity:Welcome templates]]
*[[:Category:Page creation templates]]
*[[Template:Courses]]
*[[User:JWSchmidt/Sandbox]]
*[[:Category:Import backups]]
*[[:Category:Pages moved from Meta]]
*[[Educational wikis|Educational Wikis]]
*[[Template:Browse]] - content of [[Wikiversity:Browse]]
*[[User:SBJohnny]] - imported page linking
*[[Template:WikiversityInterWikiLink]]
*[[Wikiversity:Help with the migration of Wikiversity pages from Wikibooks|Help with the migration of Wikiversity pages from Wikibooks]]
*[[Wikiversity:I Want to Know]]
*[[Wikiversity:Userboxes]] - they need to help, not hurt the community

==External links==
*[http://www.aboutus.org/WikiVersity.org Wikiversity page] at Aboutus.org
*[http://enwiki-trust.cse.ucsc.edu/index.php/Convolution color coded] wiki words to show how much they are changed by editors through time. Also forms the basis for ranking editors according to how long their edits are retained by other editors.
*[http://wikimediafoundation.org/wiki/Meetings/June_1-3%2C_2007 what do to with the wikipedia.com portal, and agreed to the concept of setting up a specific portal, where we could explain more about the projects, about the Foundation, welcome services etc... all things to be discussed in length as wikipedians know perfectly to do]?
*[http://lists.wikimedia.org/pipermail/foundation-l/2007-July/031049.html Kronberg Declaration on the Future of Knowledge Acquisition and Sharing]
*[http://globalvision.wikia.com wiki documentary]
*[[w:Wikipedia:Wikipedia Signpost/2007-06-04/Admin desysopped|could there be automated sock puppet checking for votes]]?
*[http://en.wikipedia.org/w/index.php?title=Wikipedia:Deletion_policy&amp;diff=prev&amp;oldid=133886326 Admins are encouraged to ignore the results of idiotic votes, and to listen to thoughtful discussions] - six years along and we have to keep saying this?
*[http://schools-wikipedia.org/ 2007 Wikipedia Selection for schools]
*[[w:Wikipedia:Song/The RfA Candidate's Song]] - the Wikipedia audio template is looking good
*[[m:User:Dapete/Catgraph]] - category graph tool
*[[m:Open Source Toolset|Open Source Toolset]]
*[http://www.gizmoproject.com/ Gizmo] - like Skype but allows recording.
*[[w:Template:Listen]] - beta release of in-browser ogg file play
*[http://wikimediafoundation.org/wiki/Advisory_Board Wikimedia advisory board]
[[Image:Hist article size enwiki.png|thumb|right|300px]]
*[[w:Wikipedia:WikiXRay]]
*[http://www.mediawiki.org/wiki/Extension:ImageMap image maps]
*[[w:Wikipedia:Syndication]] - RSS feeds
*[http://www.archive.org/download/Wikiversity_Reports_Editing_Tutorial/Editing_tutorial.mov Editing tutorial]
*[http://wiki.laptop.org/go/Educational_content_ideas One Laptop per Child] - educational content
*[[b:Lucid Dreaming|Lucid Dreaming]] - possible Wikiversity cognitive science project
*[[m:Multimedia|Multimedia]] - Multimedia for Wikimedia
*[[w:Wikipedia:Media|Audio and video media]] - at Wikipedia; ogg formats
*[http://beta.wikiversity.org/wiki/Main_Page Metaversity] 13:33 24 August, 2006
*[http://www.alexa.com/data/details/contact_info?url=en.wikiversity.org%2Fwiki%2FMain_Page Alexa] - October 14 2006 Traffic Rank for wikiversity.org:  66,733,  Jan 5 2007 raffic Rank for wikiversity.org:  26,382
*[http://www.uwm.edu/Libraries/courses/wiki/ slide shows about Wikipedia and wikis] - University of Wisconsin-Milwaukee
*[http://meta.wikimedia.org/wiki/Report_from_Frankfurt_-_October_2006 Board retreat] - October 2006
*[http://de.wikiversity.org/wiki/Benutzer:JWSchmidt German Wikiversity] - unique namespaces
*[http://es.wikiversity.org/wiki/Usuario:JWSchmidt Spanish Wikiversity] - bad sidebar links
*[http://fr.wikiversity.org/wiki/Utilisateur:JWSchmidt French Wikiversity] - cool category link
*[http://wikimediafoundation.org/wiki/Our_projects List of Wikimedia Foundation Projects]
*[http://wikimediafoundation.org/wiki/Resolution_ombudsperson_checkuser checkuser ombudsperson]
*[[w:Wikipedia:Wikipedia Signpost/2006-12-04/Arbitration report|pseudoscience arbcom case]]
*[[w:Lawrence Lessig|Lawrence Lessig]] - Free culture
*[http://www.scholarpedia.org/ scholarpedia] - sp
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    <title>Educational Media Awareness Campaign</title>
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__NOTOC__

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&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;

== Goals ==
The goal of the [[Educational Media Awareness Campaign]] is twofold:

# To help educators effectively use the internet to find suitable media for learning resources.
# To assist educators in integrating media correctly and legally, ensuring reusability and proper permission handling.

This campaign is primarily targeted at the primary and secondary education sectors, but it is also beneficial for tertiary, informal, and other educational fields.

== Background ==
One of the biggest challenges faced by creators of digital educational resources is the ''illegal use of images and graphic content''.

Publishers—both digital and print—regularly receive submissions that must be rejected due to copyright violations. Image copyright infringement is especially common in schools.

This situation is paradoxical. In recent years, the availability of well-documented, reusable, and redistributable media has increased significantly:
* [[c:|Wikimedia Commons]] provides millions of legally reusable, well-categorized images.
* [[w:Flickr|Flickr]] also offers a vast collection of reusable images, though less structured.

In addition, although somewhat controversial, [[w:Artificial intelligence|artificial intelligence]] is increasingly effective in helping to develop free to use educational media.

With such resources available, there is no need for illegal use of images.

The [[Educational Media Awareness Campaign]] aims to solve this issue by:
* Showcasing galleries of reusable images
* Providing case studies on finding appropriate media
* Listing trusted media repositories
* Offering tutorials on licensing and documentation

A key focus is introducing educators to the effective educational use of Wikimedia Commons.

== A Multi-site Effort ==
The [[Educational Media Awareness Campaign]] is a subproject of Wikiversity Outreach. It collaborates with various educational and media platforms to expand its impact.

== Galleries and Pictures of the Day ==
The campaign includes:

* 100+ featured images  
* Coverage across 12 major school subjects  
* Carefully written captions with source links  
* Access to thousands of related images  

These images serve as entry points for educators.

Additionally:

* Images are organized into dynamic pages  
* They can be used as &quot;Picture of the Day&quot; on wikis  
* Separate rotating sections exist for each subject  

== Investigation and Analysis of Digital Educational Resources ==
With technological advancement, digital educational resources have become increasingly diverse.

A survey of 82 undergraduate students revealed:

* Students prefer digital resources for self-directed learning  
* Search engines are the most commonly used access method  
* Traditional computers are still the preferred device  

To maximize effectiveness, educational providers should:

* Understand user behavior  
* Align resources with user needs  
* Optimize accessibility and usability

== See also ==
* [[meta:Education]]
&lt;/div&gt;

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[[:Category:Wikiversitans|Wikiversitan]] since March, 2008

''A loose, personal (i.e., somewhat idiosynchratic) organisation of Wikiversity-related how-tos and links.''

==To sort==
{|style=&quot;background:transparent;&quot;
|valign=top|
* [http://tools.wikimedia.de/~magnus/commonshelper.php commonshelper]
* [[User:Jtneill/Wikification|Wikification]]
* [[w:Help:Interwiki_linking#Project_titles_and_shortcuts|Interwiki linking]]
* [[Wikiversity:Activity bars]]
* [[Wikiversity:Percent complete]]
|valign=top|
* [[Wikiversity:Import|import]]
* [[Wikiversity:Maintenance]]
* [[Wikiversity:Namespaces]]
* [[Wikiversity:Naming conventions]]
|valign=top|
* [[Wikiversity:Participants]]
* [[Wikiversity:Peer review]]
* [[Wikiversity:Review board]]
* [[Wikiversity:Searching]]
* [[How to be a Wikimedia sysop]]
|}

==Anchor==
* [[Template:Anchor]], e.g., [[#test]] will go to &lt;code&gt;&lt;nowiki&gt;{{anchor|test}}&lt;/nowiki&gt;&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{anchor|anchor=test}}&lt;/nowiki&gt;&lt;/code&gt; (should go to end of page)

==Archiving==
* Example of autoarchiving: [[User talk:Terra]]

==Blogging==
* [[Wikiversity Blog howto]]

==Boxes==
[[User:Jtneill/Sandbox/Tables and boxes]]

The simplest of boxes

{| class=&quot;messagebox&quot;
|-
| ABC
XYZ
|}

&lt;blockquote style=&quot;padding-left:1.0em; padding-right:1.0em; background-color:#eaf8f4;&quot;&gt;
Its good that it works in practice, because it certainly doesn’t work in theory[https://blogs.ch.cam.ac.uk/pmr/2007/10/14/the-thing-about-wikipedia-is-that-it-only-works-in-practice-in-theory-it-can-never-work/] 
&lt;/blockquote&gt;

==Categories==
===Listing pages that intersect categories===
&lt;pre&gt;
&lt;dynamicpagelist&gt;
category = Resources needing improved grammar
category = Motivation and emotion/Book/2025
mode     = bullet
&lt;/dynamicpagelist&gt;
&lt;/pre&gt;
{{collapse top|2025 book chapters that need grammar improvements}}
&lt;dynamicpagelist&gt;
category = Resources needing improved grammar
category = Motivation and emotion/Book/2025
mode     = bullet
&lt;/dynamicpagelist&gt;
{{collapse bottom}}

===Order/sort===
wikitext. Wikis with a consensus to do so can [[m:Special:MyLanguage/Requesting wiki configuration changes|request]] a configuration change to display them in alphabetical order. [https://phabricator.wikimedia.org/T373480]

Using titleparts
&lt;nowiki&gt;[[Category:{{#titleparts:{{PAGENAME}}|1}}]]&lt;/nowiki&gt;

==[[/Centering/]]==
{{User:Jtneill/Wikiversity/Centering}}

==Chat==
* [[irc:wikiversity-en|#wikiversity-en]]

==Citations and referencing==
* [[w:Help:Citation tools|Citation tools]]
* [[:Category:Citation templates]]
* [[mw:Help:Cite]]
* [[Template:Citation]]
* [[WV:REF]]
* Example: Outward Bound Process Model&lt;ref&gt;Walsh, V., &amp; Golins, G. L. (1976). ''[http://wilderdom.com/theory/OutwardBoundProcessModel.html The exploration of the Outward Bound process]''.  Denver, CO: Colorado Outward Bound School.&lt;/ref&gt;

;References
{{reflist|1}}

==Collapse boxes==
{{collapse top|Mary had a little lamb}}
Mary had a little lamb,
Little lamb, little lamb,
Mary had a little lamb,
Its fleece was white as snow

And everywhere that Mary went,
Mary went, Mary went,
Everywhere that Mary went
The lamb was sure to go

It followed her to school one day
School one day, school one day
It followed her to school one day
Which was against the rules.

It made the children laugh and play,
Laugh and play, laugh and play,
It made the children laugh and play
To see a lamb at school

And so the teacher turned it out,
Turned it out, turned it out,
And so the teacher turned it out,
But still it lingered near

And waited patiently about,
Patiently about, patiently about,
And waited patiently about
Till Mary did appear

&quot;Why does the lamb love Mary so?&quot;
Love Mary so? Love Mary so?
&quot;Why does the lamb love Mary so?&quot;
The eager children cry

&quot;Why, Mary loves the lamb, you know.&quot;
Loves the lamb, you know, loves the lamb, you know
&quot;Why, Mary loves the lamb, you know.&quot;
The teacher did reply
{{collapse bottom}}

==Colour==
* [[Wikiversity web page colors|Color tables]] | [[Wikiversity:Color names|Color names]]
* e.g., Font: {{font|color=green|Green}}, Background: &lt;span style=&quot;background:hotpink; color:white;&quot;&gt;Pink&lt;/span&gt;

==Columns==
===Column breaks===
{|
|-
| Works on all browsers (col-begin/break/end):
{{col-begin}}
{{col-break}}
* Col1
{{col-break}}
* Col2
{{col-break}}
* Col3
{{col-end}}

Works on all browsers (col/break/colend):
{{col}}
{{break}}
* Col1
{{break}}
* Col2
{{break}}
* Col3
{{col/end}}
|}

===Moz-column===
Easier to use, but doesn't work on all browsers:
&lt;div style=&quot;column-count:3;-moz-column-count:3;-webkit-column-count:3&quot;&gt;
* Ant
* Bee
* Buzzard
* Cat
* Dog
* Egret
* Elephant
* Tiger
* Whale
* Worm
&lt;/div&gt;

==Conversions==
===HTML===
* [[w:Wikipedia:Tools/Editing_tools#From_HTML]]
* [http://www.ebruni.it/en/software/os/i_love_wiki/index.mpl i love wiki]
* {{tick}} [http://diberri.dyndns.org/wikipedia/html2wiki/index.html HTML::WikiConverter]
* {{tick}} [http://openfacts2.berlios.de/html2wiki/index.php HTML::WikiConverter]] Add URL

==CSS==
* [[MediaWiki:Common.css]]

==Custodianship==
* [[Wikiversity:Custodianship]]
** [[Wikiversity:Candidates for Custodianship]]
** [[Wikiversity:Notices for custodians]]
** [[Wikiversity:Request custodian action]]
** [[:Category:Wikiversity custodians]]

==Diffs==
Some ways of showing diffs:
* [[Wikipedia:User:NguoiDungKhongDinhDanh/FormattedEditRequest|Proposed edits side by side]] (using script)
* {{tl|Text diff}}: e.g.,
{{text diff|old text|new text}}
{{Text diff|[[file:Question book-new.svg|50x40px|alt=icon]]|[[file:Question book-new.svg|50x40px|alt=icon|link=]]}}
{| class=&quot;diff&quot;
! Before
! After
|-
| class=&quot;diff-deletedline&quot; |
&lt;div&gt;
The course page should list all enrolled students.&lt;br&gt;
Student names should be updated weekly.
&lt;/div&gt;
| class=&quot;diff-addedline&quot; |
&lt;div&gt;
The course page should identify participating student editors.&lt;br&gt;
Student editor names may be added as appropriate.
&lt;/div&gt;
|}

&lt;syntaxhighlight lang=&quot;diff&quot;&gt;
-The course page should list all enrolled students.
+The course page should identify participating student editors.
&lt;/syntaxhighlight&gt;

==Edit page==
Create an internal link to the edit source page using:
[[Special:EditPage/{{FULLPAGENAME}}|Edit source]]

&lt;nowiki&gt;
{{edit page}}
&lt;/nowiki&gt;

gives:
{{edit page}}

&lt;nowiki&gt;
{{edit page box}}
&lt;/nowiki&gt;

gives:
{{edit page box}}

==Extensions==
* [[Special:Version#Extensions]]
* [[/CategoryTree|CategoryTree]]
* [http://www.sandboxserver.org/wiki/index.php?title=Testing_Mediawiki_extensions Sandbox server - testing extensions]
* [[User:Jtneill/WYSIWIG|WYSIWIG]]

==Font==
&lt;p&gt;{{font|face=&quot;courier&quot;|size=medium|courier size 3}}&lt;/p&gt;
&lt;p&gt;{{font|face=&quot;verdana&quot;|size=large|verdana size 4}}&lt;/p&gt;
&lt;p&gt;{{font|face=&quot;arial&quot;|size=x-large|arial size 5}}&lt;/p&gt;
&lt;p&gt;{{font|face=&quot;times new roman&quot;|size=xx-large|times new roman size 6}}&lt;/p&gt;
&lt;p&gt;&lt;b&gt;{{font|face=&quot;verdana&quot;|size=xx-large|verdana bold size 6}}&lt;/b&gt;&lt;/p&gt;
&lt;p&gt;{{font|face=&quot;lucida calligraphy&quot;|size=xx-large|lucida calligraphy size 7}}&lt;/p&gt;

==Formatting==
===Justification===
&lt;div style=&quot;text-align: justify&quot;&gt; This text is right justified (but it doesn't look like unless the paragraph is long enough to go over one line on the page, so this is intentionally a particularly and unnecessarily long sentence in order to demonstrate right justification using &lt;nowiki&gt;&lt;div style=&quot;text-align: justify&quot;&gt;...&lt;/div&gt;&lt;/nowiki&gt;).&lt;/div&gt;
==Line height==
{{center top}}&lt;p style=&quot;line-height: 36px;&quot;&gt;
&lt;big&gt;&lt;big&gt;&lt;big&gt;&lt;big&gt;This uses a&lt;br&gt;line height of 36px&lt;/big&gt;&lt;/big&gt;&lt;/big&gt;&lt;/big&gt;&lt;/p&gt;
&lt;pre&gt;&lt;p style=&quot;line-height: 36px;&quot;&gt;...&lt;/p&gt;&lt;/pre&gt;
{{center bottom}}
===Mouse-over===
* [[Help:Mouse-over]]
* [[Template:H:title]]

==Getting started==
* [[Wikiversity:Guided tour|Guided tour]]
* [[Wikiversity:Introduction|Introduction]] (Wikiversity)
* [[/Introduction|Introduction]] (Jtneill)
* [[/Welcome|Welcome]] (Jtneill)
* [[Introduction to Wiki]] - [[Wiki 101]]
* [[How to use wiki technology as a free learner]]
* [[:Image:Short.ogg|Wikiversity - short intro]] (10 sec. video)
* [[:Image:Editing_tutorial-large.ogg|Wikiversity editing tutorial]] (2 min video)
* [[Wikiversity:Community Portal]]
* [[Wikiversity:Content development]]
* [[Help:Edit summary]]
* [[Making links]]

==Good design==
* [[User:Jtneill/Good design]]

==Icons==
* [[Help:Icons]]
* [[User:McCormack/icons]]

==Images==
===[[Template:Gallery|Gallery]]===
{{Gallery
|title=Gallery of images
|footer=Uses this [[Template:Gallery|template]]
|width=150
|lines=2
||Comment
|File:Wikiversity-logo-Snorky.svg|[[Help:Contents/Links|Links]] can be put in captions.
|File:Wikiversity-logo-Snorky.svg|Full [[MediaWiki]]&lt;br /&gt;[[syntax]] may be used…
|File:Wikiversity-logo-Snorky.svg|
}}

&lt;!-- Fixed image in bottom right which is linked --&gt;
&lt;div id=&quot;template-navbar&quot; style=&quot;position: fixed; left:1; right:0; bottom:0; padding:0; font-size:122%;&quot;&gt;[[Image:Happy.png|right|50px|link=en:Happiness|Happiness]]&lt;/div&gt;

===ImageMap===
* [[mw:Extension:ImageMap|Extension ImageMap]] e.g.,
{{center top}}
&lt;imagemap&gt;File:Treasurchest.svg|center|80px
default [[Special:Random/|Random Wikiversity mainspace page]]
desc none&lt;/imagemap&gt;Click the treasure box to go to a random [[Wikiversity]] page{{center bottom}}

;Explanation

The ImageMap extension allows, among other things, an image to link directly to a page e.g., as an internal link:
&lt;imagemap&gt;
File:Treasurchest.svg|center|150px|alt=Alt text
default [[Motivation and emotion/Book/2015|Motivation and emotion Book - 2015]]
&lt;/imagemap&gt;

The syntax is:
&lt;pre style=&quot;overflow:auto&quot;&gt;
&lt;imagemap&gt;
File:Treasurchest.svg|center|150px|alt=Alt text
default [[Motivation and emotion/Book/2015|Motivation and emotion Book - 2015]]
&lt;/imagemap&gt;
&lt;/pre&gt;

or as an external link:
&lt;imagemap&gt;
File:Treasurchest.svg|center|150px|alt=Alt text
default [https://www.psychologytoday.com/basics/motivation Motivation (Psychology Today)]
&lt;/imagemap&gt;

The syntax is:
&lt;pre style=&quot;overflow:auto&quot;&gt;
&lt;imagemap&gt;
File:Treasurchest.svg|center|150px|alt=Alt text
default [https://www.psychologytoday.com/basics/motivation Motivation (Psychology Today)]
&lt;/imagemap&gt;
&lt;/pre&gt;

== Creating multilingual svg diagrams ==
* Create diagrams in '''SVG format with translatable labels''' rather than embedding text directly into image pixels
* Develop a single master SVG image that can be reused across multiple languages
* Translate only the labels to create language-specific versions, avoiding the need to redesign the entire image
* Use the [https://svgtranslate.toolforge.org SVG Translate Tool] to create translated versions
* Refer to the documentation at [[commons:Commons:SVG Translate tool]] for guidance on preparing and translating SVG files
* This workflow improves efficiency, consistency, maintainability, and collaboration when producing multilingual educational or informational graphics

==Integrations==
I'm interested to explore possible connections between WV and:
* [http://archive.org Archive.org]
* [[w:Citizendium|Citizendium]]
* [[w:Google Groups]]
* [[Moodle]]
* [[Open University]]
* [http://openlearn.open.ac.uk/course/view.php?name=Cohere Cohere]
* [[WikiMedia Sister Projects]], particularly:
** [[Wikibooks]]
** [[Wikipedia]]
** [[Simple Wikipedia]]

==Licensing==
* My teaching materials are licensed under [[Wikiversity:License tags#Free licenses|creative commons attribution 2.5]] and hosted either on http://wilderdom.com or http://ucspace.canberra.edu.au. I am thinking I should be dual licensing, but am still coming to grips with trying to understand the licensing similarities, differences, and issues. 
* I plan to gradually transfer most of my teaching materials to the various [[w:WikiMedia Foundation|WikiMedia Foundation]] wiki projects, particularly wikiversity. [[m:Polls|Let's just hope Jimbo doesn't put adds on these sites]], otherwise I will be transferring the materials somewhere else (again).
* [http://beta.wikiversity.org/wiki/Wikiversity:IRC_meeting:New_licence_for_Wikiversity_Beta New_licence_for_Wikiversity_Beta]
* {{tl|db-copyvio}}
* {{tl|hangon}}
* [[:Category:Astronomy Images]]

==Links==
* Plain links: e.g., &lt;span class=&quot;plainlinks&quot;&gt;[http://archive.org http://archive.org]&lt;/span&gt;: &lt;br&gt;&lt;nowiki&gt;&lt;span class=&quot;plainlinks&quot;&gt; ... &lt;/span&gt;&lt;/nowiki&gt;
* [[mw:Manual:Opening external links in a new window]]

==Long page warning==
* [[MediaWiki:Longpagewarning]]

==[[Main page]]==
* [[:Category:Main page templates]]
* [[Main Page/Layout 0.5]]
* &lt;span class=&quot;plainlinks&quot;&gt;[http://en.wikiversity.org/w/index.php?title=Wikiversity:Main_Page&amp;oldid=209253 Main page]&lt;/span&gt; (old)

==Map==
&lt;mapframe latitude=&quot;-28.420391&quot; longitude=&quot;136.757813&quot; zoom=&quot;2&quot; width=&quot;200&quot; height=&quot;109&quot; align=&quot;right&quot;&gt;{
  &quot;type&quot;: &quot;FeatureCollection&quot;,
  &quot;features&quot;: [
    {
      &quot;type&quot;: &quot;Feature&quot;,
      &quot;properties&quot;: {},
      &quot;geometry&quot;: {
        &quot;type&quot;: &quot;Point&quot;,
        &quot;coordinates&quot;: [
          149.12419,
          -35.308275
        ]
      }
    }
  ]
}&lt;/mapframe&gt; 
==Namespaces==
* [[Special:NamespaceInfo]]

==Navigation==
{{nav|User:Jtneill}}
* [[Template:nav]]

==Notes==
Small e.g.,
{{attention}} &lt;small&gt;For calendar due dates, see unit outline.&lt;/small&gt;

Notice templates

{{Notice|{{tl|Notice}}}}
{{Note|{{tl|Note}}}}

==Notifications==
* [[Help:Notifications]]

==Pages==
* [[Special:AllPages]]
* Number of pages in category: &lt;nowiki&gt;{{PAGESINCATEGORY:User:Jtneill}}&lt;/nowiki&gt;
* {{hitcounter}} - &lt;nowiki&gt;{{hitcounter}}&lt;/nowiki&gt;

==Page size==
* [[Motivation and emotion/Book/2025/Size]]

==[[Project:Participants|Participants]]==
*[[Wikiversity:Support staff]]

===Users===
*{{Participant|CQ}} - see Person of the Hour script

*{{Participant|Donek}}
*{{Participant|Dan Polansky}}

==Pedagogy==
* [[Learning by doing]]
* [[Wikiversity:Project incubator]]

==Policy==
* [[w:Wikipedia:Contributing_FAQ#Is_there_a_minimum_age_requirement_to_contribute_or_register.3F|Is there a minimum age requirement?]]
{{Official policies}}
{{Proposed policies}}

==Project boxes==
* [[Help:Resource attribution]]

==Purge==
To purge the cache for a given page, append this to the URL:
?action=purge

[[mw:Manual:Purge]]

==Quotes==
* [[Template:Quote]]
*

==[[Quizzes]]==
* [[Help:Quiz-Simple]]
* [http://www.qedoc.org/en/index.php?title=User:Jtneill My Qedoc user page]
** [http://eduforge.org/forum/forum.php?forum_id=1138 Qedoc now exports quizzes to Wikiversity]

==Referencing==
* [[meta:WMDE Technical Wishes/Sub-referencing]]

==Sandbox==
* http://www.sandboxserver.org/
* [[Wikiversity:Sandbox Server]]
* [[Topic:Sandbox Server 0.5]]
* [http://scratchpad.wikia.com/wiki/Scratchpad_Wiki_Labs Scratchpad]
* [[../Sandbox]]

==Searching==
* [[Help:Google]]
* [[Wikiversity:Colloquium/archives/April 2008#Google search|Google search]] - &lt;nowiki&gt;[[google:wikiversity]]&lt;/nowiki&gt; [[google:wikiversity]]
* Use a + instead of a space

==Search multiple categories==
;Dual category search including one category with subcategories
Search for chapters which [[Template:Clarification templates|need clarification]]:
&lt;inputbox&gt;	
type=search
width=33
default=incategory:&quot;Resources needing clarification&quot;
namespaces=Main**
prefix=Motivation and emotion/Book
searchbuttonlabel=Search book chapters
bgcolor=transparent
break=no
&lt;/inputbox&gt;

==Sitenotice==
* [[MediaWiki:Sitenotice]]
* [[MediaWiki:Sitenotice id]]

==Size==

===Big/small===
* Use &lt;code&gt;&lt;nowiki&gt;&lt;big&gt;...&lt;/big&gt; - could be also &lt;big&gt;&lt;big&gt;...&lt;/big&gt;&lt;/big&gt; etc. and also &lt;small&gt;...&lt;/small&gt;&lt;/nowiki&gt;&lt;/code&gt;

===CSS===
&lt;div style=&quot;font-size: 200%&quot;&gt;200% text&lt;/div&gt;&lt;code&gt;&lt;nowiki&gt;&lt;div style=&quot;font-size: 200%&quot;&gt;200% text&lt;/div&gt;&lt;/nowiki&gt;&lt;/code&gt;
&lt;div style=&quot;font-size: 150%&quot;&gt;150% text&lt;/div&gt;&lt;code&gt;&lt;nowiki&gt;&lt;div style=&quot;font-size: 150%&quot;&gt;150% text&lt;/div&gt;&lt;/nowiki&gt;&lt;/code&gt;

==Special==
* [[Special:SpecialPages]]
* Abuse
** [[Special:AbuseFilter]]
** [[Special:AbuseLog]]
* [[Special:AccountSecurity]]
* [[Special:Allpages]]
* [https://auth.wikimedia.org/enwikiversity/wiki/Special:CreateAccount Create account]
* [[meta:Special:GlobalWatchlist]]
* [[Special:ListGroupRights]]
* [[Special:PermanentLink]]
* [[Random]] - [[Special:Random]] - [[Wikiversity:Random]]
* [[Special:ShortPages]]
* [[Special:Version#Installed extensions]]

==Strategy==
* [[Wikiversity:Publicity]]
* [[Wikiversity:Vision]]
* [[Wikiversity:Vision 2009]]

==Statistics==
* [[Wikiversity:Statistics]]
* [[Google Search and Wikiversity]]
* [http://wikistics.falsikon.de/latest/wikiversity/en/ Monthly page hits for wikiversity.en]
* [http://gtools.org/tool/wikipedia-edit-counter/?str=jtneill&amp;project=en.wikiversity Jtneill edit count]
* https://xtools.wmcloud.org/pageinfo/en.wikiversity.org/
* [[Special:Impact]] - [[w:Special:Impact]]

==Sub-pages==
* [[Special:Prefixindex/User:Jtneill]]
* Transclude:
** &lt;code&gt;&lt;nowiki&gt;{{Special:Prefixindex/User:Jtneill}}&lt;/nowiki&gt;&lt;/code&gt;
** &lt;code&gt;&lt;nowiki&gt;{{Special:Prefixindex/{{NAMESPACE}}:{{PAGENAME}}}}&lt;/nowiki&gt;&lt;/code&gt;

==Stubs==
* [[:Category:Stub templates]]

==Structure==
* [[Wikiversity:Browse/Concept]]

==Symbols==
🟨🟡⭐💛🟥⭕️❌🟦🔵🟩🟢✅
* [[User:VeronicaJeanAnderson]]

==System messages==
* [[Special:AllMessages]]
* [[#Sitenotice|Site notice]]

==Style==
* [[MoS]]
* [[MediaWiki:Common.css]]

==Tables==
* [[Help:Table]]
* [[User:Jtneill/Sandbox/Tables and boxes]]

==Tagging/notification==
* &lt;nowiki&gt;@[[User:UserName|UserName]]&lt;/nowiki&gt;
* &lt;nowiki&gt;{{ping|UserName}}&lt;/nowiki&gt;

==Templates==
===Page development===
* {{tl|welcome and expand}} - {{tl|we}}
* {{tl|main welcome}}
* {{tl|search}}
* {{tl|draft}}
* {{tl|underconstruction}}
* {{tl|Learning project boilerplate}}
* {{tl|info}}
* {{tl|note}}
* {{tl|notice}}
* {{tl|Nutshell}}
* &lt;nowiki&gt;{{notice|{{findsources}}}}&lt;/nowiki&gt;

===Page navigation===
* [[Template:EasyNavBar]]
* [[Template:Recovery psychology]] (example)
* [[Workshop for Australian education policy]] (example)

===Sister projects===
* [[Template:Sisterprojectsearch]]
* [[Template:Wikibooks]]
* [[Template:Wikipedia]]
* [[Template:Wikiversity]]

===User talk===
* {{tl|Welcomeip}}
* {{tl|Welcome}}
* {{tl|Talk header}}
* [[:Category:User warning templates]]

===Administrative===
* [[Template:Category redirect]]
* [[Template:Warning]]

==Theory==
* [[Learning by engagement]]
* [[User:JWSchmidt/Wiki Scholar]]

==Thoughts==
* [[Red link]]s are doorways to the infinite library ([[w:The Library of Babel|Library of Babel]])

==Tooltips==
{{Tooltip|Tooltips allow additional text to be displayed when cursor hovers over|Pretty cool, eh?}}


==User==
* [[w:Special:GlobalRenameRequest]]
* [[Special:UserGroupRights]]
* [[Special:UserRights]]
* [[m:Steward requests/Permissions]]
* [[meta:Help:Two-factor authentication]]

==Usability==
* [[Wikiversity:Usability]]
* http://usability.wikimedia.org - [http://usability.wikimedia.org/wiki/User:Jtneill Jtneill]

==Video==
* .ogg files can be uploaded and embedded
* See [[/Video]] for examples

==wikEd==
* [[w:User_talk:Cacycle/wikEd]]

==Wiki2Reveal==
* [[Wiki2Reveal]] (slides on the fly from MediaWiki page)

==x Test anchor==
&lt;!-- Test anchor - don't delete! --&gt;
{{anchor|test}}

==See also==
* [[User:Jade Knight/Tools]]</text>
      <sha1>n5uts1oeehoa8opfarf2pbze1hln73c</sha1>
    </revision>
    <revision>
      <id>2815620</id>
      <parentid>2815619</parentid>
      <timestamp>2026-06-14T03:14:18Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Images */ Explain</comment>
      <origin>2815620</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19042" sha1="tshj1idwwfkwce15t8vommt98iz6k4n" xml:space="preserve">{{TOCright}}
[[:Category:Wikiversitans|Wikiversitan]] since March, 2008

''A loose, personal (i.e., somewhat idiosynchratic) organisation of Wikiversity-related how-tos and links.''

==To sort==
{|style=&quot;background:transparent;&quot;
|valign=top|
* [http://tools.wikimedia.de/~magnus/commonshelper.php commonshelper]
* [[User:Jtneill/Wikification|Wikification]]
* [[w:Help:Interwiki_linking#Project_titles_and_shortcuts|Interwiki linking]]
* [[Wikiversity:Activity bars]]
* [[Wikiversity:Percent complete]]
|valign=top|
* [[Wikiversity:Import|import]]
* [[Wikiversity:Maintenance]]
* [[Wikiversity:Namespaces]]
* [[Wikiversity:Naming conventions]]
|valign=top|
* [[Wikiversity:Participants]]
* [[Wikiversity:Peer review]]
* [[Wikiversity:Review board]]
* [[Wikiversity:Searching]]
* [[How to be a Wikimedia sysop]]
|}

==Anchor==
* [[Template:Anchor]], e.g., [[#test]] will go to &lt;code&gt;&lt;nowiki&gt;{{anchor|test}}&lt;/nowiki&gt;&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{anchor|anchor=test}}&lt;/nowiki&gt;&lt;/code&gt; (should go to end of page)

==Archiving==
* Example of autoarchiving: [[User talk:Terra]]

==Blogging==
* [[Wikiversity Blog howto]]

==Boxes==
[[User:Jtneill/Sandbox/Tables and boxes]]

The simplest of boxes

{| class=&quot;messagebox&quot;
|-
| ABC
XYZ
|}

&lt;blockquote style=&quot;padding-left:1.0em; padding-right:1.0em; background-color:#eaf8f4;&quot;&gt;
Its good that it works in practice, because it certainly doesn’t work in theory[https://blogs.ch.cam.ac.uk/pmr/2007/10/14/the-thing-about-wikipedia-is-that-it-only-works-in-practice-in-theory-it-can-never-work/] 
&lt;/blockquote&gt;

==Categories==
===Listing pages that intersect categories===
&lt;pre&gt;
&lt;dynamicpagelist&gt;
category = Resources needing improved grammar
category = Motivation and emotion/Book/2025
mode     = bullet
&lt;/dynamicpagelist&gt;
&lt;/pre&gt;
{{collapse top|2025 book chapters that need grammar improvements}}
&lt;dynamicpagelist&gt;
category = Resources needing improved grammar
category = Motivation and emotion/Book/2025
mode     = bullet
&lt;/dynamicpagelist&gt;
{{collapse bottom}}

===Order/sort===
wikitext. Wikis with a consensus to do so can [[m:Special:MyLanguage/Requesting wiki configuration changes|request]] a configuration change to display them in alphabetical order. [https://phabricator.wikimedia.org/T373480]

Using titleparts
&lt;nowiki&gt;[[Category:{{#titleparts:{{PAGENAME}}|1}}]]&lt;/nowiki&gt;

==[[/Centering/]]==
{{User:Jtneill/Wikiversity/Centering}}

==Chat==
* [[irc:wikiversity-en|#wikiversity-en]]

==Citations and referencing==
* [[w:Help:Citation tools|Citation tools]]
* [[:Category:Citation templates]]
* [[mw:Help:Cite]]
* [[Template:Citation]]
* [[WV:REF]]
* Example: Outward Bound Process Model&lt;ref&gt;Walsh, V., &amp; Golins, G. L. (1976). ''[http://wilderdom.com/theory/OutwardBoundProcessModel.html The exploration of the Outward Bound process]''.  Denver, CO: Colorado Outward Bound School.&lt;/ref&gt;

;References
{{reflist|1}}

==Collapse boxes==
{{collapse top|Mary had a little lamb}}
Mary had a little lamb,
Little lamb, little lamb,
Mary had a little lamb,
Its fleece was white as snow

And everywhere that Mary went,
Mary went, Mary went,
Everywhere that Mary went
The lamb was sure to go

It followed her to school one day
School one day, school one day
It followed her to school one day
Which was against the rules.

It made the children laugh and play,
Laugh and play, laugh and play,
It made the children laugh and play
To see a lamb at school

And so the teacher turned it out,
Turned it out, turned it out,
And so the teacher turned it out,
But still it lingered near

And waited patiently about,
Patiently about, patiently about,
And waited patiently about
Till Mary did appear

&quot;Why does the lamb love Mary so?&quot;
Love Mary so? Love Mary so?
&quot;Why does the lamb love Mary so?&quot;
The eager children cry

&quot;Why, Mary loves the lamb, you know.&quot;
Loves the lamb, you know, loves the lamb, you know
&quot;Why, Mary loves the lamb, you know.&quot;
The teacher did reply
{{collapse bottom}}

==Colour==
* [[Wikiversity web page colors|Color tables]] | [[Wikiversity:Color names|Color names]]
* e.g., Font: {{font|color=green|Green}}, Background: &lt;span style=&quot;background:hotpink; color:white;&quot;&gt;Pink&lt;/span&gt;

==Columns==
===Column breaks===
{|
|-
| Works on all browsers (col-begin/break/end):
{{col-begin}}
{{col-break}}
* Col1
{{col-break}}
* Col2
{{col-break}}
* Col3
{{col-end}}

Works on all browsers (col/break/colend):
{{col}}
{{break}}
* Col1
{{break}}
* Col2
{{break}}
* Col3
{{col/end}}
|}

===Moz-column===
Easier to use, but doesn't work on all browsers:
&lt;div style=&quot;column-count:3;-moz-column-count:3;-webkit-column-count:3&quot;&gt;
* Ant
* Bee
* Buzzard
* Cat
* Dog
* Egret
* Elephant
* Tiger
* Whale
* Worm
&lt;/div&gt;

==Conversions==
===HTML===
* [[w:Wikipedia:Tools/Editing_tools#From_HTML]]
* [http://www.ebruni.it/en/software/os/i_love_wiki/index.mpl i love wiki]
* {{tick}} [http://diberri.dyndns.org/wikipedia/html2wiki/index.html HTML::WikiConverter]
* {{tick}} [http://openfacts2.berlios.de/html2wiki/index.php HTML::WikiConverter]] Add URL

==CSS==
* [[MediaWiki:Common.css]]

==Custodianship==
* [[Wikiversity:Custodianship]]
** [[Wikiversity:Candidates for Custodianship]]
** [[Wikiversity:Notices for custodians]]
** [[Wikiversity:Request custodian action]]
** [[:Category:Wikiversity custodians]]

==Diffs==
Some ways of showing diffs:
* [[Wikipedia:User:NguoiDungKhongDinhDanh/FormattedEditRequest|Proposed edits side by side]] (using script)
* {{tl|Text diff}}: e.g.,
{{text diff|old text|new text}}
{{Text diff|[[file:Question book-new.svg|50x40px|alt=icon]]|[[file:Question book-new.svg|50x40px|alt=icon|link=]]}}
{| class=&quot;diff&quot;
! Before
! After
|-
| class=&quot;diff-deletedline&quot; |
&lt;div&gt;
The course page should list all enrolled students.&lt;br&gt;
Student names should be updated weekly.
&lt;/div&gt;
| class=&quot;diff-addedline&quot; |
&lt;div&gt;
The course page should identify participating student editors.&lt;br&gt;
Student editor names may be added as appropriate.
&lt;/div&gt;
|}

&lt;syntaxhighlight lang=&quot;diff&quot;&gt;
-The course page should list all enrolled students.
+The course page should identify participating student editors.
&lt;/syntaxhighlight&gt;

==Edit page==
Create an internal link to the edit source page using:
[[Special:EditPage/{{FULLPAGENAME}}|Edit source]]

&lt;nowiki&gt;
{{edit page}}
&lt;/nowiki&gt;

gives:
{{edit page}}

&lt;nowiki&gt;
{{edit page box}}
&lt;/nowiki&gt;

gives:
{{edit page box}}

==Extensions==
* [[Special:Version#Extensions]]
* [[/CategoryTree|CategoryTree]]
* [http://www.sandboxserver.org/wiki/index.php?title=Testing_Mediawiki_extensions Sandbox server - testing extensions]
* [[User:Jtneill/WYSIWIG|WYSIWIG]]

==Font==
&lt;p&gt;{{font|face=&quot;courier&quot;|size=medium|courier size 3}}&lt;/p&gt;
&lt;p&gt;{{font|face=&quot;verdana&quot;|size=large|verdana size 4}}&lt;/p&gt;
&lt;p&gt;{{font|face=&quot;arial&quot;|size=x-large|arial size 5}}&lt;/p&gt;
&lt;p&gt;{{font|face=&quot;times new roman&quot;|size=xx-large|times new roman size 6}}&lt;/p&gt;
&lt;p&gt;&lt;b&gt;{{font|face=&quot;verdana&quot;|size=xx-large|verdana bold size 6}}&lt;/b&gt;&lt;/p&gt;
&lt;p&gt;{{font|face=&quot;lucida calligraphy&quot;|size=xx-large|lucida calligraphy size 7}}&lt;/p&gt;

==Formatting==
===Justification===
&lt;div style=&quot;text-align: justify&quot;&gt; This text is right justified (but it doesn't look like unless the paragraph is long enough to go over one line on the page, so this is intentionally a particularly and unnecessarily long sentence in order to demonstrate right justification using &lt;nowiki&gt;&lt;div style=&quot;text-align: justify&quot;&gt;...&lt;/div&gt;&lt;/nowiki&gt;).&lt;/div&gt;
==Line height==
{{center top}}&lt;p style=&quot;line-height: 36px;&quot;&gt;
&lt;big&gt;&lt;big&gt;&lt;big&gt;&lt;big&gt;This uses a&lt;br&gt;line height of 36px&lt;/big&gt;&lt;/big&gt;&lt;/big&gt;&lt;/big&gt;&lt;/p&gt;
&lt;pre&gt;&lt;p style=&quot;line-height: 36px;&quot;&gt;...&lt;/p&gt;&lt;/pre&gt;
{{center bottom}}
===Mouse-over===
* [[Help:Mouse-over]]
* [[Template:H:title]]

==Getting started==
* [[Wikiversity:Guided tour|Guided tour]]
* [[Wikiversity:Introduction|Introduction]] (Wikiversity)
* [[/Introduction|Introduction]] (Jtneill)
* [[/Welcome|Welcome]] (Jtneill)
* [[Introduction to Wiki]] - [[Wiki 101]]
* [[How to use wiki technology as a free learner]]
* [[:Image:Short.ogg|Wikiversity - short intro]] (10 sec. video)
* [[:Image:Editing_tutorial-large.ogg|Wikiversity editing tutorial]] (2 min video)
* [[Wikiversity:Community Portal]]
* [[Wikiversity:Content development]]
* [[Help:Edit summary]]
* [[Making links]]

==Good design==
* [[User:Jtneill/Good design]]

==Icons==
* [[Help:Icons]]
* [[User:McCormack/icons]]

==Images==
About creating and using images on Wikiversity and sister wiki projects

===[[Template:Gallery|Gallery]]===
Display several related images in a gallery:
{{Gallery
|title=Gallery of images
|footer=Uses this [[Template:Gallery|template]]
|width=150
|lines=2
||Comment
|File:Wikiversity-logo-Snorky.svg|[[Help:Contents/Links|Links]] can be put in captions.
|File:Wikiversity-logo-Snorky.svg|Full [[MediaWiki]]&lt;br /&gt;[[syntax]] may be used…
|File:Wikiversity-logo-Snorky.svg|
}}

&lt;!-- Fixed image in bottom right which is linked --&gt;
&lt;div id=&quot;template-navbar&quot; style=&quot;position: fixed; left:1; right:0; bottom:0; padding:0; font-size:122%;&quot;&gt;[[Image:Happy.png|right|50px|link=en:Happiness|Happiness]]&lt;/div&gt;

===ImageMap===
* [[mw:Extension:ImageMap|Extension ImageMap]] e.g.,
{{center top}}
&lt;imagemap&gt;File:Treasurchest.svg|center|80px
default [[Special:Random/|Random Wikiversity mainspace page]]
desc none&lt;/imagemap&gt;Click the treasure box to go to a random [[Wikiversity]] page{{center bottom}}

;Explanation

The ImageMap extension allows, among other things, an image to link directly to a page e.g., as an internal link:
&lt;imagemap&gt;
File:Treasurchest.svg|center|150px|alt=Alt text
default [[Motivation and emotion/Book/2015|Motivation and emotion Book - 2015]]
&lt;/imagemap&gt;

The syntax is:
&lt;pre style=&quot;overflow:auto&quot;&gt;
&lt;imagemap&gt;
File:Treasurchest.svg|center|150px|alt=Alt text
default [[Motivation and emotion/Book/2015|Motivation and emotion Book - 2015]]
&lt;/imagemap&gt;
&lt;/pre&gt;

or as an external link:
&lt;imagemap&gt;
File:Treasurchest.svg|center|150px|alt=Alt text
default [https://www.psychologytoday.com/basics/motivation Motivation (Psychology Today)]
&lt;/imagemap&gt;

The syntax is:
&lt;pre style=&quot;overflow:auto&quot;&gt;
&lt;imagemap&gt;
File:Treasurchest.svg|center|150px|alt=Alt text
default [https://www.psychologytoday.com/basics/motivation Motivation (Psychology Today)]
&lt;/imagemap&gt;
&lt;/pre&gt;

== Creating multilingual svg diagrams ==
* Create diagrams in '''SVG format with translatable labels''' rather than embedding text directly into image pixels
* Develop a single master SVG image that can be reused across multiple languages
* Translate only the labels to create language-specific versions, avoiding the need to redesign the entire image
* Use the [https://svgtranslate.toolforge.org SVG Translate Tool] to create translated versions
* Refer to the documentation at [[commons:Commons:SVG Translate tool]] for guidance on preparing and translating SVG files
* This workflow improves efficiency, consistency, maintainability, and collaboration when producing multilingual educational or informational graphics

==Integrations==
I'm interested to explore possible connections between WV and:
* [http://archive.org Archive.org]
* [[w:Citizendium|Citizendium]]
* [[w:Google Groups]]
* [[Moodle]]
* [[Open University]]
* [http://openlearn.open.ac.uk/course/view.php?name=Cohere Cohere]
* [[WikiMedia Sister Projects]], particularly:
** [[Wikibooks]]
** [[Wikipedia]]
** [[Simple Wikipedia]]

==Licensing==
* My teaching materials are licensed under [[Wikiversity:License tags#Free licenses|creative commons attribution 2.5]] and hosted either on http://wilderdom.com or http://ucspace.canberra.edu.au. I am thinking I should be dual licensing, but am still coming to grips with trying to understand the licensing similarities, differences, and issues. 
* I plan to gradually transfer most of my teaching materials to the various [[w:WikiMedia Foundation|WikiMedia Foundation]] wiki projects, particularly wikiversity. [[m:Polls|Let's just hope Jimbo doesn't put adds on these sites]], otherwise I will be transferring the materials somewhere else (again).
* [http://beta.wikiversity.org/wiki/Wikiversity:IRC_meeting:New_licence_for_Wikiversity_Beta New_licence_for_Wikiversity_Beta]
* {{tl|db-copyvio}}
* {{tl|hangon}}
* [[:Category:Astronomy Images]]

==Links==
* Plain links: e.g., &lt;span class=&quot;plainlinks&quot;&gt;[http://archive.org http://archive.org]&lt;/span&gt;: &lt;br&gt;&lt;nowiki&gt;&lt;span class=&quot;plainlinks&quot;&gt; ... &lt;/span&gt;&lt;/nowiki&gt;
* [[mw:Manual:Opening external links in a new window]]

==Long page warning==
* [[MediaWiki:Longpagewarning]]

==[[Main page]]==
* [[:Category:Main page templates]]
* [[Main Page/Layout 0.5]]
* &lt;span class=&quot;plainlinks&quot;&gt;[http://en.wikiversity.org/w/index.php?title=Wikiversity:Main_Page&amp;oldid=209253 Main page]&lt;/span&gt; (old)

==Map==
&lt;mapframe latitude=&quot;-28.420391&quot; longitude=&quot;136.757813&quot; zoom=&quot;2&quot; width=&quot;200&quot; height=&quot;109&quot; align=&quot;right&quot;&gt;{
  &quot;type&quot;: &quot;FeatureCollection&quot;,
  &quot;features&quot;: [
    {
      &quot;type&quot;: &quot;Feature&quot;,
      &quot;properties&quot;: {},
      &quot;geometry&quot;: {
        &quot;type&quot;: &quot;Point&quot;,
        &quot;coordinates&quot;: [
          149.12419,
          -35.308275
        ]
      }
    }
  ]
}&lt;/mapframe&gt; 
==Namespaces==
* [[Special:NamespaceInfo]]

==Navigation==
{{nav|User:Jtneill}}
* [[Template:nav]]

==Notes==
Small e.g.,
{{attention}} &lt;small&gt;For calendar due dates, see unit outline.&lt;/small&gt;

Notice templates

{{Notice|{{tl|Notice}}}}
{{Note|{{tl|Note}}}}

==Notifications==
* [[Help:Notifications]]

==Pages==
* [[Special:AllPages]]
* Number of pages in category: &lt;nowiki&gt;{{PAGESINCATEGORY:User:Jtneill}}&lt;/nowiki&gt;
* {{hitcounter}} - &lt;nowiki&gt;{{hitcounter}}&lt;/nowiki&gt;

==Page size==
* [[Motivation and emotion/Book/2025/Size]]

==[[Project:Participants|Participants]]==
*[[Wikiversity:Support staff]]

===Users===
*{{Participant|CQ}} - see Person of the Hour script

*{{Participant|Donek}}
*{{Participant|Dan Polansky}}

==Pedagogy==
* [[Learning by doing]]
* [[Wikiversity:Project incubator]]

==Policy==
* [[w:Wikipedia:Contributing_FAQ#Is_there_a_minimum_age_requirement_to_contribute_or_register.3F|Is there a minimum age requirement?]]
{{Official policies}}
{{Proposed policies}}

==Project boxes==
* [[Help:Resource attribution]]

==Purge==
To purge the cache for a given page, append this to the URL:
?action=purge

[[mw:Manual:Purge]]

==Quotes==
* [[Template:Quote]]
*

==[[Quizzes]]==
* [[Help:Quiz-Simple]]
* [http://www.qedoc.org/en/index.php?title=User:Jtneill My Qedoc user page]
** [http://eduforge.org/forum/forum.php?forum_id=1138 Qedoc now exports quizzes to Wikiversity]

==Referencing==
* [[meta:WMDE Technical Wishes/Sub-referencing]]

==Sandbox==
* http://www.sandboxserver.org/
* [[Wikiversity:Sandbox Server]]
* [[Topic:Sandbox Server 0.5]]
* [http://scratchpad.wikia.com/wiki/Scratchpad_Wiki_Labs Scratchpad]
* [[../Sandbox]]

==Searching==
* [[Help:Google]]
* [[Wikiversity:Colloquium/archives/April 2008#Google search|Google search]] - &lt;nowiki&gt;[[google:wikiversity]]&lt;/nowiki&gt; [[google:wikiversity]]
* Use a + instead of a space

==Search multiple categories==
;Dual category search including one category with subcategories
Search for chapters which [[Template:Clarification templates|need clarification]]:
&lt;inputbox&gt;	
type=search
width=33
default=incategory:&quot;Resources needing clarification&quot;
namespaces=Main**
prefix=Motivation and emotion/Book
searchbuttonlabel=Search book chapters
bgcolor=transparent
break=no
&lt;/inputbox&gt;

==Sitenotice==
* [[MediaWiki:Sitenotice]]
* [[MediaWiki:Sitenotice id]]

==Size==

===Big/small===
* Use &lt;code&gt;&lt;nowiki&gt;&lt;big&gt;...&lt;/big&gt; - could be also &lt;big&gt;&lt;big&gt;...&lt;/big&gt;&lt;/big&gt; etc. and also &lt;small&gt;...&lt;/small&gt;&lt;/nowiki&gt;&lt;/code&gt;

===CSS===
&lt;div style=&quot;font-size: 200%&quot;&gt;200% text&lt;/div&gt;&lt;code&gt;&lt;nowiki&gt;&lt;div style=&quot;font-size: 200%&quot;&gt;200% text&lt;/div&gt;&lt;/nowiki&gt;&lt;/code&gt;
&lt;div style=&quot;font-size: 150%&quot;&gt;150% text&lt;/div&gt;&lt;code&gt;&lt;nowiki&gt;&lt;div style=&quot;font-size: 150%&quot;&gt;150% text&lt;/div&gt;&lt;/nowiki&gt;&lt;/code&gt;

==Special==
* [[Special:SpecialPages]]
* Abuse
** [[Special:AbuseFilter]]
** [[Special:AbuseLog]]
* [[Special:AccountSecurity]]
* [[Special:Allpages]]
* [https://auth.wikimedia.org/enwikiversity/wiki/Special:CreateAccount Create account]
* [[meta:Special:GlobalWatchlist]]
* [[Special:ListGroupRights]]
* [[Special:PermanentLink]]
* [[Random]] - [[Special:Random]] - [[Wikiversity:Random]]
* [[Special:ShortPages]]
* [[Special:Version#Installed extensions]]

==Strategy==
* [[Wikiversity:Publicity]]
* [[Wikiversity:Vision]]
* [[Wikiversity:Vision 2009]]

==Statistics==
* [[Wikiversity:Statistics]]
* [[Google Search and Wikiversity]]
* [http://wikistics.falsikon.de/latest/wikiversity/en/ Monthly page hits for wikiversity.en]
* [http://gtools.org/tool/wikipedia-edit-counter/?str=jtneill&amp;project=en.wikiversity Jtneill edit count]
* https://xtools.wmcloud.org/pageinfo/en.wikiversity.org/
* [[Special:Impact]] - [[w:Special:Impact]]

==Sub-pages==
* [[Special:Prefixindex/User:Jtneill]]
* Transclude:
** &lt;code&gt;&lt;nowiki&gt;{{Special:Prefixindex/User:Jtneill}}&lt;/nowiki&gt;&lt;/code&gt;
** &lt;code&gt;&lt;nowiki&gt;{{Special:Prefixindex/{{NAMESPACE}}:{{PAGENAME}}}}&lt;/nowiki&gt;&lt;/code&gt;

==Stubs==
* [[:Category:Stub templates]]

==Structure==
* [[Wikiversity:Browse/Concept]]

==Symbols==
🟨🟡⭐💛🟥⭕️❌🟦🔵🟩🟢✅
* [[User:VeronicaJeanAnderson]]

==System messages==
* [[Special:AllMessages]]
* [[#Sitenotice|Site notice]]

==Style==
* [[MoS]]
* [[MediaWiki:Common.css]]

==Tables==
* [[Help:Table]]
* [[User:Jtneill/Sandbox/Tables and boxes]]

==Tagging/notification==
* &lt;nowiki&gt;@[[User:UserName|UserName]]&lt;/nowiki&gt;
* &lt;nowiki&gt;{{ping|UserName}}&lt;/nowiki&gt;

==Templates==
===Page development===
* {{tl|welcome and expand}} - {{tl|we}}
* {{tl|main welcome}}
* {{tl|search}}
* {{tl|draft}}
* {{tl|underconstruction}}
* {{tl|Learning project boilerplate}}
* {{tl|info}}
* {{tl|note}}
* {{tl|notice}}
* {{tl|Nutshell}}
* &lt;nowiki&gt;{{notice|{{findsources}}}}&lt;/nowiki&gt;

===Page navigation===
* [[Template:EasyNavBar]]
* [[Template:Recovery psychology]] (example)
* [[Workshop for Australian education policy]] (example)

===Sister projects===
* [[Template:Sisterprojectsearch]]
* [[Template:Wikibooks]]
* [[Template:Wikipedia]]
* [[Template:Wikiversity]]

===User talk===
* {{tl|Welcomeip}}
* {{tl|Welcome}}
* {{tl|Talk header}}
* [[:Category:User warning templates]]

===Administrative===
* [[Template:Category redirect]]
* [[Template:Warning]]

==Theory==
* [[Learning by engagement]]
* [[User:JWSchmidt/Wiki Scholar]]

==Thoughts==
* [[Red link]]s are doorways to the infinite library ([[w:The Library of Babel|Library of Babel]])

==Tooltips==
{{Tooltip|Tooltips allow additional text to be displayed when cursor hovers over|Pretty cool, eh?}}


==User==
* [[w:Special:GlobalRenameRequest]]
* [[Special:UserGroupRights]]
* [[Special:UserRights]]
* [[m:Steward requests/Permissions]]
* [[meta:Help:Two-factor authentication]]

==Usability==
* [[Wikiversity:Usability]]
* http://usability.wikimedia.org - [http://usability.wikimedia.org/wiki/User:Jtneill Jtneill]

==Video==
* .ogg files can be uploaded and embedded
* See [[/Video]] for examples

==wikEd==
* [[w:User_talk:Cacycle/wikEd]]

==Wiki2Reveal==
* [[Wiki2Reveal]] (slides on the fly from MediaWiki page)

==x Test anchor==
&lt;!-- Test anchor - don't delete! --&gt;
{{anchor|test}}

==See also==
* [[User:Jade Knight/Tools]]</text>
      <sha1>tshj1idwwfkwce15t8vommt98iz6k4n</sha1>
    </revision>
    <revision>
      <id>2815621</id>
      <parentid>2815620</parentid>
      <timestamp>2026-06-14T03:18:39Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>+ Image creation and support Wikimedia communities</comment>
      <origin>2815621</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19915" sha1="nqkfhu6p1wff3djgmjhxo83zj8cxevn" xml:space="preserve">{{TOCright}}
[[:Category:Wikiversitans|Wikiversitan]] since March, 2008

''A loose, personal (i.e., somewhat idiosynchratic) organisation of Wikiversity-related how-tos and links.''

==To sort==
{|style=&quot;background:transparent;&quot;
|valign=top|
* [http://tools.wikimedia.de/~magnus/commonshelper.php commonshelper]
* [[User:Jtneill/Wikification|Wikification]]
* [[w:Help:Interwiki_linking#Project_titles_and_shortcuts|Interwiki linking]]
* [[Wikiversity:Activity bars]]
* [[Wikiversity:Percent complete]]
|valign=top|
* [[Wikiversity:Import|import]]
* [[Wikiversity:Maintenance]]
* [[Wikiversity:Namespaces]]
* [[Wikiversity:Naming conventions]]
|valign=top|
* [[Wikiversity:Participants]]
* [[Wikiversity:Peer review]]
* [[Wikiversity:Review board]]
* [[Wikiversity:Searching]]
* [[How to be a Wikimedia sysop]]
|}

==Anchor==
* [[Template:Anchor]], e.g., [[#test]] will go to &lt;code&gt;&lt;nowiki&gt;{{anchor|test}}&lt;/nowiki&gt;&lt;/code&gt; or &lt;code&gt;&lt;nowiki&gt;{{anchor|anchor=test}}&lt;/nowiki&gt;&lt;/code&gt; (should go to end of page)

==Archiving==
* Example of autoarchiving: [[User talk:Terra]]

==Blogging==
* [[Wikiversity Blog howto]]

==Boxes==
[[User:Jtneill/Sandbox/Tables and boxes]]

The simplest of boxes

{| class=&quot;messagebox&quot;
|-
| ABC
XYZ
|}

&lt;blockquote style=&quot;padding-left:1.0em; padding-right:1.0em; background-color:#eaf8f4;&quot;&gt;
Its good that it works in practice, because it certainly doesn’t work in theory[https://blogs.ch.cam.ac.uk/pmr/2007/10/14/the-thing-about-wikipedia-is-that-it-only-works-in-practice-in-theory-it-can-never-work/] 
&lt;/blockquote&gt;

==Categories==
===Listing pages that intersect categories===
&lt;pre&gt;
&lt;dynamicpagelist&gt;
category = Resources needing improved grammar
category = Motivation and emotion/Book/2025
mode     = bullet
&lt;/dynamicpagelist&gt;
&lt;/pre&gt;
{{collapse top|2025 book chapters that need grammar improvements}}
&lt;dynamicpagelist&gt;
category = Resources needing improved grammar
category = Motivation and emotion/Book/2025
mode     = bullet
&lt;/dynamicpagelist&gt;
{{collapse bottom}}

===Order/sort===
wikitext. Wikis with a consensus to do so can [[m:Special:MyLanguage/Requesting wiki configuration changes|request]] a configuration change to display them in alphabetical order. [https://phabricator.wikimedia.org/T373480]

Using titleparts
&lt;nowiki&gt;[[Category:{{#titleparts:{{PAGENAME}}|1}}]]&lt;/nowiki&gt;

==[[/Centering/]]==
{{User:Jtneill/Wikiversity/Centering}}

==Chat==
* [[irc:wikiversity-en|#wikiversity-en]]

==Citations and referencing==
* [[w:Help:Citation tools|Citation tools]]
* [[:Category:Citation templates]]
* [[mw:Help:Cite]]
* [[Template:Citation]]
* [[WV:REF]]
* Example: Outward Bound Process Model&lt;ref&gt;Walsh, V., &amp; Golins, G. L. (1976). ''[http://wilderdom.com/theory/OutwardBoundProcessModel.html The exploration of the Outward Bound process]''.  Denver, CO: Colorado Outward Bound School.&lt;/ref&gt;

;References
{{reflist|1}}

==Collapse boxes==
{{collapse top|Mary had a little lamb}}
Mary had a little lamb,
Little lamb, little lamb,
Mary had a little lamb,
Its fleece was white as snow

And everywhere that Mary went,
Mary went, Mary went,
Everywhere that Mary went
The lamb was sure to go

It followed her to school one day
School one day, school one day
It followed her to school one day
Which was against the rules.

It made the children laugh and play,
Laugh and play, laugh and play,
It made the children laugh and play
To see a lamb at school

And so the teacher turned it out,
Turned it out, turned it out,
And so the teacher turned it out,
But still it lingered near

And waited patiently about,
Patiently about, patiently about,
And waited patiently about
Till Mary did appear

&quot;Why does the lamb love Mary so?&quot;
Love Mary so? Love Mary so?
&quot;Why does the lamb love Mary so?&quot;
The eager children cry

&quot;Why, Mary loves the lamb, you know.&quot;
Loves the lamb, you know, loves the lamb, you know
&quot;Why, Mary loves the lamb, you know.&quot;
The teacher did reply
{{collapse bottom}}

==Colour==
* [[Wikiversity web page colors|Color tables]] | [[Wikiversity:Color names|Color names]]
* e.g., Font: {{font|color=green|Green}}, Background: &lt;span style=&quot;background:hotpink; color:white;&quot;&gt;Pink&lt;/span&gt;

==Columns==
===Column breaks===
{|
|-
| Works on all browsers (col-begin/break/end):
{{col-begin}}
{{col-break}}
* Col1
{{col-break}}
* Col2
{{col-break}}
* Col3
{{col-end}}

Works on all browsers (col/break/colend):
{{col}}
{{break}}
* Col1
{{break}}
* Col2
{{break}}
* Col3
{{col/end}}
|}

===Moz-column===
Easier to use, but doesn't work on all browsers:
&lt;div style=&quot;column-count:3;-moz-column-count:3;-webkit-column-count:3&quot;&gt;
* Ant
* Bee
* Buzzard
* Cat
* Dog
* Egret
* Elephant
* Tiger
* Whale
* Worm
&lt;/div&gt;

==Conversions==
===HTML===
* [[w:Wikipedia:Tools/Editing_tools#From_HTML]]
* [http://www.ebruni.it/en/software/os/i_love_wiki/index.mpl i love wiki]
* {{tick}} [http://diberri.dyndns.org/wikipedia/html2wiki/index.html HTML::WikiConverter]
* {{tick}} [http://openfacts2.berlios.de/html2wiki/index.php HTML::WikiConverter]] Add URL

==CSS==
* [[MediaWiki:Common.css]]

==Custodianship==
* [[Wikiversity:Custodianship]]
** [[Wikiversity:Candidates for Custodianship]]
** [[Wikiversity:Notices for custodians]]
** [[Wikiversity:Request custodian action]]
** [[:Category:Wikiversity custodians]]

==Diffs==
Some ways of showing diffs:
* [[Wikipedia:User:NguoiDungKhongDinhDanh/FormattedEditRequest|Proposed edits side by side]] (using script)
* {{tl|Text diff}}: e.g.,
{{text diff|old text|new text}}
{{Text diff|[[file:Question book-new.svg|50x40px|alt=icon]]|[[file:Question book-new.svg|50x40px|alt=icon|link=]]}}
{| class=&quot;diff&quot;
! Before
! After
|-
| class=&quot;diff-deletedline&quot; |
&lt;div&gt;
The course page should list all enrolled students.&lt;br&gt;
Student names should be updated weekly.
&lt;/div&gt;
| class=&quot;diff-addedline&quot; |
&lt;div&gt;
The course page should identify participating student editors.&lt;br&gt;
Student editor names may be added as appropriate.
&lt;/div&gt;
|}

&lt;syntaxhighlight lang=&quot;diff&quot;&gt;
-The course page should list all enrolled students.
+The course page should identify participating student editors.
&lt;/syntaxhighlight&gt;

==Edit page==
Create an internal link to the edit source page using:
[[Special:EditPage/{{FULLPAGENAME}}|Edit source]]

&lt;nowiki&gt;
{{edit page}}
&lt;/nowiki&gt;

gives:
{{edit page}}

&lt;nowiki&gt;
{{edit page box}}
&lt;/nowiki&gt;

gives:
{{edit page box}}

==Extensions==
* [[Special:Version#Extensions]]
* [[/CategoryTree|CategoryTree]]
* [http://www.sandboxserver.org/wiki/index.php?title=Testing_Mediawiki_extensions Sandbox server - testing extensions]
* [[User:Jtneill/WYSIWIG|WYSIWIG]]

==Font==
&lt;p&gt;{{font|face=&quot;courier&quot;|size=medium|courier size 3}}&lt;/p&gt;
&lt;p&gt;{{font|face=&quot;verdana&quot;|size=large|verdana size 4}}&lt;/p&gt;
&lt;p&gt;{{font|face=&quot;arial&quot;|size=x-large|arial size 5}}&lt;/p&gt;
&lt;p&gt;{{font|face=&quot;times new roman&quot;|size=xx-large|times new roman size 6}}&lt;/p&gt;
&lt;p&gt;&lt;b&gt;{{font|face=&quot;verdana&quot;|size=xx-large|verdana bold size 6}}&lt;/b&gt;&lt;/p&gt;
&lt;p&gt;{{font|face=&quot;lucida calligraphy&quot;|size=xx-large|lucida calligraphy size 7}}&lt;/p&gt;

==Formatting==
===Justification===
&lt;div style=&quot;text-align: justify&quot;&gt; This text is right justified (but it doesn't look like unless the paragraph is long enough to go over one line on the page, so this is intentionally a particularly and unnecessarily long sentence in order to demonstrate right justification using &lt;nowiki&gt;&lt;div style=&quot;text-align: justify&quot;&gt;...&lt;/div&gt;&lt;/nowiki&gt;).&lt;/div&gt;
==Line height==
{{center top}}&lt;p style=&quot;line-height: 36px;&quot;&gt;
&lt;big&gt;&lt;big&gt;&lt;big&gt;&lt;big&gt;This uses a&lt;br&gt;line height of 36px&lt;/big&gt;&lt;/big&gt;&lt;/big&gt;&lt;/big&gt;&lt;/p&gt;
&lt;pre&gt;&lt;p style=&quot;line-height: 36px;&quot;&gt;...&lt;/p&gt;&lt;/pre&gt;
{{center bottom}}
===Mouse-over===
* [[Help:Mouse-over]]
* [[Template:H:title]]

==Getting started==
* [[Wikiversity:Guided tour|Guided tour]]
* [[Wikiversity:Introduction|Introduction]] (Wikiversity)
* [[/Introduction|Introduction]] (Jtneill)
* [[/Welcome|Welcome]] (Jtneill)
* [[Introduction to Wiki]] - [[Wiki 101]]
* [[How to use wiki technology as a free learner]]
* [[:Image:Short.ogg|Wikiversity - short intro]] (10 sec. video)
* [[:Image:Editing_tutorial-large.ogg|Wikiversity editing tutorial]] (2 min video)
* [[Wikiversity:Community Portal]]
* [[Wikiversity:Content development]]
* [[Help:Edit summary]]
* [[Making links]]

==Good design==
* [[User:Jtneill/Good design]]

==Icons==
* [[Help:Icons]]
* [[User:McCormack/icons]]

==Images==
About creating and using images on Wikiversity and sister wiki projects

===[[Template:Gallery|Gallery]]===
Display several related images in a gallery:
{{Gallery
|title=Gallery of images
|footer=Uses this [[Template:Gallery|template]]
|width=150
|lines=2
||Comment
|File:Wikiversity-logo-Snorky.svg|[[Help:Contents/Links|Links]] can be put in captions.
|File:Wikiversity-logo-Snorky.svg|Full [[MediaWiki]]&lt;br /&gt;[[syntax]] may be used…
|File:Wikiversity-logo-Snorky.svg|
}}

&lt;!-- Fixed image in bottom right which is linked --&gt;
&lt;div id=&quot;template-navbar&quot; style=&quot;position: fixed; left:1; right:0; bottom:0; padding:0; font-size:122%;&quot;&gt;[[Image:Happy.png|right|50px|link=en:Happiness|Happiness]]&lt;/div&gt;

===ImageMap===
* [[mw:Extension:ImageMap|Extension ImageMap]] e.g.,
{{center top}}
&lt;imagemap&gt;File:Treasurchest.svg|center|80px
default [[Special:Random/|Random Wikiversity mainspace page]]
desc none&lt;/imagemap&gt;Click the treasure box to go to a random [[Wikiversity]] page{{center bottom}}

;Explanation

The ImageMap extension allows, among other things, an image to link directly to a page e.g., as an internal link:
&lt;imagemap&gt;
File:Treasurchest.svg|center|150px|alt=Alt text
default [[Motivation and emotion/Book/2015|Motivation and emotion Book - 2015]]
&lt;/imagemap&gt;

The syntax is:
&lt;pre style=&quot;overflow:auto&quot;&gt;
&lt;imagemap&gt;
File:Treasurchest.svg|center|150px|alt=Alt text
default [[Motivation and emotion/Book/2015|Motivation and emotion Book - 2015]]
&lt;/imagemap&gt;
&lt;/pre&gt;

or as an external link:
&lt;imagemap&gt;
File:Treasurchest.svg|center|150px|alt=Alt text
default [https://www.psychologytoday.com/basics/motivation Motivation (Psychology Today)]
&lt;/imagemap&gt;

The syntax is:
&lt;pre style=&quot;overflow:auto&quot;&gt;
&lt;imagemap&gt;
File:Treasurchest.svg|center|150px|alt=Alt text
default [https://www.psychologytoday.com/basics/motivation Motivation (Psychology Today)]
&lt;/imagemap&gt;
&lt;/pre&gt;

=== Creating multilingual svg diagrams ===
* Create diagrams in '''SVG format with translatable labels''' rather than embedding text directly into image pixels
* Develop a single master SVG image that can be reused across multiple languages
* Translate only the labels to create language-specific versions, avoiding the need to redesign the entire image
* Use the [https://svgtranslate.toolforge.org SVG Translate Tool] to create translated versions
* Refer to the documentation at [[commons:Commons:SVG Translate tool]] for guidance on preparing and translating SVG files
* This workflow improves efficiency, consistency, maintainability, and collaboration when producing multilingual educational or informational graphics

=== Image creation and support Wikimedia communities ===
* Engaging with established graphics communities can improve image quality, ensure adherence to Wikimedia standards, and provide access to specialised knowledge and skills
* Wikipedia [[w:WP:GLI|Graphics Lab for Images]] to request help with creating, modifying, or improving illustrations
** Consult experienced volunteer graphic contributors for technical guidance on image creation and editing
** Seek advice about Wikimedia image styles, conventions, accessibility, and best practices
** Request assistance with image design, formatting, vectorisation, restoration, diagram creation, and other graphical tasks
** Consider using graphics community feedback to improve the quality and consistency of visual materials
* Commons [[Commons:Graphic Lab/Illustration workshop]] for additional expertise and support.

==Integrations==
I'm interested to explore possible connections between WV and:
* [http://archive.org Archive.org]
* [[w:Citizendium|Citizendium]]
* [[w:Google Groups]]
* [[Moodle]]
* [[Open University]]
* [http://openlearn.open.ac.uk/course/view.php?name=Cohere Cohere]
* [[WikiMedia Sister Projects]], particularly:
** [[Wikibooks]]
** [[Wikipedia]]
** [[Simple Wikipedia]]

==Licensing==
* My teaching materials are licensed under [[Wikiversity:License tags#Free licenses|creative commons attribution 2.5]] and hosted either on http://wilderdom.com or http://ucspace.canberra.edu.au. I am thinking I should be dual licensing, but am still coming to grips with trying to understand the licensing similarities, differences, and issues. 
* I plan to gradually transfer most of my teaching materials to the various [[w:WikiMedia Foundation|WikiMedia Foundation]] wiki projects, particularly wikiversity. [[m:Polls|Let's just hope Jimbo doesn't put adds on these sites]], otherwise I will be transferring the materials somewhere else (again).
* [http://beta.wikiversity.org/wiki/Wikiversity:IRC_meeting:New_licence_for_Wikiversity_Beta New_licence_for_Wikiversity_Beta]
* {{tl|db-copyvio}}
* {{tl|hangon}}
* [[:Category:Astronomy Images]]

==Links==
* Plain links: e.g., &lt;span class=&quot;plainlinks&quot;&gt;[http://archive.org http://archive.org]&lt;/span&gt;: &lt;br&gt;&lt;nowiki&gt;&lt;span class=&quot;plainlinks&quot;&gt; ... &lt;/span&gt;&lt;/nowiki&gt;
* [[mw:Manual:Opening external links in a new window]]

==Long page warning==
* [[MediaWiki:Longpagewarning]]

==[[Main page]]==
* [[:Category:Main page templates]]
* [[Main Page/Layout 0.5]]
* &lt;span class=&quot;plainlinks&quot;&gt;[http://en.wikiversity.org/w/index.php?title=Wikiversity:Main_Page&amp;oldid=209253 Main page]&lt;/span&gt; (old)

==Map==
&lt;mapframe latitude=&quot;-28.420391&quot; longitude=&quot;136.757813&quot; zoom=&quot;2&quot; width=&quot;200&quot; height=&quot;109&quot; align=&quot;right&quot;&gt;{
  &quot;type&quot;: &quot;FeatureCollection&quot;,
  &quot;features&quot;: [
    {
      &quot;type&quot;: &quot;Feature&quot;,
      &quot;properties&quot;: {},
      &quot;geometry&quot;: {
        &quot;type&quot;: &quot;Point&quot;,
        &quot;coordinates&quot;: [
          149.12419,
          -35.308275
        ]
      }
    }
  ]
}&lt;/mapframe&gt; 
==Namespaces==
* [[Special:NamespaceInfo]]

==Navigation==
{{nav|User:Jtneill}}
* [[Template:nav]]

==Notes==
Small e.g.,
{{attention}} &lt;small&gt;For calendar due dates, see unit outline.&lt;/small&gt;

Notice templates

{{Notice|{{tl|Notice}}}}
{{Note|{{tl|Note}}}}

==Notifications==
* [[Help:Notifications]]

==Pages==
* [[Special:AllPages]]
* Number of pages in category: &lt;nowiki&gt;{{PAGESINCATEGORY:User:Jtneill}}&lt;/nowiki&gt;
* {{hitcounter}} - &lt;nowiki&gt;{{hitcounter}}&lt;/nowiki&gt;

==Page size==
* [[Motivation and emotion/Book/2025/Size]]

==[[Project:Participants|Participants]]==
*[[Wikiversity:Support staff]]

===Users===
*{{Participant|CQ}} - see Person of the Hour script

*{{Participant|Donek}}
*{{Participant|Dan Polansky}}

==Pedagogy==
* [[Learning by doing]]
* [[Wikiversity:Project incubator]]

==Policy==
* [[w:Wikipedia:Contributing_FAQ#Is_there_a_minimum_age_requirement_to_contribute_or_register.3F|Is there a minimum age requirement?]]
{{Official policies}}
{{Proposed policies}}

==Project boxes==
* [[Help:Resource attribution]]

==Purge==
To purge the cache for a given page, append this to the URL:
?action=purge

[[mw:Manual:Purge]]

==Quotes==
* [[Template:Quote]]
*

==[[Quizzes]]==
* [[Help:Quiz-Simple]]
* [http://www.qedoc.org/en/index.php?title=User:Jtneill My Qedoc user page]
** [http://eduforge.org/forum/forum.php?forum_id=1138 Qedoc now exports quizzes to Wikiversity]

==Referencing==
* [[meta:WMDE Technical Wishes/Sub-referencing]]

==Sandbox==
* http://www.sandboxserver.org/
* [[Wikiversity:Sandbox Server]]
* [[Topic:Sandbox Server 0.5]]
* [http://scratchpad.wikia.com/wiki/Scratchpad_Wiki_Labs Scratchpad]
* [[../Sandbox]]

==Searching==
* [[Help:Google]]
* [[Wikiversity:Colloquium/archives/April 2008#Google search|Google search]] - &lt;nowiki&gt;[[google:wikiversity]]&lt;/nowiki&gt; [[google:wikiversity]]
* Use a + instead of a space

==Search multiple categories==
;Dual category search including one category with subcategories
Search for chapters which [[Template:Clarification templates|need clarification]]:
&lt;inputbox&gt;	
type=search
width=33
default=incategory:&quot;Resources needing clarification&quot;
namespaces=Main**
prefix=Motivation and emotion/Book
searchbuttonlabel=Search book chapters
bgcolor=transparent
break=no
&lt;/inputbox&gt;

==Sitenotice==
* [[MediaWiki:Sitenotice]]
* [[MediaWiki:Sitenotice id]]

==Size==

===Big/small===
* Use &lt;code&gt;&lt;nowiki&gt;&lt;big&gt;...&lt;/big&gt; - could be also &lt;big&gt;&lt;big&gt;...&lt;/big&gt;&lt;/big&gt; etc. and also &lt;small&gt;...&lt;/small&gt;&lt;/nowiki&gt;&lt;/code&gt;

===CSS===
&lt;div style=&quot;font-size: 200%&quot;&gt;200% text&lt;/div&gt;&lt;code&gt;&lt;nowiki&gt;&lt;div style=&quot;font-size: 200%&quot;&gt;200% text&lt;/div&gt;&lt;/nowiki&gt;&lt;/code&gt;
&lt;div style=&quot;font-size: 150%&quot;&gt;150% text&lt;/div&gt;&lt;code&gt;&lt;nowiki&gt;&lt;div style=&quot;font-size: 150%&quot;&gt;150% text&lt;/div&gt;&lt;/nowiki&gt;&lt;/code&gt;

==Special==
* [[Special:SpecialPages]]
* Abuse
** [[Special:AbuseFilter]]
** [[Special:AbuseLog]]
* [[Special:AccountSecurity]]
* [[Special:Allpages]]
* [https://auth.wikimedia.org/enwikiversity/wiki/Special:CreateAccount Create account]
* [[meta:Special:GlobalWatchlist]]
* [[Special:ListGroupRights]]
* [[Special:PermanentLink]]
* [[Random]] - [[Special:Random]] - [[Wikiversity:Random]]
* [[Special:ShortPages]]
* [[Special:Version#Installed extensions]]

==Strategy==
* [[Wikiversity:Publicity]]
* [[Wikiversity:Vision]]
* [[Wikiversity:Vision 2009]]

==Statistics==
* [[Wikiversity:Statistics]]
* [[Google Search and Wikiversity]]
* [http://wikistics.falsikon.de/latest/wikiversity/en/ Monthly page hits for wikiversity.en]
* [http://gtools.org/tool/wikipedia-edit-counter/?str=jtneill&amp;project=en.wikiversity Jtneill edit count]
* https://xtools.wmcloud.org/pageinfo/en.wikiversity.org/
* [[Special:Impact]] - [[w:Special:Impact]]

==Sub-pages==
* [[Special:Prefixindex/User:Jtneill]]
* Transclude:
** &lt;code&gt;&lt;nowiki&gt;{{Special:Prefixindex/User:Jtneill}}&lt;/nowiki&gt;&lt;/code&gt;
** &lt;code&gt;&lt;nowiki&gt;{{Special:Prefixindex/{{NAMESPACE}}:{{PAGENAME}}}}&lt;/nowiki&gt;&lt;/code&gt;

==Stubs==
* [[:Category:Stub templates]]

==Structure==
* [[Wikiversity:Browse/Concept]]

==Symbols==
🟨🟡⭐💛🟥⭕️❌🟦🔵🟩🟢✅
* [[User:VeronicaJeanAnderson]]

==System messages==
* [[Special:AllMessages]]
* [[#Sitenotice|Site notice]]

==Style==
* [[MoS]]
* [[MediaWiki:Common.css]]

==Tables==
* [[Help:Table]]
* [[User:Jtneill/Sandbox/Tables and boxes]]

==Tagging/notification==
* &lt;nowiki&gt;@[[User:UserName|UserName]]&lt;/nowiki&gt;
* &lt;nowiki&gt;{{ping|UserName}}&lt;/nowiki&gt;

==Templates==
===Page development===
* {{tl|welcome and expand}} - {{tl|we}}
* {{tl|main welcome}}
* {{tl|search}}
* {{tl|draft}}
* {{tl|underconstruction}}
* {{tl|Learning project boilerplate}}
* {{tl|info}}
* {{tl|note}}
* {{tl|notice}}
* {{tl|Nutshell}}
* &lt;nowiki&gt;{{notice|{{findsources}}}}&lt;/nowiki&gt;

===Page navigation===
* [[Template:EasyNavBar]]
* [[Template:Recovery psychology]] (example)
* [[Workshop for Australian education policy]] (example)

===Sister projects===
* [[Template:Sisterprojectsearch]]
* [[Template:Wikibooks]]
* [[Template:Wikipedia]]
* [[Template:Wikiversity]]

===User talk===
* {{tl|Welcomeip}}
* {{tl|Welcome}}
* {{tl|Talk header}}
* [[:Category:User warning templates]]

===Administrative===
* [[Template:Category redirect]]
* [[Template:Warning]]

==Theory==
* [[Learning by engagement]]
* [[User:JWSchmidt/Wiki Scholar]]

==Thoughts==
* [[Red link]]s are doorways to the infinite library ([[w:The Library of Babel|Library of Babel]])

==Tooltips==
{{Tooltip|Tooltips allow additional text to be displayed when cursor hovers over|Pretty cool, eh?}}


==User==
* [[w:Special:GlobalRenameRequest]]
* [[Special:UserGroupRights]]
* [[Special:UserRights]]
* [[m:Steward requests/Permissions]]
* [[meta:Help:Two-factor authentication]]

==Usability==
* [[Wikiversity:Usability]]
* http://usability.wikimedia.org - [http://usability.wikimedia.org/wiki/User:Jtneill Jtneill]

==Video==
* .ogg files can be uploaded and embedded
* See [[/Video]] for examples

==wikEd==
* [[w:User_talk:Cacycle/wikEd]]

==Wiki2Reveal==
* [[Wiki2Reveal]] (slides on the fly from MediaWiki page)

==x Test anchor==
&lt;!-- Test anchor - don't delete! --&gt;
{{anchor|test}}

==See also==
* [[User:Jade Knight/Tools]]</text>
      <sha1>nqkfhu6p1wff3djgmjhxo83zj8cxevn</sha1>
    </revision>
  </page>
  <page>
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    <ns>2</ns>
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      <timestamp>2026-06-14T02:10:09Z</timestamp>
      <contributor>
        <username>PieWriter</username>
        <id>3039865</id>
      </contributor>
      <minor />
      <comment>Remove template/category (via JWB)</comment>
      <origin>2815601</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1637" sha1="0jqu92vxpshrtd2fmp0pp1o2b9e7z93" xml:space="preserve">{{#babel:ru|en-3|nl-3|de-2}}
&lt;div style=&quot;float:right&quot;&gt;
&lt;div style=&quot;border: solid #bbb 1px; margin: 1px;&quot;&gt;
{| cellspacing=&quot;0&quot; style=&quot;width: 238px; background: #f6f6f6&quot;
| style=&quot;width: 45px; height: 45px; background: #fff; text-align: center; font-size: 14pt; color: #fff&quot; | [[Image:Wikipedia-logo-v2.svg|40px]]
| style=&quot;font-size: 8pt; padding: 4pt; line-height: 1.25em;&quot; | This user has [[w:User:Ru.spider|a page]] on [[w:Wikipedia|English Wikipedia]].
|}
&lt;/div&gt;
&lt;div style=&quot;border: solid #bbb 1px; margin: 1px;&quot;&gt;
{| cellspacing=&quot;0&quot; style=&quot;width: 238px; background: #f6f6f6&quot;
| style=&quot;width: 45px; height: 45px; background: #fff; text-align: center; font-size: 14pt; color: #fff&quot; | [[Image:Wikipedia-logo-v2.svg|40px]]
| style=&quot;font-size: 8pt; padding: 4pt; line-height: 1.25em;&quot; | This user has [[w:nl:User:Spider|a page]] on [[w:nl:|Dutch Wikipedia]].
|}
&lt;/div&gt;
&lt;div style=&quot;border: solid #bbb 1px; margin: 1px;&quot;&gt;
{| cellspacing=&quot;0&quot; style=&quot;width: 238px; background: #f6f6f6&quot;
| style=&quot;width: 45px; height: 45px; background: #fff; text-align: center; font-size: 14pt; color: #fff&quot; | [[Image:Wikipedia-logo-v2.svg|40px]]
| style=&quot;font-size: 8pt; padding: 4pt; line-height: 1.25em;&quot; | This user has [[w:ru:User:Spider|a page]] on [[w:ru:|Russian Wikipedia]].
|}
&lt;/div&gt;
{{User Commons}}
&lt;/div&gt;
Just registered &amp; looking around.

== Links ==
* [[User:Juan de Vojníkov/From the Wikiversity content to its conflicts|From the Wikiversity content to its conflicts]] (Jan's presentation on [[:w:en:wikimania|Wikimania 2011]])
* [[The Future of Wikiversity]] (my thoughts after the presentation and several discussions with Jan)

[[ru:User:Spider]]</text>
      <sha1>0jqu92vxpshrtd2fmp0pp1o2b9e7z93</sha1>
    </revision>
  </page>
  <page>
    <title>Reed–Solomon codes for coders</title>
    <ns>0</ns>
    <id>118943</id>
    <revision>
      <id>2815604</id>
      <parentid>2718361</parentid>
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      <contributor>
        <username>~2026-34857-36</username>
        <id>3093657</id>
      </contributor>
      <comment>/* Division */  Removed %255 because it's unnecessary, the range of gf_exp is already extended to 511.</comment>
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      <text bytes="103699" sha1="tu8cqfeg7at89d89ehgaef0rcw38wda" xml:space="preserve">[[w:Error_detection_and_correction|Error correcting codes]] are a signal processing technique to correct errors. They are nowadays ubiquitous, such as in communications (mobile phone, internet), data storage and archival (hard drives, optical discs CD/DVD/BluRay, archival tapes), warehouse management (barcodes) and advertisement (QR codes).

[[w:Reed–Solomon error correction|Reed–Solomon error correction]] is a specific type of error correction code. It is one of the oldest but it is still widely used, as it is very well defined and several efficient algorithms are now available under the public domain.

Usually, error correction codes are hidden and most users do not even know about them, nor when they are used. Yet, they are a critical component for some applications to be viable, such as communication or data storage. Indeed, a hard drive that would randomly lose data every few days would be useless, and a phone being able to call only on days with a cloud-less weather would be seldom used. Using error correction codes allows to recover a corrupted message into the full original message.

Barcodes and QR codes are interesting applications to study, as they have the specificity of displaying visually the error correction code, rendering these codes readily accessible to the curious user.

In this essay, we will attempt to introduce the principles of Reed–Solomon codes from the point of view of a programmer rather than a mathematician, which means that we will focus more on the practice than the theory, although we will also explain the theory, but only the necessary knowledge for intuition and implementation. Notable references in the domain will be provided, so that the interested reader can dig deeper into the mathematical theory at will. We will provide real-world examples taken from the popular [[w:QR code|QR code]] barcode system as well as working code samples. We chose to use [[w:Python (programming language)|Python]] for the samples (mainly because it looks pretty and similar to [[w:pseudocode|pseudocode]]), but we will try to explain any non-obvious features for those who are not familiar with it. The mathematics involved is advanced in the sense that it is not usually taught below the university level, but it should be understandable to someone with a good grasp of high-school algebra.

We will first gently introduce the intuitions behind error correction codes principles, then in a second section we will introduce the structural design of QR codes, in other words how information is stored in a QR code and how to read and produce it, and in a third section we will study error correction codes via the implementation of a Reed&amp;ndash;Solomon decoder, with a quick introduction of the bigger BCH codes family, in order to reliably read damaged QR codes.

Note for the curious readers that [[Reed–Solomon codes for coders/Additional information|extended information can be found in the appendix]] and on the [[Talk:Reed%E2%80%93Solomon_codes_for_coders|discussion page]].

==Principles of error correction==
Before detailing the code, it might be useful to understand the intuition behind error correction. Indeed, although error correcting codes may seem daunting mathematically-wise, most of the mathematical operations are high school grade (with the exception of Galois Fields, but which are in fact easy and common for any programmer: it's simply doing operations on integers modulo a number). However, the complexity of the mathematical ingenuity behind error correction codes hide the quite intuitive goal and mechanisms at play.

Error correcting codes might seem like a difficult mathematical concept, but they are in fact based on an intuitive idea with an ingenious mathematical implementation: '''let's make the data structured, in a way that we can &quot;guess&quot; what the data was if it gets corrupted, just by &quot;fixing&quot; the structure'''. Mathematically-wise, we use polynomials from the Galois Field to implement this structure.

Let's take a more practical analogy: let's say you want to communicate messages to someone else, but these messages can get corrupted along the way. The main insight of error correcting codes is that, '''instead of using a whole dictionary of words, we can use a smaller set of carefully selected words, a &quot;reduced dictionary&quot;, so that each word is as different as any other'''. This way, when we get a message, we just have to lookup inside our reduced dictionary to '''1) detect''' which words are corrupted (as they are not in our reduced dictionary); '''2) correct''' corrupted words by finding the most similar word in our dictionary.

Let's take a simple example: we have a reduced dictionary with only three words of 4 letters: &lt;kbd&gt;this&lt;/kbd&gt;, &lt;kbd&gt;that&lt;/kbd&gt; and &lt;kbd&gt;corn&lt;/kbd&gt;. Let's say we receive a corrupted word: &lt;kbd&gt;co**&lt;/kbd&gt;, where &lt;kbd&gt;*&lt;/kbd&gt; is an erasure. Since we have only 3 words in our dictionary, we can easily compare our received word with our dictionary to find the word that is the closest. In this case, it's &lt;kbd&gt;corn&lt;/kbd&gt;. Thus the missing letters are &lt;kbd&gt;rn&lt;/kbd&gt;.

Now let's say we receive the word &lt;kbd&gt;th**&lt;/kbd&gt;. Here the problem is that we have two words in our dictionary that match the received word: &lt;kbd&gt;this&lt;/kbd&gt; and &lt;kbd&gt;that&lt;/kbd&gt;. In this case, we cannot be sure which one it is, and thus we cannot decode. This means that our dictionary is not very good, and we should replace &lt;kbd&gt;that&lt;/kbd&gt; with another more different word, such as &lt;kbd&gt;dash&lt;/kbd&gt; to maximize the difference between each word. This difference, or more precisely the minimum number of different letters between any 2 words of our dictionary, is called the '''maximum Hamming distance''' of our dictionary. Making sure that any 2 words of the dictionary share a minimum number of letters at the same position is called '''maximum separability'''.

The same principle is used for most error correcting codes: we generate a reduced dictionary containing only words with maximum separability (we will detail more how to do that in the third section), and then we communicate only with the words of this reduced dictionary. What Galois Fields provide is the structure (ie, reduced dictionary basis), and Reed&amp;ndash;Solomon is a way to automatically create a suitable structure (make a reduced dictionary with maximum separability tailored for a dataset), as well as provide the automated methods to detect and correct errors (ie, lookups in the reduced dictionary). To be more precise, Galois Fields are the structure (thanks to their cyclic nature, the modulo an integer) and Reed&amp;ndash;Solomon is the codec (encoder/decoder) based on Galois Fields.

If a word gets corrupted in the communication, that's no big deal since we can easily fix it by looking inside our dictionary and find the closest word, which is probably the correct one (there is however a chance of choosing a wrong one if the input message is too heavily corrupted, but the probability is very small). Also, the longer our words are, the more separable they are, since more characters can be corrupted without any impact.

The simplest way to generate a dictionary of maximally separable words is to make words longer than they really are.

Let's take again our example:

    t h i s
    t h a t
    c o r n

Append a unique set of characters so that there are no duplicated characters at any of the appended positions, and add one more word to help with the explanation:

    t h i s a b c d
    t h a t b c d e
    c o r n c d e f

Note that each word in this dictionary differs from every other word by at least 6 characters, so the distance is 6. This allows up to 5 errors in known positions (which are called erasures), or 3 errors in unknown positions, to be corrected.

Assume that 4 erasures occur:

    t * * * a b * d

Then a search of the dictionary for the 4 non-erased characters can be done to find the only entry that matches those 4 characters, since the distance is 5. Here it gives: &lt;kbd&gt;t h i s a b c d&lt;/kbd&gt;

Assume that 2 errors occur as in one of these patterns:

   t h o s b c d e

The issue here is the location of the errors is unknown. The erasures might have happened in any 2 positions meaning that there are &lt;math&gt;\tbinom{8}{6}&lt;/math&gt; or 28 possible sub-sets of 6 characters:

   t h o s b c * *
   t h o s b * d *
   t h o s b * * e
   ...

If we do a dictionary search on each of these sub-sequences, we find that there is only one sub-set that matches 6 characters. &lt;kbd&gt;t h * * b c d e&lt;/kbd&gt; matches &lt;kbd&gt;t h a t b c d e&lt;/kbd&gt;.

With these examples, one can see the advantage of redundancy in recovering lost information: redundant characters help you recover your original data. The previous examples show how a crude error correcting scheme could work. Reed&amp;ndash;Solomon's core idea is similar, append redundant data to a message based on Galois Field mathematics. The original error correcting decoder was similar to the error example above, search sub-sets of a received message that correspond to a valid message, and choose the one with the most matches as the corrected message. This isn't practical for larger messages, so mathematical algorithms were developed to perform error correction in a reasonable time.

==QR code structure==
This section introduces the structure of QR codes, which is how data is stored in a QR code. The information in this section is deliberately incomplete. Only the most common features of the small 21&amp;times;21 size symbols (also known as version 1) are presented here, but see the [[Reed–Solomon codes for coders/Additional information|appendix]] for additional information.

Here is a QR symbol that will be used as an example. It consists of dark and light squares, known as modules in the barcoding world. The three square locator patterns in the corners are a visually distinctive feature of QR symbols.

[[File:QR Code Example.svg]]

===Masking===
A masking process is used to avoid features in the symbol that might confuse a scanner, such as misleading shapes that look like the locator patterns and large blank areas. Masking inverts certain modules (white becomes black and black becomes white) while leaving others alone.

In the diagram below, the red areas encode format information and use a fixed masking pattern. The data area (in black and white) is masked with a variable pattern. When the code is created, the encoder tries a number of different masks and chooses the one that minimizes undesirable features in the result. The chosen mask pattern is then indicated in the format information so that the decoder knows which one to use. The light gray areas are fixed patterns which do not encode any information. In addition to the obvious locator patterns, there are also timing patterns which contain alternating light and dark modules.

[[File:QR Code Masking Example.svg]]

The masking transformation is easily applied (or removed) using the [[w:Exclusive or|exclusive-or]] operation (denoted by a caret ^ in many programming languages). The unmasking of the format information is shown below. Reading counter-clockwise around the upper-left locator pattern, we have the following sequence of bits. White modules represent 0 and black modules represent 1.

 Input       101101101001011
 Mask      ^ &lt;u&gt;101010000010010&lt;/u&gt;
 Output      000111101011001

===Formatting information===
There are two identical copies of the formatting information, so that the symbol can still be decoded even if it is damaged. The second copy is broken in two pieces and placed around the other two locators, and is read in a clockwise direction (upwards in the lower-left corner, then left-to-right in the upper-right corner).

The first two bits of formatting information give the error correction level used for the message data. A QR symbol this size contains 26 bytes of information. Some of these are used to store the message and some are used for error correction, as shown in the table below. The left-hand column is simply a name given to that level.

{|class=&quot;wikitable&quot;
|-
! Error Correction Level !! Level Indicator !! Error Correction Bytes !! Message Data Bytes
|- align=&quot;center&quot;
| L || 01 || 7 || 19
|- align=&quot;center&quot;
| M || 00 || 10 || 16
|- align=&quot;center&quot;
| Q || 11 || 13 || 13
|- align=&quot;center&quot;
| H || 10 || 17 || 9
|}

The next three bits of format information select the masking pattern to be used in the data area. The patterns are illustrated below, including the mathematical formula that tells whether a module is black (i and j are the row and column numbers, respectively, and start with 0 in the upper-left hand corner).

[[File:QR Code Mask Patterns.svg]]

The remaining ten bits of formatting information are for correcting errors in the format itself. This will be explained in a [[#BCH codes|later section]].

===Message data===

Here is a larger diagram showing the &quot;unmasked&quot; QR code. Different regions of the symbol are indicated, including the boundaries of the message data bytes.

[[File:QR Code Unmasked.svg]]

Data bits are read starting from the lower-right corner and moving up the two right-hand columns in a zig-zag pattern. The first three bytes are 01000000 11010010 01110101. The next two columns are read in a downward direction, so the next byte is 01000111. Upon reaching the bottom, the two columns after that are read upward. Proceed in this up-and-down fashion all the way to the left side of the symbol (skipping over the timing pattern where necessary). Here is the complete message in [[w:Hexadecimal|hexadecimal]] notation.

:Message data bytes: 40 d2 75 47 76 17 32 06 27 26 96 c6 c6 96 70 ec
:Error correction bytes: bc 2a 90 13 6b af ef fd 4b e0

===Decoding===

The final step is to decode the message bytes into something readable. The first four bits indicate how the message is encoded. QR codes use several different encoding schemes, so that different kinds of messages can be stored efficiently. These are summarized in the table below. After the mode indicator is a length field, which tells how many characters are stored. The size of the length field depends on the specific encoding.

{|class=&quot;wikitable&quot;
|-
! Mode Name !! Mode Indicator !! Length Bits !! Data Bits
|- align=&quot;center&quot;
| Numeric || 0001 || 10 || 10 bits per 3 digits
|- align=&quot;center&quot;
| Alphanumeric || 0010 || 9 || 11 bits per 2 characters
|- align=&quot;center&quot;
| Byte || 0100 || 8 || 8 bits per character
|- align=&quot;center&quot;
| Kanji || 1000 || 8 || 13 bits per character
|}

(The length field sizes above are valid only for smaller QR codes.)

Our sample message starts with 0100 (hex 4), indicating that there are 8 bits per character. The next 8 bits (hex 0d) are the length field, 13 in decimal notation. The bits after that can be rearranged in bytes representing the actual characters of the messageː 27 54 77 61 73 20 62 72 69 6c 6c 69 67, and additionally 0e c. The first two, hex 27 and 54 are the [[w:ASCII|ASCII]] codes for apostrophe and T. The whole message is &quot;'Twas brillig&quot; (from [[w:Jabberwocky#Lexicon]]).

After the last of the data bits is another 4-bit mode indicator. It can be different from the first one, allowing different encodings to be mixed within the same QR symbol. When there is no more data to store, the special end-of-message code 0000 is given. (Note that the standard allows the end-of-message code to be omitted if it wouldn't fit in the available number of data bytes.)

At this point, we know how to decode, or read, a whole QR code. However, in real life conditions, a QR code is rarely whole: usually, it is scanned via a phone's camera, under potentially poor lighting conditions, or on a scratched surface where part of the QR code was ripped, or on a stained surface, etc.

To make our QR code decoder **reliable**, we need to be able to **correct** errors. The next part of this article will describe how to correct errors, by constructing a BCH decoder, and more specifically a Reed&amp;ndash;Solomon decoder.

==BCH codes==
In this section, we introduce a general class of error correction codes: the [[w:BCH code|BCH codes]], the parent family of modern Reed&amp;ndash;Solomon codes, and the basic detection and correction mechanisms.

The formatting information is encoded with a [[w:BCH code|BCH code]] which allows a certain number of bit-errors to be detected and corrected. BCH codes are a generalization of Reed&amp;ndash;Solomon codes (modern Reed&amp;ndash;Solomon codes are BCH codes). In the case of QR codes, the BCH code used for the format information is much simpler than the Reed&amp;ndash;Solomon code used for the message data, so it makes sense to start with the BCH code for format information.

===BCH error detection===
The process for checking the encoded information is similar to long division, but uses exclusive-or instead of subtraction. The format code should produce a remainder of zero when it is &quot;divided&quot; by the so-called generator of the code. QR format codes use the generator 10100110111. This process is demonstrated for the format information in the example code (000111101011001) below.

               000111101011001
                ^ &lt;u&gt;101001101110 &lt;/u&gt;
                  010100110111
                 ^ &lt;u&gt;10100110111&lt;/u&gt;
                   00000000000

Here is a Python function which implements this calculation.

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def qr_check_format(fmt):
   g = 0x537 # = 0b10100110111 in python 2.6+
   for i in range(4,-1,-1):
      if fmt &amp; (1 &lt;&lt; (i+10)):
         fmt ^= g &lt;&lt; i
   return fmt
&lt;/syntaxhighlight&gt;

''Python note:'' The &lt;kbd&gt;range&lt;/kbd&gt; function may not be clear to non-Python programmers. It produces a list of numbers counting down from 4 to 0 (the code has &quot;-1&quot; because the interval returned by &quot;range&quot; includes the start but not the end value). In C-derived languages, the for loop might be written as &lt;kbd style=&quot;white-space:nowrap&quot;&gt;for (i = 4; i &gt;= 0; i--)&lt;/kbd&gt;; in Pascal-derived languages, &lt;kbd style=&quot;white-space:nowrap&quot;&gt;for i := 4 downto 0&lt;/kbd&gt;.

''Python note 2:'' The &lt;kbd&gt;&amp;&lt;/kbd&gt; operator performs [[w:Bitwise operation#AND|bitwise and]], while &lt;kbd&gt;&amp;lt;&amp;lt;&lt;/kbd&gt; is a [[w:Bitwise operation#Bit shifts|left bit-shift]]. This is consistent with C-like languages.

This function can also be used to encode the 5-bit format information.
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
encoded_format = (format&lt;&lt;10) + qr_check_format(format&lt;&lt;10)
&lt;/syntaxhighlight&gt;

Readers may find it an interesting exercise to generalize this function to divide by different numbers. For example, larger QR codes contain six bits of version information with 12 error correction bits using the generator 1111100100101.

In mathematical formalism, these binary numbers are described as polynomials whose coefficients are [[w:Modular arithmetic|integers mod 2]]. Each bit of the number is a coefficient of one term. For example:

:10100110111 = 1 ''x''&lt;sup&gt;10&lt;/sup&gt; + 0 ''x''&lt;sup&gt;9&lt;/sup&gt; + 1 ''x''&lt;sup&gt;8&lt;/sup&gt; + 0 ''x''&lt;sup&gt;7&lt;/sup&gt; + 0 ''x''&lt;sup&gt;6&lt;/sup&gt; + 1 ''x''&lt;sup&gt;5&lt;/sup&gt; + 1 ''x''&lt;sup&gt;4&lt;/sup&gt; + 0 ''x''&lt;sup&gt;3&lt;/sup&gt; + 1 ''x''&lt;sup&gt;2&lt;/sup&gt; + 1 ''x'' + 1 = ''x''&lt;sup&gt;10&lt;/sup&gt; + ''x''&lt;sup&gt;8&lt;/sup&gt; + ''x''&lt;sup&gt;5&lt;/sup&gt; + ''x''&lt;sup&gt;4&lt;/sup&gt; + ''x''&lt;sup&gt;2&lt;/sup&gt; + ''x'' + 1

If the remainder produced by &lt;kbd&gt;qr_check_format&lt;/kbd&gt; is not zero, then the code has been damaged or misread. The next step is to determine which format code is most likely the one that was intended (ie, lookup in our reduced dictionary).

===BCH error correction===
Although sophisticated algorithms for decoding BCH codes exist, they are probably overkill in this case. Since there are only 32 possible format codes, it's much easier to simply try each one and pick the one that has the smallest number of bits different from the code in question (the number of different bits is known as the [[w:Hamming distance|Hamming distance]]). This method of finding the closest code is known as exhaustive search, and is possible only because we have very few codes (a code is a valid message, and here there are only 32, all other binary numbers aren't correct).

(Note that Reed&amp;ndash;Solomon is also based on this principle, but since the number of possible codewords is simply too big, we can't afford to do an exhaustive search, and that's why clever but complicated algorithms have been devised, such as Berlekamp-Massey.)

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def hamming_weight(x):
   weight = 0
   while x &gt; 0:
      weight += x &amp; 1
      x &gt;&gt;= 1
   return weight

def qr_decode_format(fmt):
   best_fmt = -1
   best_dist = 15
   for test_fmt in range(0,32):
      test_code = (test_fmt&lt;&lt;10) ^ qr_check_format(test_fmt&lt;&lt;10)
      test_dist = hamming_weight(fmt ^ test_code)
      if test_dist &lt; best_dist:
         best_dist = test_dist
         best_fmt = test_fmt
      elif test_dist == best_dist:
         best_fmt = -1
   return best_fmt
&lt;/syntaxhighlight&gt;

The function &lt;kbd&gt;qr_decode_format&lt;/kbd&gt; returns -1 if the format code could not be unambiguously decoded. This happens when two or more format codes have the same distance from the input.

To run this code in Python, first start [[w:IDLE (Python)|IDLE]], Python's integrated development environment. You should see a version message and the interactive input prompt &lt;kbd&gt;&gt;&gt;&gt;&lt;/kbd&gt;. Open a new window, copy the functions &lt;kbd&gt;qr_check_format&lt;/kbd&gt;, &lt;kbd&gt;hamming_weight&lt;/kbd&gt;, and &lt;kbd&gt;qr_decode_format&lt;/kbd&gt; into it, and save as &lt;kbd&gt;qr.py&lt;/kbd&gt;. Return to the prompt and type the lines following &lt;kbd&gt;&gt;&gt;&gt;&lt;/kbd&gt; below.

&lt;pre&gt;&gt;&gt;&gt; from qr import *
&gt;&gt;&gt; qr_decode_format(int(&quot;000111101011001&quot;,2))  # no errors
3
&gt;&gt;&gt; qr_decode_format(int(&quot;111111101011001&quot;,2))  # 3 bit-errors
3
&gt;&gt;&gt; qr_decode_format(int(&quot;111011101011001&quot;,2))  # 4 bit-errors
-1
&lt;/pre&gt;

You can also start Python by typing &lt;kbd&gt;python&lt;/kbd&gt; at a command prompt.

In the next sections, we will study Finite Field Arithmetics and Reed&amp;ndash;Solomon code, which is a subtype of BCH codes. The basic idea (ie, '''using a limited words dictionary with maximum separability''') is the same, but since we will encode longer words (256 bytes instead of 2 bytes), with more symbols available (encoded on all 8bits, thus 256 different possible values), we cannot use this naive, exhaustive approach, because it would take way too much time: we need to use cleverer algorithms, and Finite Field mathematics will help us do just that, by giving us a '''structure'''.

==Finite field arithmetic==

===Introduction to mathematical fields===
Before discussing the Reed&amp;ndash;Solomon codes used for the message, it will be useful to introduce a bit more mathematics. 

We'd like to define addition, subtraction, multiplication, and division for 8-bit bytes and always produce 8-bit bytes as a result, so as to avoid any overflow. Naively, we might attempt to use the normal definitions for these operations, and then mod by 256 to keep results from overflowing. And this is exactly what we will be doing, and is what is called a Galois Field 2^8. You can easily imagine why it works for everything, except for division: what is 5/4?

Here's a brief introduction to Galois Fields: a finite field is a set of numbers, and a field needs to have six properties governing addition, subtraction, multiplication and division: Closure, Associative, Commutative, Distributive, Identity and Inverse. More simply put, using a field allows us to study the relationship between numbers of this field, and apply the result to any other field that follows the same properties. For example, the set of reals ℝ is a field. In other words, mathematical fields studies the structure of a set of numbers.

However, integers ℤ aren't a field, because as we said above, not all divisions are defined (such as 5/4), which violates multiplicative inverse property (x such that x*4=5 does not exist). One simple way to fix that is to do modulo using a prime number, such as 257, or any positive integer power of a prime number: in this way, we are guaranteed that x*4=5 exists since we will just wrap around. ℤ modulo any prime number is called a Galois Field, and modulo 2 is an extra interesting Galois Field: since an 8-bit string can express a total of 256 = 2^8 values, we say that we use a Galois Field of 2^8, or GF(2^8). In spoken language, 2 is the characteristic of the field, 8 is the exponent, and 256 is the field's cardinality. More information on [http://research.swtch.com/field finite fields can be found here].

Here we will define the usual mathematical operations that you are used to doing with integers, but adapted to GF(2^8), which is basically doing usual operations but modulo 2^8.

Another way to consider the link between GF(2) and GF(2^8) is to think that GF(2^8) represents a polynomial of 8 binary coefficients. For example, in GF(2^8), 170 is equivalent to &lt;kbd&gt;10101010 = 1*x^7 + 0*x^6 + 1*x^5 + 0*x^4 + 1*x^3 + 0*x^2 + 1*x + 0 = x^7 + x^5 + x^3 + x&lt;/kbd&gt;. Both representations are equivalent, it's just that in the first case, 170, the representation is decimal, and in the other case it's binary, which can be thought as representing a polynomial [[w:Finite_field_arithmetic#Effective_polynomial_representation|by convention (only used in GF(2^p) as explained here)]]. The latter is often the representation used in academic books and in hardware implementations (because of logical gates and registers, which work at the binary level). For a software implementation, the decimal representation can be preferred for clearer and more close-to-the-mathematics code (this is what we will use for the code in this tutorial, except for some examples that will use the binary representation).

In any case, try to not confuse the polynomial representing a single GF(2^p) symbol (each coefficient is a bit/boolean: either 0 or 1), and the polynomial representing a list of GF(2^p) symbols (in this case the polynomial is equivalent to the message+RScode, each coefficient is a value between 0 and 2^p and represent one character of the message+RScode). We will first describe operations on single symbol, then polynomial operations on a list of symbols.

===Addition and Subtraction===
Both addition and subtraction are replaced with exclusive-or in Galois Field base 2. This is logical: addition modulo 2 is exactly like an XOR, and subtraction modulo 2 is exactly the same as addition modulo 2. This is possible because additions and subtractions in this Galois Field are carry-less.

Thinking of our 8-bit values as polynomials with coefficients mod 2:

    0101 + 0110 = 0101 - 0110 = 0101 XOR 0110 = 0011

The same way (in binary representation of two single GF(2^8) integers):

:(''x''&lt;sup&gt;2&lt;/sup&gt; + 1) + (''x''&lt;sup&gt;2&lt;/sup&gt; + ''x'') = 2 ''x''&lt;sup&gt;2&lt;/sup&gt; + ''x'' + 1 = 0 ''x''&lt;sup&gt;2&lt;/sup&gt; + ''x'' + 1 = ''x'' + 1

Since &lt;kbd&gt;(a ^ a) = 0&lt;/kbd&gt;, every number is its own opposite, so (''x'' - ''y'') is the same as (''x'' + ''y'').

Note that in books, you will find additions and subtractions to define some mathematical operations on GF integers, but in practice, you can just XOR (as long as you are in a Galois Field base 2; this is not true in other fields).

Here is the equivalent Python code:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_add(x, y):
    return x ^ y

def gf_sub(x, y):
    return x ^ y # in binary galois field, subtraction is just the same as addition (since we mod 2)
&lt;/syntaxhighlight&gt;

===Multiplication===
Multiplication is likewise based on polynomial multiplication. Simply write the inputs as polynomials and multiply them out using the distributive law as normal. As an example, 10001001 times 00101010 is calculated as follows.

:(''x''&lt;sup&gt;7&lt;/sup&gt; + ''x''&lt;sup&gt;3&lt;/sup&gt; + 1) (''x''&lt;sup&gt;5&lt;/sup&gt; + ''x''&lt;sup&gt;3&lt;/sup&gt; + ''x'') = ''x''&lt;sup&gt;7&lt;/sup&gt; (''x''&lt;sup&gt;5&lt;/sup&gt; + ''x''&lt;sup&gt;3&lt;/sup&gt; + ''x'') + ''x''&lt;sup&gt;3&lt;/sup&gt; (''x''&lt;sup&gt;5&lt;/sup&gt; + ''x''&lt;sup&gt;3&lt;/sup&gt; + ''x'') + 1 (''x''&lt;sup&gt;5&lt;/sup&gt; + ''x''&lt;sup&gt;3&lt;/sup&gt; + ''x'')
:= ''x''&lt;sup&gt;12&lt;/sup&gt; + ''x''&lt;sup&gt;10&lt;/sup&gt; + 2 ''x''&lt;sup&gt;8&lt;/sup&gt; + ''x''&lt;sup&gt;6&lt;/sup&gt; + ''x''&lt;sup&gt;5&lt;/sup&gt; + ''x''&lt;sup&gt;4&lt;/sup&gt; +  ''x''&lt;sup&gt;3&lt;/sup&gt; + ''x''
:= ''x''&lt;sup&gt;12&lt;/sup&gt; + ''x''&lt;sup&gt;10&lt;/sup&gt; + ''x''&lt;sup&gt;6&lt;/sup&gt; + ''x''&lt;sup&gt;5&lt;/sup&gt; + ''x''&lt;sup&gt;4&lt;/sup&gt; +  ''x''&lt;sup&gt;3&lt;/sup&gt; + ''x''

The same result can be obtained by a modified version of the standard grade-school multiplication procedure, in which we replace addition with exclusive-or.

        10001001
 *      &lt;u&gt;00101010&lt;/u&gt;
       10001001
 ^   10001001
 ^ &lt;u&gt;10001001&lt;/u&gt;
   1010001111010

Note: the XOR multiplication here is carry-less! If you do it with-carry, you will get the wrong result 1011001111010 with the extra term ''x''&lt;sup&gt;9&lt;/sup&gt; instead of the correct result 1010001111010.

Here is a Python function which implements this polynomial multiplication on single GF(2^8) integers.

Note: this function (and some other functions below) use a lot of bitwise operators such as &gt;&gt; and &lt;&lt;, because they are both faster and more concise to do what we want to do. These operators are available in most languages, they are not specific to Python, and [https://wiki.python.org/moin/BitwiseOperators you can get more information about them here].

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def cl_mul(x,y):
    '''Bitwise carry-less multiplication on integers'''
    z = 0
    i = 0
    while (y&gt;&gt;i) &gt; 0:
        if y &amp; (1&lt;&lt;i):
            z ^= x&lt;&lt;i
        i += 1
    return z
&lt;/syntaxhighlight&gt;

Of course, the result no longer fits in an 8-bit byte (in this example, it is 13 bits long), so we need to perform one more step before we are finished. The result is reduced modulo 100011101 (the choice of this number is explained below the code), using the long division process described previously. In this instance, this is called &quot;modular reduction&quot;, because basically what we do is that we divide and keep only the remainder, using a modulo. This produces the final answer 11000011 in our example.

   1010001111010
 ^ &lt;u&gt;100011101&lt;/u&gt;
   0010110101010
   ^ &lt;u&gt;100011101&lt;/u&gt;
     00111011110
     ^ &lt;u&gt;100011101&lt;/u&gt;
       011000011

Here is the Python code to do the whole Galois Field multiplication with modular reduction:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_mult_noLUT(x, y, prim=0):
    '''Multiplication in Galois Fields without using a precomputed look-up table (and thus it's slower)
    by using the standard carry-less multiplication + modular reduction using an irreducible prime polynomial'''

    ### Define bitwise carry-less operations as inner functions ###
    def cl_mult(x,y):
        '''Bitwise carry-less multiplication on integers'''
        z = 0
        i = 0
        while (y&gt;&gt;i) &gt; 0:
            if y &amp; (1&lt;&lt;i):
                z ^= x&lt;&lt;i
            i += 1
        return z

    def bit_length(n):
        '''Compute the position of the most significant bit (1) of an integer. Equivalent to int.bit_length()'''
        bits = 0
        while n &gt;&gt; bits: bits += 1
        return bits

    def cl_div(dividend, divisor=None):
        '''Bitwise carry-less long division on integers and returns the remainder'''
        # Compute the position of the most significant bit for each integers
        dl1 = bit_length(dividend)
        dl2 = bit_length(divisor)
        # If the dividend is smaller than the divisor, just exit
        if dl1 &lt; dl2:
            return dividend
        # Else, align the most significant 1 of the divisor to the most significant 1 of the dividend (by shifting the divisor)
        for i in range(dl1-dl2,-1,-1):
            # Check that the dividend is divisible (useless for the first iteration but important for the next ones)
            if dividend &amp; (1 &lt;&lt; i+dl2-1):
                # If divisible, then shift the divisor to align the most significant bits and XOR (carry-less subtraction)
                dividend ^= divisor &lt;&lt; i
        return dividend
    
    ### Main GF multiplication routine ###

    # Multiply the gf numbers
    result = cl_mult(x,y)
    # Then do a modular reduction (ie, remainder from the division) with an irreducible primitive polynomial so that it stays inside GF bounds
    if prim &gt; 0:
        result = cl_div(result, prim)

    return result
&lt;/syntaxhighlight&gt;

Result:

&lt;pre&gt;
&gt;&gt;&gt; a = 0b10001001
&gt;&gt;&gt; b = 0b00101010
&gt;&gt;&gt; print bin(gf_mult_noLUT(a, b, 0)) # multiplication only
0b1010001111010
&gt;&gt;&gt; print bin(gf_mult_noLUT(a, b, 0x11d)) # multiplication + modular reduction
0b11000011
&lt;/pre&gt;

Why mod 100011101 (in hexadecimal: 0x11d)? The mathematics is a little complicated here, but in short, 100011101 represents an 8th degree polynomial which is &quot;irreducible&quot; (meaning it can't be represented as the product of two smaller polynomials). This number is called a '''primitive polynomial''' or irreducible polynomial, or prime polynomial (we will mainly use this latter name for the rest of this tutorial). This is necessary for division to be well-behaved, which is to stay in the limits of the Galois Field, but without duplicating values. There are other numbers we could have chosen, but they're all essentially the same, and 100011101 (0x11d) is a common primitive polynomial for Reed&amp;ndash;Solomon codes. If you are curious to know how to generate those prime polynomials, please see the [[Reed%E2%80%93Solomon_codes_for_coders/Additional_information#Universal_Reed-Solomon_Codec|appendix]].

Additional infos on the prime polynomial (you can skip): What is a prime polynomial? It is the equivalent of a prime number, but in the Galois Field. Remember that a Galois Field uses values that are multiples of 2 as the generator. Of course, a prime number cannot be a multiple of two in standard arithmetics, but in a Galois Field it is possible. Why do we need a prime polynomial? Because to stay in the bound of the field, we need to compute the modulo of any value above the Galois Field. Why don't we just modulo with the Galois Field size? Because we will end up with lots of duplicate values, and we want to have as many unique values as possible in the field, so that a number always has one and only projection when doing a modulo or a XOR with the prime polynomial.

Note for the interested reader: as an example of what you can achieve with clever algorithms, here is another way to achieve multiplication of GF numbers in a more concise and faster way, using the [http://www.cut-the-knot.org/Curriculum/Algebra/PeasantMultiplication.shtml Russian Peasant Multiplication algorithm]:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_mult_noLUT(x, y, prim=0, field_charac_full=256, carryless=True):
    '''Galois Field integer multiplication using Russian Peasant Multiplication algorithm (faster than the standard multiplication + modular reduction).
    If prim is 0 and carryless=False, then the function produces the result for a standard integers multiplication (no carry-less arithmetics nor modular reduction).'''
    r = 0
    while y: # while y is above 0
        if y &amp; 1: r = r ^ x if carryless else r + x # y is odd, then add the corresponding x to r (the sum of all x's corresponding to odd y's will give the final product). Note that since we're in GF(2), the addition is in fact an XOR (very important because in GF(2) the multiplication and additions are carry-less, thus it changes the result!).
        y = y &gt;&gt; 1 # equivalent to y // 2
        x = x &lt;&lt; 1 # equivalent to x*2
        if prim &gt; 0 and x &amp; field_charac_full: x = x ^ prim # GF modulo: if x &gt;= 256 then apply modular reduction using the primitive polynomial (we just subtract, but since the primitive number can be above 256 then we directly XOR).

    return r
&lt;/syntaxhighlight&gt;

Note that using this last function with parameters prim=0 and carryless=False will return the result for a standard integers multiplication (and thus you can see the difference between carryless and with-carry addition and its impact on multiplication).

===Multiplication with logarithms===

The procedure described above is not the most convenient way to implement Galois field multiplication. Multiplying two numbers takes up to eight iterations of the multiplication loop, followed by up to eight iterations of the division loop. However, we can multiply with no looping by using lookup tables. One solution would be to construct the entire multiplication table in memory, but that would require a bulky 64k table. The solution described below is much more compact.

First, notice that it is particularly easy to multiply by 2=00000010 (by convention, this number is referred to as '''&amp;alpha;''' or the '''generator number'''): simply left-shift by one place, then exclusive-or with the modulus 100011101 if necessary (why xor is sufficient for taking the mod in this case is an exercise left to the reader). Here are the first few powers of &amp;alpha;.

{| class=&quot;wikitable&quot;
|-
| &amp;alpha;&lt;sup&gt;0&lt;/sup&gt; = 00000001
| &amp;alpha;&lt;sup&gt;4&lt;/sup&gt; = 00010000
| &amp;alpha;&lt;sup&gt;8&lt;/sup&gt;&amp;nbsp; = 00011101
| &amp;alpha;&lt;sup&gt;12&lt;/sup&gt; = 11001101
|-
| &amp;alpha;&lt;sup&gt;1&lt;/sup&gt; = 00000010
| &amp;alpha;&lt;sup&gt;5&lt;/sup&gt; = 00100000
| &amp;alpha;&lt;sup&gt;9&lt;/sup&gt;&amp;nbsp; = 00111010
| &amp;alpha;&lt;sup&gt;13&lt;/sup&gt; = 10000111
|-
| &amp;alpha;&lt;sup&gt;2&lt;/sup&gt; = 00000100
| &amp;alpha;&lt;sup&gt;6&lt;/sup&gt; = 01000000
| &amp;alpha;&lt;sup&gt;10&lt;/sup&gt; = 01110100
| &amp;alpha;&lt;sup&gt;14&lt;/sup&gt; = 00010011
|-
| &amp;alpha;&lt;sup&gt;3&lt;/sup&gt; = 00001000
| &amp;alpha;&lt;sup&gt;7&lt;/sup&gt; = 10000000
| &amp;alpha;&lt;sup&gt;11&lt;/sup&gt; = 11101000
| &amp;alpha;&lt;sup&gt;15&lt;/sup&gt; = 00100110
|}

If this table is continued in the same fashion, the powers of &amp;alpha; do not repeat themselves until &amp;alpha;&lt;sup&gt;255&lt;/sup&gt; = 00000001. Thus, every element of the field except zero is equal to some power of &amp;alpha;. The element '''&amp;alpha;''', that we define, is known as a [[w:Primitive element (finite field)|primitive element]] or '''generator''' of the Galois field.

This observation suggests another way to implement multiplication: by adding the exponents of &amp;alpha;.
:10001001 * 00101010 = &amp;alpha;&lt;sup&gt;74&lt;/sup&gt; * &amp;alpha;&lt;sup&gt;142&lt;/sup&gt; = &amp;alpha;&lt;sup&gt;74 + 142&lt;/sup&gt; = &amp;alpha;&lt;sup&gt;216&lt;/sup&gt; = 11000011
The problem is, how do we find the power of &amp;alpha; that corresponds to 10001001? This is known as the [[w:Discrete logarithm|discrete logarithm]] problem, and no efficient general solution is known. However, since there are only 256 elements in this field, we can easily construct a table of logarithms. While we're at it, a corresponding table of antilogs (exponentials) will also be useful. [[w:Finite_field_arithmetic#Implementation_tricks|More mathematical information about this trick can be found here]].
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
gf_exp = [0] * 512 # Create list of 512 elements. In Python 2.6+, consider using bytearray
gf_log = [0] * 256

def init_tables(prim=0x11d):
    '''Precompute the logarithm and anti-log tables for faster computation later, using the provided primitive polynomial.'''
    # prim is the primitive (binary) polynomial. Since it's a polynomial in the binary sense,
    # it's only in fact a single galois field value between 0 and 255, and not a list of gf values.
    global gf_exp, gf_log
    gf_exp = [0] * 512 # anti-log (exponential) table
    gf_log = [0] * 256 # log table
    # For each possible value in the galois field 2^8, we will pre-compute the logarithm and anti-logarithm (exponential) of this value
    x = 1
    for i in range(0, 255):
        gf_exp[i] = x # compute anti-log for this value and store it in a table
        gf_log[x] = i # compute log at the same time
        x = gf_mult_noLUT(x, 2, prim)

        # If you use only generator==2 or a power of 2, you can use the following which is faster than gf_mult_noLUT():
        #x &lt;&lt;= 1 # multiply by 2 (change 1 by another number y to multiply by a power of 2^y)
        #if x &amp; 0x100: # similar to x &gt;= 256, but a lot faster (because 0x100 == 256)
            #x ^= prim # subtract the primary polynomial to the current value (instead of 255, so that we get a unique set made of coprime numbers), this is the core of the tables generation

    # Optimization: double the size of the anti-log table so that we don't need to mod 255 to
    # stay inside the bounds (because we will mainly use this table for the multiplication of two GF numbers, no more).
    for i in range(255, 512):
        gf_exp[i] = gf_exp[i - 255]
    return [gf_log, gf_exp]
&lt;/syntaxhighlight&gt;
''Python note:'' The &lt;kbd&gt;range&lt;/kbd&gt; operator's upper bound is exclusive, so &lt;kbd&gt;gf_exp[255]&lt;/kbd&gt; is not set twice by the above.

The &lt;kbd&gt;gf_exp&lt;/kbd&gt; table is oversized in order to simplify the multiplication function. This way, we don't have to check to make sure that &lt;kbd&gt;gf_log[x] + gf_log[y]&lt;/kbd&gt; is within the table size.
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_mul(x,y):
    if x==0 or y==0:
        return 0
    return gf_exp[gf_log[x] + gf_log[y]] # should be gf_exp[(gf_log[x]+gf_log[y])%255] if gf_exp wasn't oversized
&lt;/syntaxhighlight&gt;

===Division===
Another advantage of the logarithm table approach is that it allows us to define division using the difference of logarithms. In the code below, 255 is added to make sure the difference isn't negative.
&lt;syntaxhighlight lang=&quot;python&quot; line=&quot;1&quot;&gt;def gf_div(x,y):
    if y==0:
        raise ZeroDivisionError()
    if x==0:
        return 0
    return gf_exp[(gf_log[x] + 255 - gf_log[y])]&lt;/syntaxhighlight&gt;
''Python note:'' The &lt;kbd&gt;raise&lt;/kbd&gt; statement throws an exception and aborts execution of the &lt;kbd&gt;gf_div&lt;/kbd&gt; function.

With this definition of division, &lt;kbd&gt;gf_div(gf_mul(x,y),y)==x&lt;/kbd&gt; for any &lt;kbd&gt;x&lt;/kbd&gt; and any nonzero &lt;kbd&gt;y&lt;/kbd&gt;.

Readers who are more advanced programmers may find it interesting to write a class encapsulating Galois field arithmetic. [[w:Operator overloading|Operator overloading]] can be used to replace calls to &lt;kbd&gt;gf_mul&lt;/kbd&gt; and &lt;kbd&gt;gf_div&lt;/kbd&gt; with the familiar operators &lt;kbd&gt;*&lt;/kbd&gt; and &lt;kbd&gt;/&lt;/kbd&gt;, but this can lead to confusion as to exactly what type of operation is being performed. Certain details can be generalized in ways that would make the class more widely useful. For example, [[w:Aztec Code|Aztec codes]] use five different Galois fields with element sizes ranging from 4 to 12 bits.

===Power and Inverse===
The logarithm table approach will once again simplify and speed up our calculations when computing the power and the inverse:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_pow(x, power):
    return gf_exp[(gf_log[x] * power) % 255]

def gf_inverse(x):
    return gf_exp[255 - gf_log[x]] # gf_inverse(x) == gf_div(1, x)
&lt;/syntaxhighlight&gt;

===Polynomials===
Before moving on to Reed&amp;ndash;Solomon codes, we need to define several operations on polynomials whose coefficients are Galois field elements. This is a potential source of confusion, since the elements themselves are described as polynomials; my advice is not to think about it too much. Adding to the confusion is the fact that ''x'' is still used as the placeholder. This ''x'' has nothing to do with the ''x'' mentioned previously, so don't mix them up.

The binary notation used previously for Galois field elements starts to become inconveniently bulky at this point, so I will switch to hexadecimal instead.
:00000001 ''x''&lt;sup&gt;4&lt;/sup&gt; + 00001111 ''x''&lt;sup&gt;3&lt;/sup&gt; + 00110110 ''x''&lt;sup&gt;2&lt;/sup&gt; + 01111000 ''x'' + 01000000 = &lt;kbd&gt;01&lt;/kbd&gt; ''x''&lt;sup&gt;4&lt;/sup&gt; + &lt;kbd&gt;0f&lt;/kbd&gt; ''x''&lt;sup&gt;3&lt;/sup&gt; + &lt;kbd&gt;36&lt;/kbd&gt; ''x''&lt;sup&gt;2&lt;/sup&gt; + &lt;kbd&gt;78&lt;/kbd&gt; ''x'' + &lt;kbd&gt;40&lt;/kbd&gt;
In Python, polynomials will be represented by a list of numbers in descending order of powers of ''x'', so the polynomial above becomes &lt;kbd&gt;[ 0x01, 0x0f, 0x36, 0x78, 0x40 ]&lt;/kbd&gt;. (The reverse order could have been used instead; both choices have their advantages and disadvantages.)

The first function multiplies a polynomial by a scalar.
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_poly_scale(p,x):
    r = [0] * len(p)
    for i in range(0, len(p)):
        r[i] = gf_mul(p[i], x)
    return r
&lt;/syntaxhighlight&gt;
''Note to Python programmers:'' This function is not written in a &quot;pythonic&quot; style. It could be expressed quite elegantly as a [[w:List comprehension|list comprehension]], but I have limited myself to language features that are easier to translate to other programming languages.

This function &quot;adds&quot; two polynomials (using exclusive-or, as usual).
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_poly_add(p,q):
    r = [0] * max(len(p),len(q))
    for i in range(0,len(p)):
        r[i+len(r)-len(p)] = p[i]
    for i in range(0,len(q)):
        r[i+len(r)-len(q)] ^= q[i]
    return r
&lt;/syntaxhighlight&gt;

The next function multiplies two polynomials.
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_poly_mul(p,q):
    '''Multiply two polynomials, inside Galois Field'''
    # Pre-allocate the result array
    r = [0] * (len(p)+len(q)-1)
    # Compute the polynomial multiplication (just like the outer product of two vectors,
    # we multiply each coefficients of p with all coefficients of q)
    for j in range(0, len(q)):
        for i in range(0, len(p)):
            r[i+j] ^= gf_mul(p[i], q[j]) # equivalent to: r[i + j] = gf_add(r[i+j], gf_mul(p[i], q[j]))
                                                         # -- you can see it's your usual polynomial multiplication
    return r
&lt;/syntaxhighlight&gt;

Finally, we need a function to evaluate a polynomial at a particular value of ''x'', producing a scalar result. [[w:Horner's method|Horner's method]] is used to avoid explicitly calculating powers of ''x''. Horner's method works by factorizing consecutively the terms, so that we always deal with ''x^1'', iteratively, avoiding the computation of higher degree terms:
:&lt;kbd&gt;01&lt;/kbd&gt; ''x''&lt;sup&gt;4&lt;/sup&gt; + &lt;kbd&gt;0f&lt;/kbd&gt; ''x''&lt;sup&gt;3&lt;/sup&gt; + &lt;kbd&gt;36&lt;/kbd&gt; ''x''&lt;sup&gt;2&lt;/sup&gt; + &lt;kbd&gt;78&lt;/kbd&gt; ''x'' + &lt;kbd&gt;40&lt;/kbd&gt; = (((&lt;kbd&gt;01&lt;/kbd&gt; ''x'' + &lt;kbd&gt;0f&lt;/kbd&gt;) ''x'' + &lt;kbd&gt;36&lt;/kbd&gt;) ''x'' + &lt;kbd&gt;78&lt;/kbd&gt;) ''x'' + &lt;kbd&gt;40&lt;/kbd&gt;
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_poly_eval(poly, x):
    '''Evaluates a polynomial in GF(2^p) given the value for x. This is based on Horner's scheme for maximum efficiency.'''
    y = poly[0]
    for i in range(1, len(poly)):
        y = gf_mul(y, x) ^ poly[i]
    return y
&lt;/syntaxhighlight&gt;

There's still one missing polynomial operation that we will need: polynomial division. This is more complicated than the other operations on polynomial, so we will study it in the next chapter, along with Reed&amp;ndash;Solomon encoding.

==Reed&amp;ndash;Solomon codes==
Now that the preliminaries are out of the way, we are ready to begin looking at Reed&amp;ndash;Solomon codes.

===Insight of the coding theory===

But first, why did we have to learn about finite fields and polynomials? Because this is the main insight of error-correcting codes like Reed&amp;ndash;Solomon: instead of just seeing a message as a series of (ASCII) numbers, we see it as '''a polynomial''' following the very well-defined '''rules of finite field arithmetic'''. In other words, by representing the data using polynomials and finite fields arithmetic, '''we added a structure to the data'''. The values of the message are still the same, but this conceptual structure now allows us to operate on the message, on its characters values, using well defined mathematical rules. This structure, that we always know because it's outside and independent of the data, is what allows us to repair a corrupted message.

Thus, even if in your code implementation you may choose to not explicitly represent the polynomials and the finite field arithmetic, these notions are essential for the error-correcting codes to work, and you will find these notions to underlie (even if implicitly) any implementation.

And now we will put these notions into practice!

===RS encoding===

====Encoding outline====
Like BCH codes, Reed&amp;ndash;Solomon codes are encoded by dividing the polynomial representing the message by an irreducible generator polynomial, and then the remainder is the RS code, which we will just append to the original message.

Why? We previously said that the principle behind BCH codes, and most other error correcting codes, is to use a reduced dictionary with very different words as to maximize the distance between words, and that longer words have greater distance: here it's the same principle, first because we lengthen the original message with additional symbols (the remainder) which raises the distance, and secondly because the remainder is almost unique (thanks to the carefully designed irreducible generator polynomial), so that it can be exploited by clever algorithms to deduce parts of the original message.

To summarize, with an approximated analogy to encryption: our '''generator polynomial''' is our encoding '''dictionary''', and '''polynomial division''' is the operator to '''convert''' our message using the dictionary (the generator polynomial) into a RS code.

====Exception management====
To manage errors and cases where we can't correct a message, we will display a meaningful error message, by raising an exception. We will make our own custom exception so that users can easily catch and manage them:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
class ReedSolomonError(Exception):
    pass
&lt;/syntaxhighlight&gt;

To display an error by raising our custom exception, we can then simply do the following:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
raise ReedSolomonError(&quot;Some error message&quot;)
&lt;/syntaxhighlight&gt;

And you can easily catch this exception to manage it by using a try/except block:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
try:
    raise ReedSolomonError(&quot;Some error message&quot;)
except ReedSolomonError as e:
    pass # do something here
&lt;/syntaxhighlight&gt;

====RS generator polynomial====
Reed&amp;ndash;Solomon codes use a '''generator polynomial''' similar to BCH codes (not to be confused with the generator number alpha). The generator is the product of factors (''x'' - &amp;alpha;&lt;sup&gt;''n''&lt;/sup&gt;), starting with ''n''=0 for QR codes.

For example:
''g''&lt;sub&gt;4&lt;/sub&gt;(''x'') = (''x'' - &amp;alpha;&lt;sup&gt;0&lt;/sup&gt;) (''x'' - &amp;alpha;&lt;sup&gt;1&lt;/sup&gt;) (''x'' - &amp;alpha;&lt;sup&gt;2&lt;/sup&gt;) (''x'' - &amp;alpha;&lt;sup&gt;3&lt;/sup&gt;) 

The same as (x + a&lt;sup&gt;''i''&lt;/sup&gt;) because of GF(2^8). 

''g''&lt;sub&gt;4&lt;/sub&gt;(''x'') = ''x''&lt;sup&gt;4&lt;/sup&gt; - (&amp;alpha;&lt;sup&gt;3&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;2&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;1&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;0&lt;/sup&gt;) ''x''&lt;sup&gt;3&lt;/sup&gt; + ((&amp;alpha;&lt;sup&gt;0&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;1&lt;/sup&gt;) (&amp;alpha;&lt;sup&gt;2&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;3&lt;/sup&gt;)+(&amp;alpha;&lt;sup&gt;5&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;1&lt;/sup&gt;)) ''x''&lt;sup&gt;2&lt;/sup&gt; + (&amp;alpha;&lt;sup&gt;6&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;5&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;4&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;3&lt;/sup&gt;) ''x'' +&amp;alpha;&lt;sup&gt;6&lt;/sup&gt;

''g''&lt;sub&gt;4&lt;/sub&gt;(''x'') = ''x''&lt;sup&gt;4&lt;/sup&gt; - (&amp;alpha;&lt;sup&gt;3&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;2&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;1&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;0&lt;/sup&gt;) ''x''&lt;sup&gt;3&lt;/sup&gt; + (&amp;alpha;&lt;sup&gt;2&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;3&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;3&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;4&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;5&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;1&lt;/sup&gt;) ''x''&lt;sup&gt;2&lt;/sup&gt; + (&amp;alpha;&lt;sup&gt;6&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;5&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;4&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;3&lt;/sup&gt;) ''x'' +&amp;alpha;&lt;sup&gt;6&lt;/sup&gt; 

''g''&lt;sub&gt;4&lt;/sub&gt;(''x'') = ''x''&lt;sup&gt;4&lt;/sup&gt; - (&amp;alpha;&lt;sup&gt;3&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;2&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;1&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;0&lt;/sup&gt;) ''x''&lt;sup&gt;3&lt;/sup&gt; + (&amp;alpha;&lt;sup&gt;5&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;4&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;2&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;1&lt;/sup&gt;) ''x''&lt;sup&gt;2&lt;/sup&gt; + (&amp;alpha;&lt;sup&gt;6&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;5&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;4&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;3&lt;/sup&gt;) ''x'' +&amp;alpha;&lt;sup&gt;6&lt;/sup&gt; 

''g''&lt;sub&gt;4&lt;/sub&gt;(''x'') = &lt;kbd&gt;01&lt;/kbd&gt; ''x''&lt;sup&gt;4&lt;/sup&gt; + &lt;kbd&gt;0f&lt;/kbd&gt; ''x''&lt;sup&gt;3&lt;/sup&gt; + &lt;kbd&gt;36&lt;/kbd&gt; ''x''&lt;sup&gt;2&lt;/sup&gt; + &lt;kbd&gt;78&lt;/kbd&gt; ''x'' + &lt;kbd&gt;40&lt;/kbd&gt;

Here is a function that computes the generator polynomial for a given number of error correction symbols.
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_generator_poly(nsym):
    '''Generate an irreducible generator polynomial (necessary to encode a message into Reed-Solomon)'''
    g = [1]
    for i in range(0, nsym):
        g = gf_poly_mul(g, [1, gf_pow(2, i)])
    return g
&lt;/syntaxhighlight&gt;
This function is somewhat inefficient in that it allocates successively larger arrays for &lt;kbd&gt;g&lt;/kbd&gt;. While this is unlikely to be a performance problem in practice, readers who are inveterate optimizers may find it interesting to rewrite it so that &lt;kbd&gt;g&lt;/kbd&gt; is only allocated once, or you can compute once and memorize g since it is fixed for a given nsym, so you can reuse g.

====Polynomial division====
Several algorithms for polynomial division exist, the simplest one that is often taught in elementary school is [[w:Polynomial_long_division|long division]]. This example shows the calculation for the message &lt;kbd&gt;12 34 56&lt;/kbd&gt;.
                &lt;u&gt;              12 da df&lt;/u&gt;
 01 0f 36 78 40 ) 12 34 56 00 00 00 00
                ^ &lt;u&gt;12 ee 2b 23 f4&lt;/u&gt;
                     da 7d 23 f4 00
                   ^ &lt;u&gt;da a2 85 79 84&lt;/u&gt;
                        df a6 8d 84 00
                      ^ &lt;u&gt;df 91 6b fc d9&lt;/u&gt;
                           37 e6 78 d9

Note: The concepts of polynomial long division apply, but there are a few important differences: When computing the resulting terms/coefficients that will be Galois Field subtracted from the divisor, bitwise carryless multiplication is performed and the result &quot;bitstream&quot; is XORed from the first encountered MSB with the chosen primitive polynomial until the answer is less than the Galois Field value, in this case, 256. The XOR &quot;subtractions&quot; are then performed as usual.

To illustrate the method for one operation (0x12 * 0x36):
 
     00010010 ( 12 )
   x &lt;u&gt;00110110&lt;/u&gt; ( 36 )
    00110110
 &lt;u&gt;00110110    &lt;/u&gt;
 001100001100
  ^100011101   &lt;-- XOR with primitive polynomial value (11D)...
 000100110110
   ^100011101  &lt;-- ...until answer is less than 256.
     00101011
        2   b

The remainder is concatenated with the message, so the encoded message is &lt;kbd&gt;12 34 56 37 e6 78 d9&lt;/kbd&gt;.

However, long division is quite slow as it requires a lot of recursive iterations to terminate. More efficient strategies can be devised, such as using [[synthetic division]] (also called Horner's method, a good tutorial video can be found on [https://www.khanacademy.org/math/algebra2/polynomial_and_rational/synthetic-division/v/synthetic-division Khan Academy]). Here is a function that implements [[w:Synthetic_division#Expanded_synthetic_division|extended synthetic division]] of GF(2^p) polynomials (extended because the divisor is a polynomial instead of a monomial):

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_poly_div(dividend, divisor):
    '''Fast polynomial division by using Extended Synthetic Division and optimized for GF(2^p) computations
    (doesn't work with standard polynomials outside of this galois field, see the Wikipedia article for generic algorithm).'''
    # CAUTION: this function expects polynomials to follow the opposite convention at decoding:
    # the terms must go from the biggest to lowest degree (while most other functions here expect
    # a list from lowest to biggest degree). eg: 1 + 2x + 5x^2 = [5, 2, 1], NOT [1, 2, 5]

    msg_out = list(dividend) # Copy the dividend
    #normalizer = divisor[0] # precomputing for performance
    for i in range(0, len(dividend) - (len(divisor)-1)):
        #msg_out[i] /= normalizer # for general polynomial division (when polynomials are non-monic), the usual way of using
                                  # synthetic division is to divide the divisor g(x) with its leading coefficient, but not needed here.
        coef = msg_out[i] # precaching
        if coef != 0: # log(0) is undefined, so we need to avoid that case explicitly (and it's also a good optimization).
            for j in range(1, len(divisor)): # in synthetic division, we always skip the first coefficient of the divisior,
                                              # because it's only used to normalize the dividend coefficient
                if divisor[j] != 0: # log(0) is undefined
                    msg_out[i + j] ^= gf_mul(divisor[j], coef) # equivalent to the more mathematically correct
                                                               # (but xoring directly is faster): msg_out[i + j] += -divisor[j] * coef

    # The resulting msg_out contains both the quotient and the remainder, the remainder being the size of the divisor
    # (the remainder has necessarily the same degree as the divisor -- not length but degree == length-1 -- since it's
    # what we couldn't divide from the dividend), so we compute the index where this separation is, and return the quotient and remainder.
    separator = -(len(divisor)-1)
    return msg_out[:separator], msg_out[separator:] # return quotient, remainder.
&lt;/syntaxhighlight&gt;

====Encoding main function====

And now, here's how to encode a message to get its RS code:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_encode_msg(msg_in, nsym):
    '''Reed-Solomon main encoding function'''
    gen = rs_generator_poly(nsym)

    # Pad the message, then divide it by the irreducible generator polynomial
    _, remainder = gf_poly_div(msg_in + [0] * (len(gen)-1), gen)
    # The remainder is our RS code! Just append it to our original message to get our full codeword (this represents a polynomial of max 256 terms)
    msg_out = msg_in + remainder
    # Return the codeword
    return msg_out
&lt;/syntaxhighlight&gt;

Simple, isn't it? Encoding is in fact the easiest part in Reed&amp;ndash;Solomon, and it's always the same approach (polynomial division). Decoding is the tough part of Reed&amp;ndash;Solomon, and you will find a lot of different algorithms depending on your needs, but we will touch on that later on.

This function is quite fast, but since encoding is quite critical, here is an enhanced encoding function that inlines the polynomial synthetic division, which is the form that you will most often find in Reed&amp;ndash;Solomon software libraries:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_encode_msg(msg_in, nsym):
    '''Reed-Solomon main encoding function, using polynomial division (algorithm Extended Synthetic Division)'''
    if (len(msg_in) + nsym) &gt; 255: raise ValueError(&quot;Message is too long (%i when max is 255)&quot; % (len(msg_in)+nsym))
    gen = rs_generator_poly(nsym)
    # Init msg_out with the values inside msg_in and pad with len(gen)-1 bytes (which is the number of ecc symbols).
    msg_out = [0] * (len(msg_in) + len(gen)-1)
    # Initializing the Synthetic Division with the dividend (= input message polynomial)
    msg_out[:len(msg_in)] = msg_in

    # Synthetic division main loop
    for i in range(len(msg_in)):
        # Note that it's msg_out here, not msg_in. Thus, we reuse the updated value at each iteration
        # (this is how Synthetic Division works: instead of storing in a temporary register the intermediate values,
        # we directly commit them to the output).
        coef = msg_out[i]

        # log(0) is undefined, so we need to manually check for this case. There's no need to check
        # the divisor here because we know it can't be 0 since we generated it.
        if coef != 0:
            # in synthetic division, we always skip the first coefficient of the divisior, because it's only used to normalize the dividend coefficient (which is here useless since the divisor, the generator polynomial, is always monic)
            for j in range(1, len(gen)):
                msg_out[i+j] ^= gf_mul(gen[j], coef) # equivalent to msg_out[i+j] += gf_mul(gen[j], coef)

    # At this point, the Extended Synthetic Divison is done, msg_out contains the quotient in msg_out[:len(msg_in)]
    # and the remainder in msg_out[len(msg_in):]. Here for RS encoding, we don't need the quotient but only the remainder
    # (which represents the RS code), so we can just overwrite the quotient with the input message, so that we get
    # our complete codeword composed of the message + code.
    msg_out[:len(msg_in)] = msg_in

    return msg_out
&lt;/syntaxhighlight&gt;
This algorithm is faster, but it's still quite slow for practical use, particularly in Python. There are some ways to optimize the speed by using various tricks, such as inlining (instead of gf_mul, replace by the operation to avoid a call), by precomputing (the logarithm of gen and of coef, or even by generating a multiplication table &amp;ndash; but it seems the latter does not work well in Python), by using statically typed constructs (assign gf_log and gf_exp to &lt;kbd&gt;array.array('i', [...])&lt;/kbd&gt;), by using memoryviews (like by changing all your lists to bytearrays), by running it with PyPy, or by converting the algorithm into a Cython or a C extension&lt;ref&gt;Optimizing a reed-solomon encoder, question on StackOverflow.com http://stackoverflow.com/questions/30363903/optimizing-a-reed-solomon-encoder-polynomial-division&lt;/ref&gt;.

This example shows the encode function applied to the message in the sample QR code introduced earlier. The calculated error correction symbols (on the second line) match the values decoded from the QR code.
&lt;pre&gt;
&gt;&gt;&gt; msg_in = [ 0x40, 0xd2, 0x75, 0x47, 0x76, 0x17, 0x32, 0x06,
...            0x27, 0x26, 0x96, 0xc6, 0xc6, 0x96, 0x70, 0xec ]
&gt;&gt;&gt; msg = rs_encode_msg(msg_in, 10)
&gt;&gt;&gt; for i in range(0,len(msg)):
...    print(hex(msg[i]), end=' ')
... 
0x40 0xd2 0x75 0x47 0x76 0x17 0x32 0x6 0x27 0x26 0x96 0xc6 0xc6 0x96 0x70 0xec
0xbc 0x2a 0x90 0x13 0x6b 0xaf 0xef 0xfd 0x4b 0xe0
&lt;/pre&gt;
''Python version note:'' The syntax for the &lt;kbd&gt;print&lt;/kbd&gt; function has changed, and this example uses the Python 3.0+ version. In previous versions of Python (particularly Python 2.x), replace the &lt;kbd&gt;print&lt;/kbd&gt; line with &lt;kbd&gt;print hex(msg[i]),&lt;/kbd&gt; (including the final comma) and &lt;kbd&gt;range&lt;/kbd&gt; by &lt;kbd&gt;xrange&lt;/kbd&gt;.

===RS decoding===

====Decoding outline====

Reed&amp;ndash;Solomon decoding is the process that, from a potentially corrupted message and a RS code, returns a corrected message. In other words, decoding is the process of repairing your message using the previously computed RS code.

Although there is only one way to encode a message with Reed&amp;ndash;Solomon, there are lots of different ways to decode them, and thus there are a lot of different decoding algorithms.

However, we can generally outline the decoding process in 5 steps&lt;ref&gt;Tilavat, V., &amp; Shukla, Y. (2014). Simplification of procedure for decoding Reed&amp;ndash;Solomon codes using various algorithms: an introductory survey. International Journal of Engineering Development and Research, 2(1), 279-283.&lt;/ref&gt;&lt;ref&gt;Sarwate, D. V., &amp; Morrison, R. D. (1990). Decoder malfunction in BCH decoders. Information Theory, IEEE Transactions on, 36(4), 884-889.&lt;/ref&gt;:

# Compute the '''syndromes polynomial'''. This allows us to analyze what characters are in error using Berlekamp-Massey (or another algorithm), and also to quickly check if the input message is corrupted at all.
# Compute the erasure/error '''locator polynomial''' (from the syndromes). This is computed by Berlekamp-Massey, and is a detector that will tell us exactly what characters are corrupted.
# Compute the erasure/error '''evaluator polynomial''' (from the syndromes and erasure/error locator polynomial). Necessary to evaluate how much the characters were tampered (ie, helps to compute the magnitude).
# Compute the erasure/error '''magnitude polynomial''' (from all 3 polynomials above): this polynomial can also be called the corruption polynomial, since in fact it exactly stores the values that need to be subtracted from the received message to get the original, correct message (i.e., with correct values for erased characters). In other words, at this point, we extracted the noise and stored it in this polynomial, and we just have to remove this noise from the input message to repair it.
# '''Repair the input message''' simply by subtracting the magnitude polynomial from the input message.

We will describe each of those five steps below.

In addition, decoders can also be classified by the type of error they can repair: erasures (we know the location of the corrupted characters but not the magnitude), errors (we ignore both the location and magnitude), or a mix of errors-and-erasures. We will describe how to support all of these.

====Syndrome calculation====
Decoding a Reed&amp;ndash;Solomon message involves several steps. The first step is to calculate the &quot;syndrome&quot; of the message. Treat the message as a polynomial and evaluate it at &amp;alpha;&lt;sup&gt;0&lt;/sup&gt;, &amp;alpha;&lt;sup&gt;1&lt;/sup&gt;, &amp;alpha;&lt;sup&gt;2&lt;/sup&gt;, ..., &amp;alpha;&lt;sup&gt;''n''&lt;/sup&gt;. Since these are the zeros of the generator polynomial, the result should be zero if the scanned message is undamaged (this can be used to check if the message is corrupted, and after correction of a corrupted message if the message was completely repaired). If not, the syndromes contain all the information necessary to determine the correction that should be made. It is simple to write a function to calculate the syndromes.
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_calc_syndromes(msg, nsym):
    '''Given the received codeword msg and the number of error correcting symbols (nsym), computes the syndromes polynomial.
    Mathematically, it's essentially equivalent to a Fourrier Transform (Chien search being the inverse).
    '''
    # Note the &quot;[0] +&quot; : we add a 0 coefficient for the lowest degree (the constant). This effectively shifts the syndrome, and will shift every computations depending on the syndromes (such as the errors locator polynomial, errors evaluator polynomial, etc. but not the errors positions).
    # This is not necessary, you can adapt subsequent computations to start from 0 instead of skipping the first iteration (ie, the often seen range(1, n-k+1)),
    synd = [0] * nsym
    for i in range(0, nsym):
        synd[i] = gf_poly_eval(msg, gf_pow(2,i))
    return [0] + synd # pad with one 0 for mathematical precision (else we can end up with weird calculations sometimes)
&lt;/syntaxhighlight&gt;

Continuing the example, we see that the syndromes of the original codeword without any corruption are indeed zero. Introducing a corruption of at least one character into the message or its RS code gives nonzero syndromes.
&lt;pre&gt;
&gt;&gt;&gt; synd = rs_calc_syndromes(msg, 10)
&gt;&gt;&gt; print(synd)
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] # not corrupted message = all 0 syndromes
&gt;&gt;&gt; msg[0] = 0  # deliberately damage the message
&gt;&gt;&gt; synd = rs_calc_syndromes(msg, 10)
&gt;&gt;&gt; print(synd)
[0, 64, 192, 93, 231, 52, 92, 228, 49, 83, 245] # when corrupted, the syndromes will be non zero
&lt;/pre&gt;

Here is the code to automate this checking:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_check(msg, nsym):
    '''Returns true if the message + ecc has no error or false otherwise (may not always catch a wrong decoding or a wrong message, particularly if there are too many errors -- above the Singleton bound --, but it usually does)'''
    return ( max(rs_calc_syndromes(msg, nsym)) == 0 )
&lt;/syntaxhighlight&gt;

====Erasure correction====
It is simplest to correct mistakes in the code if the locations of the mistakes are already known. This is known as '''erasure correction'''. It is possible to correct one erased symbol (ie, character) for each error-correction symbol added to the code. If the error locations are not known, two EC symbols are needed for each symbol error (so you can correct twice less errors than erasures). This makes erasure correction useful in practice if part of the QR code being scanned is covered or physically torn away. It may be difficult for a scanner to determine that this has happened, though, so not all QR code scanners can perform erasure correction.

Now that we already have the syndromes, we need to compute the locator polynomial. This is easy:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_find_errata_locator(e_pos):
    '''Compute the erasures/errors/errata locator polynomial from the erasures/errors/errata positions
       (the positions must be relative to the x coefficient, eg: &quot;hello worldxxxxxxxxx&quot; is tampered to &quot;h_ll_ worldxxxxxxxxx&quot;
       with xxxxxxxxx being the ecc of length n-k=9, here the string positions are [1, 4], but the coefficients are reversed
       since the ecc characters are placed as the first coefficients of the polynomial, thus the coefficients of the
       erased characters are n-1 - [1, 4] = [18, 15] = erasures_loc to be specified as an argument.'''

    e_loc = [1] # just to init because we will multiply, so it must be 1 so that the multiplication starts correctly without nulling any term
    # erasures_loc = product(1 - x*alpha**i) for i in erasures_pos and where alpha is the alpha chosen to evaluate polynomials.
    for i in e_pos:
        e_loc = gf_poly_mul( e_loc, gf_poly_add([1], [gf_pow(2, i), 0]) )
    return e_loc
&lt;/syntaxhighlight&gt;

Next, computing the erasure/error evaluator polynomial from the locator polynomial is easy, it's simply a polynomial multiplication followed by a polynomial division (that you can replace by a list slicing because that's the effect we want in the end):

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_find_error_evaluator(synd, err_loc, nsym):
    '''Compute the error (or erasures if you supply sigma=erasures locator polynomial, or errata) evaluator polynomial Omega
       from the syndrome and the error/erasures/errata locator Sigma.'''

    # Omega(x) = [ Synd(x) * Error_loc(x) ] mod x^(n-k+1)
    _, remainder = gf_poly_div( gf_poly_mul(synd, err_loc), ([1] + [0]*(nsym+1)) ) # first multiply syndromes * errata_locator, then do a
                                                                                   # polynomial division to truncate the polynomial to the
                                                                                   # required length

    # Faster way that is equivalent
    #remainder = gf_poly_mul(synd, err_loc) # first multiply the syndromes with the errata locator polynomial
    #remainder = remainder[len(remainder)-(nsym+1):] # then slice the list to truncate it (which represents the polynomial), which
                                                                          # is equivalent to dividing by a polynomial of the length we want

    return remainder
&lt;/syntaxhighlight&gt;

Finally, the [[w:Forney algorithm|Forney algorithm]] is used to calculate the correction values (also called the error magnitude polynomial). It is implemented in the function below.

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_correct_errata(msg_in, synd, err_pos): # err_pos is a list of the positions of the errors/erasures/errata
    '''Forney algorithm, computes the values (error magnitude) to correct the input message.'''
    # calculate errata locator polynomial to correct both errors and erasures (by combining the errors positions given by the error locator polynomial found by BM with the erasures positions given by caller)
    coef_pos = [len(msg_in) - 1 - p for p in err_pos] # need to convert the positions to coefficients degrees for the errata locator algo to work (eg: instead of [0, 1, 2] it will become [len(msg)-1, len(msg)-2, len(msg) -3])
    err_loc = rs_find_errata_locator(coef_pos)
    # calculate errata evaluator polynomial (often called Omega or Gamma in academic papers)
    err_eval = rs_find_error_evaluator(synd[::-1], err_loc, len(err_loc)-1)[::-1]

    # Second part of Chien search to get the error location polynomial X from the error positions in err_pos (the roots of the error locator polynomial, ie, where it evaluates to 0)
    X = [] # will store the position of the errors
    for i in range(0, len(coef_pos)):
        l = 255 - coef_pos[i]
        X.append( gf_pow(2, -l) )

    # Forney algorithm: compute the magnitudes
    E = [0] * (len(msg_in)) # will store the values that need to be corrected (substracted) to the message containing errors. This is sometimes called the error magnitude polynomial.
    Xlength = len(X)
    for i, Xi in enumerate(X):

        Xi_inv = gf_inverse(Xi)

        # Compute the formal derivative of the error locator polynomial (see Blahut, Algebraic codes for data transmission, pp 196-197).
        # the formal derivative of the errata locator is used as the denominator of the Forney Algorithm, which simply says that the ith error value is given by error_evaluator(gf_inverse(Xi)) / error_locator_derivative(gf_inverse(Xi)). See Blahut, Algebraic codes for data transmission, pp 196-197.
        err_loc_prime_tmp = []
        for j in range(0, Xlength):
            if j != i:
                err_loc_prime_tmp.append( gf_sub(1, gf_mul(Xi_inv, X[j])) )
        # compute the product, which is the denominator of the Forney algorithm (errata locator derivative)
        err_loc_prime = 1
        for coef in err_loc_prime_tmp:
            err_loc_prime = gf_mul(err_loc_prime, coef)
        # equivalent to: err_loc_prime = functools.reduce(gf_mul, err_loc_prime_tmp, 1)

        # Compute y (evaluation of the errata evaluator polynomial)
        # This is a more faithful translation of the theoretical equation contrary to the old forney method. Here it is an exact reproduction:
        # Yl = omega(Xl.inverse()) / prod(1 - Xj*Xl.inverse()) for j in len(X)
        y = gf_poly_eval(err_eval[::-1], Xi_inv) # numerator of the Forney algorithm (errata evaluator evaluated)
        y = gf_mul(gf_pow(Xi, 1), y)
        
        # Check: err_loc_prime (the divisor) should not be zero.
        if err_loc_prime == 0:
            raise ReedSolomonError(&quot;Could not find error magnitude&quot;)    # Could not find error magnitude

        # Compute the magnitude
        magnitude = gf_div(y, err_loc_prime) # magnitude value of the error, calculated by the Forney algorithm (an equation in fact): dividing the errata evaluator with the errata locator derivative gives us the errata magnitude (ie, value to repair) the ith symbol
        E[err_pos[i]] = magnitude # store the magnitude for this error into the magnitude polynomial

    # Apply the correction of values to get our message corrected! (note that the ecc bytes also gets corrected!)
    # (this isn't the Forney algorithm, we just apply the result of decoding here)
    msg_in = gf_poly_add(msg_in, E) # equivalent to Ci = Ri - Ei where Ci is the correct message, Ri the received (senseword) message, and Ei the errata magnitudes (minus is replaced by XOR since it's equivalent in GF(2^p)). So in fact here we substract from the received message the errors magnitude, which logically corrects the value to what it should be.
    return msg_in
&lt;/syntaxhighlight&gt;

''Mathematics note:'' The denominator of the expression for the error value is the [[w:Formal derivative|formal derivative]] of the error locator polynomial &lt;kbd&gt;q&lt;/kbd&gt;. This is calculated by the usual procedure of replacing each term ''c''&lt;sub&gt;''n''&lt;/sub&gt;&amp;nbsp;''x''&lt;sup&gt;''n''&lt;/sup&gt; with ''n''&amp;nbsp;''c''&lt;sub&gt;''n''&lt;/sub&gt;&amp;nbsp;''x''&lt;sup&gt;''n''-1&lt;/sup&gt;. Since we're working in a field of [[w:Characteristic (algebra)|characteristic]] two, ''n''&amp;nbsp;''c''&lt;sub&gt;''n''&lt;/sub&gt; is equal to ''c''&lt;sub&gt;''n''&lt;/sub&gt; when ''n'' is odd, and 0 when ''n'' is even. Thus, we can simply remove the even coefficients (resulting in the polynomial &lt;kbd&gt;qprime&lt;/kbd&gt;) and evaluate &lt;kbd&gt;qprime(x&lt;sup&gt;2&lt;/sup&gt;)&lt;/kbd&gt;.

''Python note:'' This function uses [::-1] to inverse the order of the elements in a list. This is necessary because the functions do not all use the same ordering convention (ie, some use the least item first, others use the biggest item first). It also use a [[w:List_comprehension#Python|list comprehension]], which is simply a concise way to write a for loop where items are appended in a list, but the Python interpreter can optimize this a bit more than a loop.

Continuing the example, here we use &lt;kbd&gt;rs_correct_errata&lt;/kbd&gt; to restore the first byte of the message.

&lt;pre&gt;
&gt;&gt;&gt; msg[0] = 0
&gt;&gt;&gt; synd = rs_calc_syndromes(msg, 10)
&gt;&gt;&gt; msg = rs_correct_errata(msg, synd, [0]) # [0] is the list of the erasures locations, here it's the first character, at position 0
&gt;&gt;&gt; print(hex(msg[0]))
0x40
&lt;/pre&gt;

====Error correction====
In the more likely situation where the error locations are unknown (what we usually call '''errors''', in opposition to '''erasures''' where the locations are known), we will use the same steps as for erasures, but we now need additional steps to find the location. The [[w:Berlekamp&amp;ndash;Massey algorithm|Berlekamp&amp;ndash;Massey algorithm]] is used to calculate the error '''locator polynomial''', which we can use later on to determine the errors locations:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_find_error_locator(synd, nsym, erase_loc=None, erase_count=0):
    '''Find error/errata locator and evaluator polynomials with Berlekamp-Massey algorithm'''
    # The idea is that BM will iteratively estimate the error locator polynomial.
    # To do this, it will compute a Discrepancy term called Delta, which will tell us if the error locator polynomial needs an update or not
    # (hence why it's called discrepancy: it tells us when we are getting off board from the correct value).

    # Init the polynomials
    if erase_loc: # if the erasure locator polynomial is supplied, we init with its value, so that we include erasures in the final locator polynomial
        err_loc = list(erase_loc)
        old_loc = list(erase_loc)
    else:
        err_loc = [1] # This is the main variable we want to fill, also called Sigma in other notations or more formally the errors/errata locator polynomial.
        old_loc = [1] # BM is an iterative algorithm, and we need the errata locator polynomial of the previous iteration in order to update other necessary variables.
    #L = 0 # update flag variable, not needed here because we use an alternative equivalent way of checking if update is needed (but using the flag could potentially be faster depending on if using length(list) is taking linear time in your language, here in Python it's constant so it's as fast.

    # Fix the syndrome shifting: when computing the syndrome, some implementations may prepend a 0 coefficient for the lowest degree term (the constant). This is a case of syndrome shifting, thus the syndrome will be bigger than the number of ecc symbols (I don't know what purpose serves this shifting). If that's the case, then we need to account for the syndrome shifting when we use the syndrome such as inside BM, by skipping those prepended coefficients.
    # Another way to detect the shifting is to detect the 0 coefficients: by definition, a syndrome does not contain any 0 coefficient (except if there are no errors/erasures, in this case they are all 0). This however doesn't work with the modified Forney syndrome, which set to 0 the coefficients corresponding to erasures, leaving only the coefficients corresponding to errors.
    synd_shift = len(synd) - nsym

    for i in range(0, nsym-erase_count): # generally: nsym-erase_count == len(synd), except when you input a partial erase_loc and using the full syndrome instead of the Forney syndrome, in which case nsym-erase_count is more correct (len(synd) will fail badly with IndexError).
        if erase_loc: # if an erasures locator polynomial was provided to init the errors locator polynomial, then we must skip the FIRST erase_count iterations (not the last iterations, this is very important!)
            K = erase_count+i+synd_shift
        else: # if erasures locator is not provided, then either there's no erasures to account or we use the Forney syndromes, so we don't need to use erase_count nor erase_loc (the erasures have been trimmed out of the Forney syndromes).
            K = i+synd_shift

        # Compute the discrepancy Delta
        # Here is the close-to-the-books operation to compute the discrepancy Delta: it's a simple polynomial multiplication of error locator with the syndromes, and then we get the Kth element.
        #delta = gf_poly_mul(err_loc[::-1], synd)[K] # theoretically it should be gf_poly_add(synd[::-1], [1])[::-1] instead of just synd, but it seems it's not absolutely necessary to correctly decode.
        # But this can be optimized: since we only need the Kth element, we don't need to compute the polynomial multiplication for any other element but the Kth. Thus to optimize, we compute the polymul only at the item we need, skipping the rest (avoiding a nested loop, thus we are linear time instead of quadratic).
        # This optimization is actually described in several figures of the book &quot;Algebraic codes for data transmission&quot;, Blahut, Richard E., 2003, Cambridge university press.
        delta = synd[K]
        for j in range(1, len(err_loc)):
            delta ^= gf_mul(err_loc[-(j+1)], synd[K - j]) # delta is also called discrepancy. Here we do a partial polynomial multiplication (ie, we compute the polynomial multiplication only for the term of degree K). Should be equivalent to brownanrs.polynomial.mul_at().
        #print &quot;delta&quot;, K, delta, list(gf_poly_mul(err_loc[::-1], synd)) # debugline

        # Shift polynomials to compute the next degree
        old_loc = old_loc + [0]

        # Iteratively estimate the errata locator and evaluator polynomials
        if delta != 0: # Update only if there's a discrepancy
            if len(old_loc) &gt; len(err_loc): # Rule B (rule A is implicitly defined because rule A just says that we skip any modification for this iteration)
            #if 2*L &lt;= K+erase_count: # equivalent to len(old_loc) &gt; len(err_loc), as long as L is correctly computed
                # Computing errata locator polynomial Sigma
                new_loc = gf_poly_scale(old_loc, delta)
                old_loc = gf_poly_scale(err_loc, gf_inverse(delta)) # effectively we are doing err_loc * 1/delta = err_loc // delta
                err_loc = new_loc
                # Update the update flag
                #L = K - L # the update flag L is tricky: in Blahut's schema, it's mandatory to use `L = K - L - erase_count` (and indeed in a previous draft of this function, if you forgot to do `- erase_count` it would lead to correcting only 2*(errors+erasures) &lt;= (n-k) instead of 2*errors+erasures &lt;= (n-k)), but in this latest draft, this will lead to a wrong decoding in some cases where it should correctly decode! Thus you should try with and without `- erase_count` to update L on your own implementation and see which one works OK without producing wrong decoding failures.

            # Update with the discrepancy
            err_loc = gf_poly_add(err_loc, gf_poly_scale(old_loc, delta))

    # Check if the result is correct, that there's not too many errors to correct
    while len(err_loc) and err_loc[0] == 0: del err_loc[0] # drop leading 0s, else errs will not be of the correct size
    errs = len(err_loc) - 1
    if (errs-erase_count) * 2 + erase_count &gt; nsym:
        raise ReedSolomonError(&quot;Too many errors to correct&quot;)    # too many errors to correct

    return err_loc
&lt;/syntaxhighlight&gt;

Then, using the error locator polynomial, we simply use a brute-force approach called trial substitution to find the zeros of this polynomial, which identifies the error locations (ie, the index of the characters that need to be corrected). A more efficient algorithm called Chien search exists, which avoids recomputing the whole evaluation at each iteration step, but this algorithm is left as an exercise to the reader.

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_find_errors(err_loc, nmess): # nmess is len(msg_in)
    '''Find the roots (ie, where evaluation = zero) of error polynomial by brute-force trial, this is a sort of Chien's search
    (but less efficient, Chien's search is a way to evaluate the polynomial such that each evaluation only takes constant time).'''
    errs = len(err_loc) - 1
    err_pos = []
    for i in range(nmess): # normally we should try all 2^8 possible values, but here we optimize to just check the interesting symbols
        if gf_poly_eval(err_loc, gf_pow(2, i)) == 0: # It's a 0? Bingo, it's a root of the error locator polynomial,
                                                                                   # in other terms this is the location of an error
            err_pos.append(nmess - 1 - i)
    # Sanity check: the number of errors/errata positions found should be exactly the same as the length of the errata locator polynomial
    if len(err_pos) != errs:
        # couldn't find error locations
        raise ReedSolomonError(&quot;Too many (or few) errors found by Chien Search for the errata locator polynomial!&quot;)
    return err_pos
&lt;/syntaxhighlight&gt;
''Mathematics note:'' When the error locator polynomial is linear (&lt;kbd&gt;err_poly&lt;/kbd&gt; has length 2), it can be solved easily without resorting to a brute-force approach. The function presented above does not take advantage of this fact, but the interested reader may wish to implement the more efficient solution. Similarly, when the error locator is quadratic, it can be solved by using a [[w:Quadratic equation#Generalization of quadratic equation|generalization of the quadratic formula]]. A more ambitious reader may wish to implement this procedure as well.

Here is an example where three errors in the message are corrected:

&lt;pre&gt;
&gt;&gt;&gt; print(hex(msg[10]))
0x96
&gt;&gt;&gt; msg[0] = 6
&gt;&gt;&gt; msg[10] = 7
&gt;&gt;&gt; msg[20] = 8
&gt;&gt;&gt; synd = rs_calc_syndromes(msg, 10)
&gt;&gt;&gt; err_loc = rs_find_error_locator(synd, nsym)
&gt;&gt;&gt; pos = rs_find_errors(err_loc[::-1], len(msg)) # find the errors locations
&gt;&gt;&gt; print(pos)
[20, 10, 0]
&gt;&gt;&gt; msg = rs_correct_errata(msg, synd, pos)
&gt;&gt;&gt; print(hex(msg[10]))
0x96
&lt;/pre&gt;

====Error and erasure correction====

It is possible for a Reed&amp;ndash;Solomon decoder to decode both erasures and errors at the same time, up to a limit (called the Singleton Bound) of &lt;kbd&gt;2*e+v &lt;= (n-k)&lt;/kbd&gt;, where &lt;kbd&gt;e&lt;/kbd&gt; is the number of errors, &lt;kbd&gt;v&lt;/kbd&gt; the number of erasures and &lt;kbd&gt;(n-k)&lt;/kbd&gt; the number of RS code characters (called &lt;kbd&gt;nsym&lt;/kbd&gt; in the code). Basically, it means that for every erasures, you just need one RS code character to repair it, while for every errors you need two RS code characters (because you need to find the position in addition of the value/magnitude to correct). Such a decoder is called an errors-and-erasures decoder, or an '''errata decoder'''.

In order to correct both errors and erasures, we must prevent the erasures from interfering with the error location process. This can be done by calculating the Forney syndromes, as follows.

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_forney_syndromes(synd, pos, nmess):
    # Compute Forney syndromes, which computes a modified syndromes to compute only errors (erasures are trimmed out). Do not confuse this with Forney algorithm, which allows to correct the message based on the location of errors.
    erase_pos_reversed = [nmess-1-p for p in pos] # prepare the coefficient degree positions (instead of the erasures positions)

    # Optimized method, all operations are inlined
    fsynd = list(synd[1:])      # make a copy and trim the first coefficient which is always 0 by definition
    for i in range(0, len(pos)):
        x = gf_pow(2, erase_pos_reversed[i])
        for j in range(0, len(fsynd) - 1):
            fsynd[j] = gf_mul(fsynd[j], x) ^ fsynd[j + 1]

    # Equivalent, theoretical way of computing the modified Forney syndromes: fsynd = (erase_loc * synd) % x^(n-k)
    # See Shao, H. M., Truong, T. K., Deutsch, L. J., &amp; Reed, I. S. (1986, April). A single chip VLSI Reed-Solomon decoder. In Acoustics, Speech, and Signal Processing, IEEE International Conference on ICASSP'86. (Vol. 11, pp. 2151-2154). IEEE.ISO 690
    #erase_loc = rs_find_errata_locator(erase_pos_reversed, generator=generator) # computing the erasures locator polynomial
    #fsynd = gf_poly_mul(erase_loc[::-1], synd[1:]) # then multiply with the syndrome to get the untrimmed forney syndrome
    #fsynd = fsynd[len(pos):] # then trim the first erase_pos coefficients which are useless. Seems to be not necessary, but this reduces the computation time later in BM (thus it's an optimization).

    return fsynd
&lt;/syntaxhighlight&gt;

The Forney syndromes can then be used in place of the regular syndromes in the error location process.

The function &lt;kbd&gt;rs_correct_msg&lt;/kbd&gt; below brings the complete procedure together. Erasures are indicated by providing &lt;kbd&gt;erase_pos&lt;/kbd&gt;, a list of erasures index positions in the message &lt;kbd&gt;msg_in&lt;/kbd&gt; (the full received message: original message + ecc).

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_correct_msg(msg_in, nsym, erase_pos=None):
    '''Reed-Solomon main decoding function'''
    if len(msg_in) &gt; 255: # can't decode, message is too big
        raise ValueError(&quot;Message is too long (%i when max is 255)&quot; % len(msg_in))

    msg_out = list(msg_in)     # copy of message
    # erasures: set them to null bytes for easier decoding (but this is not necessary, they will be corrected anyway, but debugging will be easier with null bytes because the error locator polynomial values will only depend on the errors locations, not their values)
    if erase_pos is None:
        erase_pos = []
    else:
        for e_pos in erase_pos:
            msg_out[e_pos] = 0
    # check if there are too many erasures to correct (beyond the Singleton bound)
    if len(erase_pos) &gt; nsym: raise ReedSolomonError(&quot;Too many erasures to correct&quot;)
    # prepare the syndrome polynomial using only errors (ie: errors = characters that were either replaced by null byte
    # or changed to another character, but we don't know their positions)
    synd = rs_calc_syndromes(msg_out, nsym)
    # check if there's any error/erasure in the input codeword. If not (all syndromes coefficients are 0), then just return the message as-is.
    if max(synd) == 0:
        return msg_out[:-nsym], msg_out[-nsym:]  # no errors

    # compute the Forney syndromes, which hide the erasures from the original syndrome (so that BM will just have to deal with errors, not erasures)
    fsynd = rs_forney_syndromes(synd, erase_pos, len(msg_out))
    # compute the error locator polynomial using Berlekamp-Massey
    err_loc = rs_find_error_locator(fsynd, nsym, erase_count=len(erase_pos))
    # locate the message errors using Chien search (or brute-force search)
    err_pos = rs_find_errors(err_loc[::-1] , len(msg_out))
    if err_pos is None:
        raise ReedSolomonError(&quot;Could not locate error&quot;)    # error location failed

    # Find errors values and apply them to correct the message
    # compute errata evaluator and errata magnitude polynomials, then correct errors and erasures
    msg_out = rs_correct_errata(msg_out, synd, (erase_pos + err_pos)) # note that we here use the original syndrome, not the forney syndrome
                                                                                                                                  # (because we will correct both errors and erasures, so we need the full syndrome)
    # check if the final message is fully repaired
    synd = rs_calc_syndromes(msg_out, nsym)
    if max(synd) &gt; 0:
        raise ReedSolomonError(&quot;Could not correct message&quot;)     # message could not be repaired
    # return the successfully decoded message
    return msg_out[:-nsym], msg_out[-nsym:] # also return the corrected ecc block so that the user can check()
&lt;/syntaxhighlight&gt;
''Python note:'' The lists &lt;kbd&gt;erase_pos&lt;/kbd&gt; and &lt;kbd&gt;err_pos&lt;/kbd&gt; are concatenated with the &lt;kbd&gt;+&lt;/kbd&gt; operator.

This is the last piece needed for a fully operational error-and-erasure correcting Reed&amp;ndash;Solomon decoder. If you want to delve more into the inner workings of errata (errors-and-erasures) decoders, you can read the excellent book &quot;Algebraic Codes for Data Transmission&quot; (2003) by Richard E. Blahut.

Mathematics note: in some software implementations, particularly the ones using a language optimized for linear algebra and matrix operations, you will find that the algorithms (encoding, Berlekamp-Massey, etc.) will seem totally different and use the Fourier Transform. This is because this is totally equivalent: when stated in the jargon of spectral estimation, decoding Reed&amp;ndash;Solomon consists of a Fourier transform (syndrome computer), followed by a spectral analysis (Berlekamp-Massey or Euclidian algorithm), followed by an inverse Fourier transform (Chien search). See the Blahut book for more info&lt;ref&gt;Richard E. Blahut, &quot;Algebraic Codes for Data Transmission&quot;, 2003, chapter 7.6 &quot;Decoding in Time Domain&quot;&lt;/ref&gt;. Indeed, if you are using a programming language optimized for linear algebra, or if you want to use fast linear algebra libraries, it can be a very good idea to use Fourier Transform since it's very fast nowadays (particularly the Fast Fourier Transform or Number Theoretic Transform&lt;ref name=&quot;ntt&quot;/&gt;).

===Wrapping up with an example===

Here's an example of how to use the functions you have just made, and how to decode both errors-and-erasures:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Configuration of the parameters and input message
prim = 0x11d
n = 20 # set the size you want, it must be &gt; k, the remaining n-k symbols will be the ECC code (more is better)
k = 11 # k = len(message)
message = &quot;hello world&quot; # input message

# Initializing the log/antilog tables
init_tables(prim)

# Encoding the input message
mesecc = rs_encode_msg([ord(x) for x in message], n-k)
print(&quot;Original: %s&quot; % mesecc)

# Tampering 6 characters of the message (over 9 ecc symbols, so we are above the Singleton Bound)
mesecc[0] = 0
mesecc[1] = 2
mesecc[2] = 2
mesecc[3] = 2
mesecc[4] = 2
mesecc[5] = 2
print(&quot;Corrupted: %s&quot; % mesecc)

# Decoding/repairing the corrupted message, by providing the locations of a few erasures, we get below the Singleton Bound
# Remember that the Singleton Bound is: 2*e+v &lt;= (n-k)
corrected_message, corrected_ecc = rs_correct_msg(mesecc, n-k, erase_pos=[0, 1, 2])
print(&quot;Repaired: %s&quot; % (corrected_message+corrected_ecc))
print(''.join([chr(x) for x in corrected_message]))
&lt;/syntaxhighlight&gt;

This should output the following:

&lt;pre&gt;
Original:  [104, 101, 108, 108, 111,  32, 119, 111, 114, 108, 100, 145, 124, 96, 105, 94, 31, 179, 149, 163]
Corrupted: [  0,   2,   2,   2,   2,   2, 119, 111, 114, 108, 100, 145, 124, 96, 105, 94, 31, 179, 149, 163]
Repaired:  [104, 101, 108, 108, 111,  32, 119, 111, 114, 108, 100, 145, 124, 96, 105, 94, 31, 179, 149, 163]
hello world
&lt;/pre&gt;

==Conclusion and going further==

The basic principles of Reed–Solomon codes have been presented in this essay. Working Python code for a particular implementation (QR codes using a generic Reed&amp;ndash;Solomon codec to correct misreadings) has been included. The code presented here is quite generic and can be used for any purpose beyond QR codes where you need to correct errors/erasures, such as file protection, networking, etc. Many variations and refinements of these ideas are possible, since coding theory is a very rich field of study.

If your code is just intended for your own data (eg, you want to be able to generate and read your own QR codes), then you're fine, but if you intend to work with data provided by others (eg, you want to read and decode QR codes of other apps), then this decoder probably won't be enough, because there are some hidden parameters that were here fixed for simplicity (namely: the generator/alpha number and the first consecutive root). If you want to decode Reed&amp;ndash;Solomon codes generated by other libraries, you will need to use a '''universal''' Reed&amp;ndash;Solomon codec, which will allow you to specify your own parameters, and even go beyond the field 2^8.

[[Reed–Solomon codes for coders/Additional information#Universal_Reed-Solomon_Codec|On the complementary resource page, you will find an extended, universal version]] of the code presented here that you can use to decode almost any Reed&amp;ndash;Solomon code, with also a function to generate the list of prime polynomials, and [[Reed–Solomon codes for coders/Additional information#Autodetecting_the_Reed-Solomon_parameters|an algorithm to detect the parameters of an unknown RS code]]. Note that whatever the parameters you use, the repairing capabilities will always be the same: the generated values for the log/antilog tables and for the generator polynomial do not change the structure of Reed&amp;ndash;Solomon code, so that you always get the same functionality whatever the parameters. Indeed, modifying any of the available parameter will not change the theoretical Singleton bound which defines the maximal repairing capacity of Reed-Solomon (and in theory of any error correction code).

One immediate issue that you may have noticed is that we can only encode messages of up to 256 characters. This limit can be circumvented by several ways, the three most common being:

* using a higher Galois Field, for example 2&lt;sup&gt;16&lt;/sup&gt; which would allow for 65536 characters, or 2&lt;sup&gt;32&lt;/sup&gt;, 2&lt;sup&gt;64&lt;/sup&gt;, 2&lt;sup&gt;128&lt;/sup&gt;, etc. The issue here is that polynomial computations required to encode and decode Reed&amp;ndash;Solomon become very costly with big polynomials (most algorithms being in quadratic time, the most efficient being in ''n'' log ''n'' such as with number theoretic transform&lt;ref name=&quot;ntt&quot;&gt;Lin, S. J., Chung, W. H., &amp; Han, Y. S. (2014, October). Novel polynomial basis and its application to reed-solomon erasure codes. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on (pp. 316-325). IEEE.&lt;/ref&gt;).
* by &quot;chunking&quot;, which means that you simply encode your big data stream by chunks of 256 characters.
* using a variant algorithm that includes a packet size such as Cauchy Reed&amp;ndash;Solomon (see below).

If you want to go further, there are a lot of books and scientific articles on Reed&amp;ndash;Solomon codes, a good starting point is the author Richard Blahut who is notable in the domain. Also, there are a lot of different ways that Reed&amp;ndash;Solomon codes can be encoded and decoded, and thus you will find many different algorithms, in particular for decoding (Berlekamp-Massey, Berlekamp-Welch, Euclidian algorithm, etc.).

If you are looking for more performance, you will find in the literature several variants of the algorithms presented here, such as Cauchy&amp;ndash;Reed&amp;ndash;Solomon. The programming implementation also plays a big role in the performance of your Reed&amp;ndash;Solomon codec, which can lead into a 1000x speed difference. For more information, please read the [[Reed–Solomon codes for coders/Additional information#Optimizing performances|&quot;Optimizing performances&quot; section of the additional resources]].

Even if near-optimal forward error correction algorithms are all the rage nowadays (such as LDPC codes, Turbo codes, etc.) because of their great speed, Reed&amp;ndash;Solomon is an optimal FEC, which means that it can attain the theoretical limit known as the [[w:Singleton_bound|Singleton bound]]. In practice, this means that RS can correct up to &lt;kbd&gt;2*e+v &lt;= (n-k)&lt;/kbd&gt; errors and erasures at the same time, where e is the number of errors, v the number of erasures, k the message size, n the message+code size and &lt;kbd&gt;(n-k)&lt;/kbd&gt; the [[w:Minimum_distance|minimum distance]]. This is not to say that near-optimal FEC are useless: they are unimaginably faster than Reed&amp;ndash;Solomon could ever be, and they may suffer less from the [[w:Forward_error_correction#Averaging_noise_to_reduce_errors|cliff effect]] (which means they may still partially decode parts of the message even if there are too many errors to correct all errors, contrary to RS which will surely fail and even silently by decoding wrong messages without any detection&lt;ref&gt;Sofair, Isaac. &quot;Probability of miscorrection for Reed-Solomon codes.&quot; Information Technology: Coding and Computing, 2000. Proceedings. International Conference on. IEEE, 2000.&lt;/ref&gt;), but they surely can't correct as many errors as Reed&amp;ndash;Solomon. Choosing between a near-optimal and an optimal FEC is mainly a concern of speed.

Lately, the research field on Reed&amp;ndash;Solomon has regained some vitality since the discovery of [[w:List_decoding]] (not to confuse with soft decoding), which allows to decode/repair more symbols than the theoretical optimal limit. The core idea is that, instead of standard Reed&amp;ndash;Solomon which only do a unique decoding (meaning that it always results in a single solution, if it cannot because it's above the theoretical limit the decoder will return an error or a wrong result), Reed&amp;ndash;Solomon with list decoding will still try to decode beyond the limit and get several possible results, but by a clever examination of the different results, it's often possible to discriminate only one polynomial that is probably the correct one.

A few list decoding algorithms are already available that allows to repair up to &lt;kbd&gt;n - sqrt(n*k)&lt;/kbd&gt;&lt;ref&gt;&quot;Reed-Solomon Error-correcting Codes - The Deep Hole Problem&quot;, by Matt Keti, Nov 2012&lt;/ref&gt; instead of &lt;kbd&gt;2*e+v &lt;= (n-k)&lt;/kbd&gt;, and other list decoding algorithms (more efficient or decoding more symbols) are currently being investigated.

==Third-party implementations==

Here are a few implementations of Reed&amp;ndash;Solomon if you want to see practical examples:

* [https://github.com/tomerfiliba/reedsolomon Purely functional pure-Python Reedsolomon library] by Tomer Filiba and LRQ3000, inspired and expanding on this tutorial by supporting more features.
* [https://github.com/lrq3000/unireedsolomon Object-oriented Reed Solomon library in pure-Python] by Andrew Brown and LRQ3000 (same features as Tomer Filiba's lib, but object-oriented so closer to mathematical nomenclatura).
* [http://lxr.free-electrons.com/source/lib/reed_solomon/ Reed-Solomon in the Linux Kernel] (with a [https://github.com/tierney/reed-solomon userspace port here], initially ported from Phil Karn's library [http://www.ka9q.net/code/fec libfec] and [https://github.com/quiet/libfec libfec clone]).
* [https://github.com/zxing/zxing/ ZXing (Zebra Crossing)], a full-blown library to generate and decode QR codes.
* [https://github.com/catid/wirehair/blob/master/wirehair-mobile/wirehair_codec_8.cpp Speed-optimized Reed-Solomon] and [https://github.com/catid/longhair Cauchy-Reed-Solomon] with lots of comments and an associated [http://catid.mechafetus.com/news/news.php blog] for more details.
* [https://github.com/klauspost/reedsolomon Another high speed-optimized Reed-Solomon] in Go language.
* [https://github.com/mersinvald/reed-solomon-rs Port of code in the article] in Rust language.
* [https://github.com/mersinvald/Reed-Solomon C++ Reed Solomon implementation] with on-stack memory allocation and compile-time changable msg\ecc sizes for embedded, inspired by this tutorial.
* [https://github.com/NinjaDevelper/ReedSolomon Interleaved Reed Solomon implementation in C++] by NinjaDevelper.
* [https://github.com/Bulat-Ziganshin/FastECC FastECC, C++ Reed Solomon implementation in O(n log n) using Number Theoretic Transforms (NTT)] (open source, Apache License 2.0). Claims to have fast encoding rates even for large data.
* [https://github.com/catid/leopard Leopard-RS], another library in C++ for fast large data encoding, with a similar (but a bit different) algorithm as FastECC.
* [https://github.com/colin-davis/reedSolomon Pure Go Implementation] by Colin Davis (open source, GLPv3 License).
* [https://github.com/catid/shorthair Shorthair], an implementation of error correction code combined with UDP for fast reliable networking to replace the TCP stack or UDP duplication technique (which can be seen as a low efficiency redundancy scheme). [https://github.com/catid/shorthair/blob/master/docs/ErasureCodesInSoftware.pdf Slides] are provided, describing this approach for realtime game networking.
*[https://github.com/jackchouchani/reedsolomon Pure C Implementation] optimised using uint8_t and very efficient. 
*[https://github.com/hqm/rscode hqm rscode] ANSI C implementation, for 8-bit symbols 

==External links==
* [[w:Reed–Solomon_error_correction]]
* [[w:Finite_field_arithmetic]]
* [http://research.swtch.com/field Short tutorial on Reed-Solomon encoding with an introduction to finite fields]
* [https://www.academia.edu/31243287/Reed_Solomon_Encoding_Simplified_Explanation_for_Programmers A practical tutorial article to implement the core mathematical (galois field) operators].

==References==

[[Category:Essays]]
[[Category:Applied mathematics]]
[[Category:Algorithms]]</text>
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      <text bytes="103690" sha1="f44jorsn2gkz857rnazg6cg4n9h4fgo" xml:space="preserve">[[w:Error_detection_and_correction|Error correcting codes]] are a signal processing technique to correct errors. They are nowadays ubiquitous, such as in communications (mobile phone, internet), data storage and archival (hard drives, optical discs CD/DVD/BluRay, archival tapes), warehouse management (barcodes) and advertisement (QR codes).

[[w:Reed–Solomon error correction|Reed–Solomon error correction]] is a specific type of error correction code. It is one of the oldest but it is still widely used, as it is very well defined and several efficient algorithms are now available under the public domain.

Usually, error correction codes are hidden and most users do not even know about them, nor when they are used. Yet, they are a critical component for some applications to be viable, such as communication or data storage. Indeed, a hard drive that would randomly lose data every few days would be useless, and a phone being able to call only on days with a cloud-less weather would be seldom used. Using error correction codes allows to recover a corrupted message into the full original message.

Barcodes and QR codes are interesting applications to study, as they have the specificity of displaying visually the error correction code, rendering these codes readily accessible to the curious user.

In this essay, we will attempt to introduce the principles of Reed–Solomon codes from the point of view of a programmer rather than a mathematician, which means that we will focus more on the practice than the theory, although we will also explain the theory, but only the necessary knowledge for intuition and implementation. Notable references in the domain will be provided, so that the interested reader can dig deeper into the mathematical theory at will. We will provide real-world examples taken from the popular [[w:QR code|QR code]] barcode system as well as working code samples. We chose to use [[w:Python (programming language)|Python]] for the samples (mainly because it looks pretty and similar to [[w:pseudocode|pseudocode]]), but we will try to explain any non-obvious features for those who are not familiar with it. The mathematics involved is advanced in the sense that it is not usually taught below the university level, but it should be understandable to someone with a good grasp of high-school algebra.

We will first gently introduce the intuitions behind error correction codes principles, then in a second section we will introduce the structural design of QR codes, in other words how information is stored in a QR code and how to read and produce it, and in a third section we will study error correction codes via the implementation of a Reed&amp;ndash;Solomon decoder, with a quick introduction of the bigger BCH codes family, in order to reliably read damaged QR codes.

Note for the curious readers that [[Reed–Solomon codes for coders/Additional information|extended information can be found in the appendix]] and on the [[Talk:Reed%E2%80%93Solomon_codes_for_coders|discussion page]].

==Principles of error correction==
Before detailing the code, it might be useful to understand the intuition behind error correction. Indeed, although error correcting codes may seem daunting mathematically-wise, most of the mathematical operations are high school grade (with the exception of Galois Fields, but which are in fact easy and common for any programmer: it's simply doing operations on integers modulo a number). However, the complexity of the mathematical ingenuity behind error correction codes hide the quite intuitive goal and mechanisms at play.

Error correcting codes might seem like a difficult mathematical concept, but they are in fact based on an intuitive idea with an ingenious mathematical implementation: '''let's make the data structured, in a way that we can &quot;guess&quot; what the data was if it gets corrupted, just by &quot;fixing&quot; the structure'''. Mathematically-wise, we use polynomials from the Galois Field to implement this structure.

Let's take a more practical analogy: let's say you want to communicate messages to someone else, but these messages can get corrupted along the way. The main insight of error correcting codes is that, '''instead of using a whole dictionary of words, we can use a smaller set of carefully selected words, a &quot;reduced dictionary&quot;, so that each word is as different as any other'''. This way, when we get a message, we just have to lookup inside our reduced dictionary to '''1) detect''' which words are corrupted (as they are not in our reduced dictionary); '''2) correct''' corrupted words by finding the most similar word in our dictionary.

Let's take a simple example: we have a reduced dictionary with only three words of 4 letters: &lt;kbd&gt;this&lt;/kbd&gt;, &lt;kbd&gt;that&lt;/kbd&gt; and &lt;kbd&gt;corn&lt;/kbd&gt;. Let's say we receive a corrupted word: &lt;kbd&gt;co**&lt;/kbd&gt;, where &lt;kbd&gt;*&lt;/kbd&gt; is an erasure. Since we have only 3 words in our dictionary, we can easily compare our received word with our dictionary to find the word that is the closest. In this case, it's &lt;kbd&gt;corn&lt;/kbd&gt;. Thus the missing letters are &lt;kbd&gt;rn&lt;/kbd&gt;.

Now let's say we receive the word &lt;kbd&gt;th**&lt;/kbd&gt;. Here the problem is that we have two words in our dictionary that match the received word: &lt;kbd&gt;this&lt;/kbd&gt; and &lt;kbd&gt;that&lt;/kbd&gt;. In this case, we cannot be sure which one it is, and thus we cannot decode. This means that our dictionary is not very good, and we should replace &lt;kbd&gt;that&lt;/kbd&gt; with another more different word, such as &lt;kbd&gt;dash&lt;/kbd&gt; to maximize the difference between each word. This difference, or more precisely the minimum number of different letters between any 2 words of our dictionary, is called the '''maximum Hamming distance''' of our dictionary. Making sure that any 2 words of the dictionary share a minimum number of letters at the same position is called '''maximum separability'''.

The same principle is used for most error correcting codes: we generate a reduced dictionary containing only words with maximum separability (we will detail more how to do that in the third section), and then we communicate only with the words of this reduced dictionary. What Galois Fields provide is the structure (ie, reduced dictionary basis), and Reed&amp;ndash;Solomon is a way to automatically create a suitable structure (make a reduced dictionary with maximum separability tailored for a dataset), as well as provide the automated methods to detect and correct errors (ie, lookups in the reduced dictionary). To be more precise, Galois Fields are the structure (thanks to their cyclic nature, the modulo an integer) and Reed&amp;ndash;Solomon is the codec (encoder/decoder) based on Galois Fields.

If a word gets corrupted in the communication, that's no big deal since we can easily fix it by looking inside our dictionary and find the closest word, which is probably the correct one (there is however a chance of choosing a wrong one if the input message is too heavily corrupted, but the probability is very small). Also, the longer our words are, the more separable they are, since more characters can be corrupted without any impact.

The simplest way to generate a dictionary of maximally separable words is to make words longer than they really are.

Let's take again our example:

    t h i s
    t h a t
    c o r n

Append a unique set of characters so that there are no duplicated characters at any of the appended positions, and add one more word to help with the explanation:

    t h i s a b c d
    t h a t b c d e
    c o r n c d e f

Note that each word in this dictionary differs from every other word by at least 6 characters, so the distance is 6. This allows up to 5 errors in known positions (which are called erasures), or 3 errors in unknown positions, to be corrected.

Assume that 4 erasures occur:

    t * * * a b * d

Then a search of the dictionary for the 4 non-erased characters can be done to find the only entry that matches those 4 characters, since the distance is 5. Here it gives: &lt;kbd&gt;t h i s a b c d&lt;/kbd&gt;

Assume that 2 errors occur as in one of these patterns:

   t h o s b c d e

The issue here is the location of the errors is unknown. The erasures might have happened in any 2 positions meaning that there are &lt;math&gt;\tbinom{8}{6}&lt;/math&gt; or 28 possible sub-sets of 6 characters:

   t h o s b c * *
   t h o s b * d *
   t h o s b * * e
   ...

If we do a dictionary search on each of these sub-sequences, we find that there is only one sub-set that matches 6 characters. &lt;kbd&gt;t h * * b c d e&lt;/kbd&gt; matches &lt;kbd&gt;t h a t b c d e&lt;/kbd&gt;.

With these examples, one can see the advantage of redundancy in recovering lost information: redundant characters help you recover your original data. The previous examples show how a crude error correcting scheme could work. Reed&amp;ndash;Solomon's core idea is similar, append redundant data to a message based on Galois Field mathematics. The original error correcting decoder was similar to the error example above, search sub-sets of a received message that correspond to a valid message, and choose the one with the most matches as the corrected message. This isn't practical for larger messages, so mathematical algorithms were developed to perform error correction in a reasonable time.

==QR code structure==
This section introduces the structure of QR codes, which is how data is stored in a QR code. The information in this section is deliberately incomplete. Only the most common features of the small 21&amp;times;21 size symbols (also known as version 1) are presented here, but see the [[Reed–Solomon codes for coders/Additional information|appendix]] for additional information.

Here is a QR symbol that will be used as an example. It consists of dark and light squares, known as modules in the barcoding world. The three square locator patterns in the corners are a visually distinctive feature of QR symbols.

[[File:QR Code Example.svg]]

===Masking===
A masking process is used to avoid features in the symbol that might confuse a scanner, such as misleading shapes that look like the locator patterns and large blank areas. Masking inverts certain modules (white becomes black and black becomes white) while leaving others alone.

In the diagram below, the red areas encode format information and use a fixed masking pattern. The data area (in black and white) is masked with a variable pattern. When the code is created, the encoder tries a number of different masks and chooses the one that minimizes undesirable features in the result. The chosen mask pattern is then indicated in the format information so that the decoder knows which one to use. The light gray areas are fixed patterns which do not encode any information. In addition to the obvious locator patterns, there are also timing patterns which contain alternating light and dark modules.

[[File:QR Code Masking Example.svg]]

The masking transformation is easily applied (or removed) using the [[w:Exclusive or|exclusive-or]] operation (denoted by a caret ^ in many programming languages). The unmasking of the format information is shown below. Reading counter-clockwise around the upper-left locator pattern, we have the following sequence of bits. White modules represent 0 and black modules represent 1.

 Input       101101101001011
 Mask      ^ &lt;u&gt;101010000010010&lt;/u&gt;
 Output      000111101011001

===Formatting information===
There are two identical copies of the formatting information, so that the symbol can still be decoded even if it is damaged. The second copy is broken in two pieces and placed around the other two locators, and is read in a clockwise direction (upwards in the lower-left corner, then left-to-right in the upper-right corner).

The first two bits of formatting information give the error correction level used for the message data. A QR symbol this size contains 26 bytes of information. Some of these are used to store the message and some are used for error correction, as shown in the table below. The left-hand column is simply a name given to that level.

{|class=&quot;wikitable&quot;
|-
! Error Correction Level !! Level Indicator !! Error Correction Bytes !! Message Data Bytes
|- align=&quot;center&quot;
| L || 01 || 7 || 19
|- align=&quot;center&quot;
| M || 00 || 10 || 16
|- align=&quot;center&quot;
| Q || 11 || 13 || 13
|- align=&quot;center&quot;
| H || 10 || 17 || 9
|}

The next three bits of format information select the masking pattern to be used in the data area. The patterns are illustrated below, including the mathematical formula that tells whether a module is black (i and j are the row and column numbers, respectively, and start with 0 in the upper-left hand corner).

[[File:QR Code Mask Patterns.svg]]

The remaining ten bits of formatting information are for correcting errors in the format itself. This will be explained in a [[#BCH codes|later section]].

===Message data===

Here is a larger diagram showing the &quot;unmasked&quot; QR code. Different regions of the symbol are indicated, including the boundaries of the message data bytes.

[[File:QR Code Unmasked.svg]]

Data bits are read starting from the lower-right corner and moving up the two right-hand columns in a zig-zag pattern. The first three bytes are 01000000 11010010 01110101. The next two columns are read in a downward direction, so the next byte is 01000111. Upon reaching the bottom, the two columns after that are read upward. Proceed in this up-and-down fashion all the way to the left side of the symbol (skipping over the timing pattern where necessary). Here is the complete message in [[w:Hexadecimal|hexadecimal]] notation.

:Message data bytes: 40 d2 75 47 76 17 32 06 27 26 96 c6 c6 96 70 ec
:Error correction bytes: bc 2a 90 13 6b af ef fd 4b e0

===Decoding===

The final step is to decode the message bytes into something readable. The first four bits indicate how the message is encoded. QR codes use several different encoding schemes, so that different kinds of messages can be stored efficiently. These are summarized in the table below. After the mode indicator is a length field, which tells how many characters are stored. The size of the length field depends on the specific encoding.

{|class=&quot;wikitable&quot;
|-
! Mode Name !! Mode Indicator !! Length Bits !! Data Bits
|- align=&quot;center&quot;
| Numeric || 0001 || 10 || 10 bits per 3 digits
|- align=&quot;center&quot;
| Alphanumeric || 0010 || 9 || 11 bits per 2 characters
|- align=&quot;center&quot;
| Byte || 0100 || 8 || 8 bits per character
|- align=&quot;center&quot;
| Kanji || 1000 || 8 || 13 bits per character
|}

(The length field sizes above are valid only for smaller QR codes.)

Our sample message starts with 0100 (hex 4), indicating that there are 8 bits per character. The next 8 bits (hex 0d) are the length field, 13 in decimal notation. The bits after that can be rearranged in bytes representing the actual characters of the messageː 27 54 77 61 73 20 62 72 69 6c 6c 69 67, and additionally 0e c. The first two, hex 27 and 54 are the [[w:ASCII|ASCII]] codes for apostrophe and T. The whole message is &quot;'Twas brillig&quot; (from [[w:Jabberwocky#Lexicon]]).

After the last of the data bits is another 4-bit mode indicator. It can be different from the first one, allowing different encodings to be mixed within the same QR symbol. When there is no more data to store, the special end-of-message code 0000 is given. (Note that the standard allows the end-of-message code to be omitted if it wouldn't fit in the available number of data bytes.)

At this point, we know how to decode, or read, a whole QR code. However, in real life conditions, a QR code is rarely whole: usually, it is scanned via a phone's camera, under potentially poor lighting conditions, or on a scratched surface where part of the QR code was ripped, or on a stained surface, etc.

To make our QR code decoder **reliable**, we need to be able to **correct** errors. The next part of this article will describe how to correct errors, by constructing a BCH decoder, and more specifically a Reed&amp;ndash;Solomon decoder.

==BCH codes==
In this section, we introduce a general class of error correction codes: the [[w:BCH code|BCH codes]], the parent family of modern Reed&amp;ndash;Solomon codes, and the basic detection and correction mechanisms.

The formatting information is encoded with a [[w:BCH code|BCH code]] which allows a certain number of bit-errors to be detected and corrected. BCH codes are a generalization of Reed&amp;ndash;Solomon codes (modern Reed&amp;ndash;Solomon codes are BCH codes). In the case of QR codes, the BCH code used for the format information is much simpler than the Reed&amp;ndash;Solomon code used for the message data, so it makes sense to start with the BCH code for format information.

===BCH error detection===
The process for checking the encoded information is similar to long division, but uses exclusive-or instead of subtraction. The format code should produce a remainder of zero when it is &quot;divided&quot; by the so-called generator of the code. QR format codes use the generator 10100110111. This process is demonstrated for the format information in the example code (000111101011001) below.

               000111101011001
                ^ &lt;u&gt;101001101110 &lt;/u&gt;
                  010100110111
                 ^ &lt;u&gt;10100110111&lt;/u&gt;
                   00000000000

Here is a Python function which implements this calculation.

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def qr_check_format(fmt):
   g = 0x537 # = 0b10100110111 in python 2.6+
   for i in range(4,-1,-1):
      if fmt &amp; (1 &lt;&lt; (i+10)):
         fmt ^= g &lt;&lt; i
   return fmt
&lt;/syntaxhighlight&gt;

''Python note:'' The &lt;kbd&gt;range&lt;/kbd&gt; function may not be clear to non-Python programmers. It produces a list of numbers counting down from 4 to 0 (the code has &quot;-1&quot; because the interval returned by &quot;range&quot; includes the start but not the end value). In C-derived languages, the for loop might be written as &lt;kbd style=&quot;white-space:nowrap&quot;&gt;for (i = 4; i &gt;= 0; i--)&lt;/kbd&gt;; in Pascal-derived languages, &lt;kbd style=&quot;white-space:nowrap&quot;&gt;for i := 4 downto 0&lt;/kbd&gt;.

''Python note 2:'' The &lt;kbd&gt;&amp;&lt;/kbd&gt; operator performs [[w:Bitwise operation#AND|bitwise and]], while &lt;kbd&gt;&amp;lt;&amp;lt;&lt;/kbd&gt; is a [[w:Bitwise operation#Bit shifts|left bit-shift]]. This is consistent with C-like languages.

This function can also be used to encode the 5-bit format information.
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
encoded_format = (format&lt;&lt;10) + qr_check_format(format&lt;&lt;10)
&lt;/syntaxhighlight&gt;

Readers may find it an interesting exercise to generalize this function to divide by different numbers. For example, larger QR codes contain six bits of version information with 12 error correction bits using the generator 1111100100101.

In mathematical formalism, these binary numbers are described as polynomials whose coefficients are [[w:Modular arithmetic|integers mod 2]]. Each bit of the number is a coefficient of one term. For example:

:10100110111 = 1 ''x''&lt;sup&gt;10&lt;/sup&gt; + 0 ''x''&lt;sup&gt;9&lt;/sup&gt; + 1 ''x''&lt;sup&gt;8&lt;/sup&gt; + 0 ''x''&lt;sup&gt;7&lt;/sup&gt; + 0 ''x''&lt;sup&gt;6&lt;/sup&gt; + 1 ''x''&lt;sup&gt;5&lt;/sup&gt; + 1 ''x''&lt;sup&gt;4&lt;/sup&gt; + 0 ''x''&lt;sup&gt;3&lt;/sup&gt; + 1 ''x''&lt;sup&gt;2&lt;/sup&gt; + 1 ''x'' + 1 = ''x''&lt;sup&gt;10&lt;/sup&gt; + ''x''&lt;sup&gt;8&lt;/sup&gt; + ''x''&lt;sup&gt;5&lt;/sup&gt; + ''x''&lt;sup&gt;4&lt;/sup&gt; + ''x''&lt;sup&gt;2&lt;/sup&gt; + ''x'' + 1

If the remainder produced by &lt;kbd&gt;qr_check_format&lt;/kbd&gt; is not zero, then the code has been damaged or misread. The next step is to determine which format code is most likely the one that was intended (ie, lookup in our reduced dictionary).

===BCH error correction===
Although sophisticated algorithms for decoding BCH codes exist, they are probably overkill in this case. Since there are only 32 possible format codes, it's much easier to simply try each one and pick the one that has the smallest number of bits different from the code in question (the number of different bits is known as the [[w:Hamming distance|Hamming distance]]). This method of finding the closest code is known as exhaustive search, and is possible only because we have very few codes (a code is a valid message, and here there are only 32, all other binary numbers aren't correct).

(Note that Reed&amp;ndash;Solomon is also based on this principle, but since the number of possible codewords is simply too big, we can't afford to do an exhaustive search, and that's why clever but complicated algorithms have been devised, such as Berlekamp-Massey.)

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def hamming_weight(x):
   weight = 0
   while x &gt; 0:
      weight += x &amp; 1
      x &gt;&gt;= 1
   return weight

def qr_decode_format(fmt):
   best_fmt = -1
   best_dist = 15
   for test_fmt in range(0,32):
      test_code = (test_fmt&lt;&lt;10) ^ qr_check_format(test_fmt&lt;&lt;10)
      test_dist = hamming_weight(fmt ^ test_code)
      if test_dist &lt; best_dist:
         best_dist = test_dist
         best_fmt = test_fmt
      elif test_dist == best_dist:
         best_fmt = -1
   return best_fmt
&lt;/syntaxhighlight&gt;

The function &lt;kbd&gt;qr_decode_format&lt;/kbd&gt; returns -1 if the format code could not be unambiguously decoded. This happens when two or more format codes have the same distance from the input.

To run this code in Python, first start [[w:IDLE (Python)|IDLE]], Python's integrated development environment. You should see a version message and the interactive input prompt &lt;kbd&gt;&gt;&gt;&gt;&lt;/kbd&gt;. Open a new window, copy the functions &lt;kbd&gt;qr_check_format&lt;/kbd&gt;, &lt;kbd&gt;hamming_weight&lt;/kbd&gt;, and &lt;kbd&gt;qr_decode_format&lt;/kbd&gt; into it, and save as &lt;kbd&gt;qr.py&lt;/kbd&gt;. Return to the prompt and type the lines following &lt;kbd&gt;&gt;&gt;&gt;&lt;/kbd&gt; below.

&lt;pre&gt;&gt;&gt;&gt; from qr import *
&gt;&gt;&gt; qr_decode_format(int(&quot;000111101011001&quot;,2))  # no errors
3
&gt;&gt;&gt; qr_decode_format(int(&quot;111111101011001&quot;,2))  # 3 bit-errors
3
&gt;&gt;&gt; qr_decode_format(int(&quot;111011101011001&quot;,2))  # 4 bit-errors
-1
&lt;/pre&gt;

You can also start Python by typing &lt;kbd&gt;python&lt;/kbd&gt; at a command prompt.

In the next sections, we will study Finite Field Arithmetics and Reed&amp;ndash;Solomon code, which is a subtype of BCH codes. The basic idea (ie, '''using a limited words dictionary with maximum separability''') is the same, but since we will encode longer words (256 bytes instead of 2 bytes), with more symbols available (encoded on all 8bits, thus 256 different possible values), we cannot use this naive, exhaustive approach, because it would take way too much time: we need to use cleverer algorithms, and Finite Field mathematics will help us do just that, by giving us a '''structure'''.

==Finite field arithmetic==

===Introduction to mathematical fields===
Before discussing the Reed&amp;ndash;Solomon codes used for the message, it will be useful to introduce a bit more mathematics. 

We'd like to define addition, subtraction, multiplication, and division for 8-bit bytes and always produce 8-bit bytes as a result, so as to avoid any overflow. Naively, we might attempt to use the normal definitions for these operations, and then mod by 256 to keep results from overflowing. And this is exactly what we will be doing, and is what is called a Galois Field 2^8. You can easily imagine why it works for everything, except for division: what is 5/4?

Here's a brief introduction to Galois Fields: a finite field is a set of numbers, and a field needs to have six properties governing addition, subtraction, multiplication and division: Closure, Associative, Commutative, Distributive, Identity and Inverse. More simply put, using a field allows us to study the relationship between numbers of this field, and apply the result to any other field that follows the same properties. For example, the set of reals ℝ is a field. In other words, mathematical fields studies the structure of a set of numbers.

However, integers ℤ aren't a field, because as we said above, not all divisions are defined (such as 5/4), which violates multiplicative inverse property (x such that x*4=5 does not exist). One simple way to fix that is to do modulo using a prime number, such as 257, or any positive integer power of a prime number: in this way, we are guaranteed that x*4=5 exists since we will just wrap around. ℤ modulo any prime number is called a Galois Field, and modulo 2 is an extra interesting Galois Field: since an 8-bit string can express a total of 256 = 2^8 values, we say that we use a Galois Field of 2^8, or GF(2^8). In spoken language, 2 is the characteristic of the field, 8 is the exponent, and 256 is the field's cardinality. More information on [http://research.swtch.com/field finite fields can be found here].

Here we will define the usual mathematical operations that you are used to doing with integers, but adapted to GF(2^8), which is basically doing usual operations but modulo 2^8.

Another way to consider the link between GF(2) and GF(2^8) is to think that GF(2^8) represents a polynomial of 8 binary coefficients. For example, in GF(2^8), 170 is equivalent to &lt;kbd&gt;10101010 = 1*x^7 + 0*x^6 + 1*x^5 + 0*x^4 + 1*x^3 + 0*x^2 + 1*x + 0 = x^7 + x^5 + x^3 + x&lt;/kbd&gt;. Both representations are equivalent, it's just that in the first case, 170, the representation is decimal, and in the other case it's binary, which can be thought as representing a polynomial [[w:Finite_field_arithmetic#Effective_polynomial_representation|by convention (only used in GF(2^p) as explained here)]]. The latter is often the representation used in academic books and in hardware implementations (because of logical gates and registers, which work at the binary level). For a software implementation, the decimal representation can be preferred for clearer and more close-to-the-mathematics code (this is what we will use for the code in this tutorial, except for some examples that will use the binary representation).

In any case, try to not confuse the polynomial representing a single GF(2^p) symbol (each coefficient is a bit/boolean: either 0 or 1), and the polynomial representing a list of GF(2^p) symbols (in this case the polynomial is equivalent to the message+RScode, each coefficient is a value between 0 and 2^p and represent one character of the message+RScode). We will first describe operations on single symbol, then polynomial operations on a list of symbols.

===Addition and Subtraction===
Both addition and subtraction are replaced with exclusive-or in Galois Field base 2. This is logical: addition modulo 2 is exactly like an XOR, and subtraction modulo 2 is exactly the same as addition modulo 2. This is possible because additions and subtractions in this Galois Field are carry-less.

Thinking of our 8-bit values as polynomials with coefficients mod 2:

    0101 + 0110 = 0101 - 0110 = 0101 XOR 0110 = 0011

The same way (in binary representation of two single GF(2^8) integers):

:(''x''&lt;sup&gt;2&lt;/sup&gt; + 1) + (''x''&lt;sup&gt;2&lt;/sup&gt; + ''x'') = 2 ''x''&lt;sup&gt;2&lt;/sup&gt; + ''x'' + 1 = 0 ''x''&lt;sup&gt;2&lt;/sup&gt; + ''x'' + 1 = ''x'' + 1

Since &lt;kbd&gt;(a ^ a) = 0&lt;/kbd&gt;, every number is its own opposite, so (''x'' - ''y'') is the same as (''x'' + ''y'').

Note that in books, you will find additions and subtractions to define some mathematical operations on GF integers, but in practice, you can just XOR (as long as you are in a Galois Field base 2; this is not true in other fields).

Here is the equivalent Python code:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_add(x, y):
    return x ^ y

def gf_sub(x, y):
    return x ^ y # in binary galois field, subtraction is just the same as addition (since we mod 2)
&lt;/syntaxhighlight&gt;

===Multiplication===
Multiplication is likewise based on polynomial multiplication. Simply write the inputs as polynomials and multiply them out using the distributive law as normal. As an example, 10001001 times 00101010 is calculated as follows.

:(''x''&lt;sup&gt;7&lt;/sup&gt; + ''x''&lt;sup&gt;3&lt;/sup&gt; + 1) (''x''&lt;sup&gt;5&lt;/sup&gt; + ''x''&lt;sup&gt;3&lt;/sup&gt; + ''x'') = ''x''&lt;sup&gt;7&lt;/sup&gt; (''x''&lt;sup&gt;5&lt;/sup&gt; + ''x''&lt;sup&gt;3&lt;/sup&gt; + ''x'') + ''x''&lt;sup&gt;3&lt;/sup&gt; (''x''&lt;sup&gt;5&lt;/sup&gt; + ''x''&lt;sup&gt;3&lt;/sup&gt; + ''x'') + 1 (''x''&lt;sup&gt;5&lt;/sup&gt; + ''x''&lt;sup&gt;3&lt;/sup&gt; + ''x'')
:= ''x''&lt;sup&gt;12&lt;/sup&gt; + ''x''&lt;sup&gt;10&lt;/sup&gt; + 2 ''x''&lt;sup&gt;8&lt;/sup&gt; + ''x''&lt;sup&gt;6&lt;/sup&gt; + ''x''&lt;sup&gt;5&lt;/sup&gt; + ''x''&lt;sup&gt;4&lt;/sup&gt; +  ''x''&lt;sup&gt;3&lt;/sup&gt; + ''x''
:= ''x''&lt;sup&gt;12&lt;/sup&gt; + ''x''&lt;sup&gt;10&lt;/sup&gt; + ''x''&lt;sup&gt;6&lt;/sup&gt; + ''x''&lt;sup&gt;5&lt;/sup&gt; + ''x''&lt;sup&gt;4&lt;/sup&gt; +  ''x''&lt;sup&gt;3&lt;/sup&gt; + ''x''

The same result can be obtained by a modified version of the standard grade-school multiplication procedure, in which we replace addition with exclusive-or.

        10001001
 *      &lt;u&gt;00101010&lt;/u&gt;
       10001001
 ^   10001001
 ^ &lt;u&gt;10001001&lt;/u&gt;
   1010001111010

Note: the XOR multiplication here is carry-less! If you do it with-carry, you will get the wrong result 1011001111010 with the extra term ''x''&lt;sup&gt;9&lt;/sup&gt; instead of the correct result 1010001111010.

Here is a Python function which implements this polynomial multiplication on single GF(2^8) integers.

Note: this function (and some other functions below) use a lot of bitwise operators such as &gt;&gt; and &lt;&lt;, because they are both faster and more concise to do what we want to do. These operators are available in most languages, they are not specific to Python, and [https://wiki.python.org/moin/BitwiseOperators you can get more information about them here].

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def cl_mul(x,y):
    '''Bitwise carry-less multiplication on integers'''
    z = 0
    i = 0
    while (y&gt;&gt;i) &gt; 0:
        if y &amp; (1&lt;&lt;i):
            z ^= x&lt;&lt;i
        i += 1
    return z
&lt;/syntaxhighlight&gt;

Of course, the result no longer fits in an 8-bit byte (in this example, it is 13 bits long), so we need to perform one more step before we are finished. The result is reduced modulo 100011101 (the choice of this number is explained below the code), using the long division process described previously. In this instance, this is called &quot;modular reduction&quot;, because basically what we do is that we divide and keep only the remainder, using a modulo. This produces the final answer 11000011 in our example.

   1010001111010
 ^ &lt;u&gt;100011101&lt;/u&gt;
   0010110101010
   ^ &lt;u&gt;100011101&lt;/u&gt;
     00111011110
     ^ &lt;u&gt;100011101&lt;/u&gt;
       011000011

Here is the Python code to do the whole Galois Field multiplication with modular reduction:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_mult_noLUT(x, y, prim=0):
    '''Multiplication in Galois Fields without using a precomputed look-up table (and thus it's slower)
    by using the standard carry-less multiplication + modular reduction using an irreducible prime polynomial'''

    ### Define bitwise carry-less operations as inner functions ###
    def cl_mult(x,y):
        '''Bitwise carry-less multiplication on integers'''
        z = 0
        i = 0
        while (y&gt;&gt;i) &gt; 0:
            if y &amp; (1&lt;&lt;i):
                z ^= x&lt;&lt;i
            i += 1
        return z

    def bit_length(n):
        '''Compute the position of the most significant bit (1) of an integer. Equivalent to int.bit_length()'''
        bits = 0
        while n &gt;&gt; bits: bits += 1
        return bits

    def cl_div(dividend, divisor=None):
        '''Bitwise carry-less long division on integers and returns the remainder'''
        # Compute the position of the most significant bit for each integers
        dl1 = bit_length(dividend)
        dl2 = bit_length(divisor)
        # If the dividend is smaller than the divisor, just exit
        if dl1 &lt; dl2:
            return dividend
        # Else, align the most significant 1 of the divisor to the most significant 1 of the dividend (by shifting the divisor)
        for i in range(dl1-dl2,-1,-1):
            # Check that the dividend is divisible (useless for the first iteration but important for the next ones)
            if dividend &amp; (1 &lt;&lt; i+dl2-1):
                # If divisible, then shift the divisor to align the most significant bits and XOR (carry-less subtraction)
                dividend ^= divisor &lt;&lt; i
        return dividend
    
    ### Main GF multiplication routine ###

    # Multiply the gf numbers
    result = cl_mult(x,y)
    # Then do a modular reduction (ie, remainder from the division) with an irreducible primitive polynomial so that it stays inside GF bounds
    if prim &gt; 0:
        result = cl_div(result, prim)

    return result
&lt;/syntaxhighlight&gt;

Result:

&lt;pre&gt;
&gt;&gt;&gt; a = 0b10001001
&gt;&gt;&gt; b = 0b00101010
&gt;&gt;&gt; print bin(gf_mult_noLUT(a, b, 0)) # multiplication only
0b1010001111010
&gt;&gt;&gt; print bin(gf_mult_noLUT(a, b, 0x11d)) # multiplication + modular reduction
0b11000011
&lt;/pre&gt;

Why mod 100011101 (in hexadecimal: 0x11d)? The mathematics is a little complicated here, but in short, 100011101 represents an 8th degree polynomial which is &quot;irreducible&quot; (meaning it can't be represented as the product of two smaller polynomials). This number is called a '''primitive polynomial''' or irreducible polynomial, or prime polynomial (we will mainly use this latter name for the rest of this tutorial). This is necessary for division to be well-behaved, which is to stay in the limits of the Galois Field, but without duplicating values. There are other numbers we could have chosen, but they're all essentially the same, and 100011101 (0x11d) is a common primitive polynomial for Reed&amp;ndash;Solomon codes. If you are curious to know how to generate those prime polynomials, please see the [[Reed%E2%80%93Solomon_codes_for_coders/Additional_information#Universal_Reed-Solomon_Codec|appendix]].

Additional infos on the prime polynomial (you can skip): What is a prime polynomial? It is the equivalent of a prime number, but in the Galois Field. Remember that a Galois Field uses values that are multiples of 2 as the generator. Of course, a prime number cannot be a multiple of two in standard arithmetics, but in a Galois Field it is possible. Why do we need a prime polynomial? Because to stay in the bound of the field, we need to compute the modulo of any value above the Galois Field. Why don't we just modulo with the Galois Field size? Because we will end up with lots of duplicate values, and we want to have as many unique values as possible in the field, so that a number always has one and only projection when doing a modulo or a XOR with the prime polynomial.

Note for the interested reader: as an example of what you can achieve with clever algorithms, here is another way to achieve multiplication of GF numbers in a more concise and faster way, using the [http://www.cut-the-knot.org/Curriculum/Algebra/PeasantMultiplication.shtml Russian Peasant Multiplication algorithm]:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_mult_noLUT(x, y, prim=0, field_charac_full=256, carryless=True):
    '''Galois Field integer multiplication using Russian Peasant Multiplication algorithm (faster than the standard multiplication + modular reduction).
    If prim is 0 and carryless=False, then the function produces the result for a standard integers multiplication (no carry-less arithmetics nor modular reduction).'''
    r = 0
    while y: # while y is above 0
        if y &amp; 1: r = r ^ x if carryless else r + x # y is odd, then add the corresponding x to r (the sum of all x's corresponding to odd y's will give the final product). Note that since we're in GF(2), the addition is in fact an XOR (very important because in GF(2) the multiplication and additions are carry-less, thus it changes the result!).
        y = y &gt;&gt; 1 # equivalent to y // 2
        x = x &lt;&lt; 1 # equivalent to x*2
        if prim &gt; 0 and x &amp; field_charac_full: x = x ^ prim # GF modulo: if x &gt;= 256 then apply modular reduction using the primitive polynomial (we just subtract, but since the primitive number can be above 256 then we directly XOR).

    return r
&lt;/syntaxhighlight&gt;

Note that using this last function with parameters prim=0 and carryless=False will return the result for a standard integers multiplication (and thus you can see the difference between carryless and with-carry addition and its impact on multiplication).

===Multiplication with logarithms===

The procedure described above is not the most convenient way to implement Galois field multiplication. Multiplying two numbers takes up to eight iterations of the multiplication loop, followed by up to eight iterations of the division loop. However, we can multiply with no looping by using lookup tables. One solution would be to construct the entire multiplication table in memory, but that would require a bulky 64k table. The solution described below is much more compact.

First, notice that it is particularly easy to multiply by 2=00000010 (by convention, this number is referred to as '''&amp;alpha;''' or the '''generator number'''): simply left-shift by one place, then exclusive-or with the modulus 100011101 if necessary (why xor is sufficient for taking the mod in this case is an exercise left to the reader). Here are the first few powers of &amp;alpha;.

{| class=&quot;wikitable&quot;
|-
| &amp;alpha;&lt;sup&gt;0&lt;/sup&gt; = 00000001
| &amp;alpha;&lt;sup&gt;4&lt;/sup&gt; = 00010000
| &amp;alpha;&lt;sup&gt;8&lt;/sup&gt;&amp;nbsp; = 00011101
| &amp;alpha;&lt;sup&gt;12&lt;/sup&gt; = 11001101
|-
| &amp;alpha;&lt;sup&gt;1&lt;/sup&gt; = 00000010
| &amp;alpha;&lt;sup&gt;5&lt;/sup&gt; = 00100000
| &amp;alpha;&lt;sup&gt;9&lt;/sup&gt;&amp;nbsp; = 00111010
| &amp;alpha;&lt;sup&gt;13&lt;/sup&gt; = 10000111
|-
| &amp;alpha;&lt;sup&gt;2&lt;/sup&gt; = 00000100
| &amp;alpha;&lt;sup&gt;6&lt;/sup&gt; = 01000000
| &amp;alpha;&lt;sup&gt;10&lt;/sup&gt; = 01110100
| &amp;alpha;&lt;sup&gt;14&lt;/sup&gt; = 00010011
|-
| &amp;alpha;&lt;sup&gt;3&lt;/sup&gt; = 00001000
| &amp;alpha;&lt;sup&gt;7&lt;/sup&gt; = 10000000
| &amp;alpha;&lt;sup&gt;11&lt;/sup&gt; = 11101000
| &amp;alpha;&lt;sup&gt;15&lt;/sup&gt; = 00100110
|}

If this table is continued in the same fashion, the powers of &amp;alpha; do not repeat themselves until &amp;alpha;&lt;sup&gt;255&lt;/sup&gt; = 00000001. Thus, every element of the field except zero is equal to some power of &amp;alpha;. The element '''&amp;alpha;''', that we define, is known as a [[w:Primitive element (finite field)|primitive element]] or '''generator''' of the Galois field.

This observation suggests another way to implement multiplication: by adding the exponents of &amp;alpha;.
:10001001 * 00101010 = &amp;alpha;&lt;sup&gt;74&lt;/sup&gt; * &amp;alpha;&lt;sup&gt;142&lt;/sup&gt; = &amp;alpha;&lt;sup&gt;74 + 142&lt;/sup&gt; = &amp;alpha;&lt;sup&gt;216&lt;/sup&gt; = 11000011
The problem is, how do we find the power of &amp;alpha; that corresponds to 10001001? This is known as the [[w:Discrete logarithm|discrete logarithm]] problem, and no efficient general solution is known. However, since there are only 256 elements in this field, we can easily construct a table of logarithms. While we're at it, a corresponding table of antilogs (exponentials) will also be useful. [[w:Finite_field_arithmetic#Implementation_tricks|More mathematical information about this trick can be found here]].
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
gf_exp = [0] * 512 # Create list of 512 elements. In Python 2.6+, consider using bytearray
gf_log = [0] * 256

def init_tables(prim=0x11d):
    '''Precompute the logarithm and anti-log tables for faster computation later, using the provided primitive polynomial.'''
    # prim is the primitive (binary) polynomial. Since it's a polynomial in the binary sense,
    # it's only in fact a single galois field value between 0 and 255, and not a list of gf values.
    global gf_exp, gf_log
    gf_exp = [0] * 512 # anti-log (exponential) table
    gf_log = [0] * 256 # log table
    # For each possible value in the galois field 2^8, we will pre-compute the logarithm and anti-logarithm (exponential) of this value
    x = 1
    for i in range(0, 255):
        gf_exp[i] = x # compute anti-log for this value and store it in a table
        gf_log[x] = i # compute log at the same time
        x = gf_mult_noLUT(x, 2, prim)

        # If you use only generator==2 or a power of 2, you can use the following which is faster than gf_mult_noLUT():
        #x &lt;&lt;= 1 # multiply by 2 (change 1 by another number y to multiply by a power of 2^y)
        #if x &amp; 0x100: # similar to x &gt;= 256, but a lot faster (because 0x100 == 256)
            #x ^= prim # subtract the primary polynomial to the current value (instead of 255, so that we get a unique set made of coprime numbers), this is the core of the tables generation

    # Optimization: double the size of the anti-log table so that we don't need to mod 255 to
    # stay inside the bounds (because we will mainly use this table for the multiplication of two GF numbers, no more).
    for i in range(255, 512):
        gf_exp[i] = gf_exp[i - 255]
    return [gf_log, gf_exp]
&lt;/syntaxhighlight&gt;
''Python note:'' The &lt;kbd&gt;range&lt;/kbd&gt; operator's upper bound is exclusive, so &lt;kbd&gt;gf_exp[255]&lt;/kbd&gt; is not set twice by the above.

The &lt;kbd&gt;gf_exp&lt;/kbd&gt; table is oversized in order to simplify the multiplication function. This way, we don't have to check to make sure that &lt;kbd&gt;gf_log[x] + gf_log[y]&lt;/kbd&gt; is within the table size.
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_mul(x,y):
    if x==0 or y==0:
        return 0
    return gf_exp[gf_log[x] + gf_log[y]] # should be gf_exp[(gf_log[x]+gf_log[y])%255] if gf_exp wasn't oversized
&lt;/syntaxhighlight&gt;

===Division===
Another advantage of the logarithm table approach is that it allows us to define division using the difference of logarithms. In the code below, 255 is added to make sure the difference isn't negative.
&lt;syntaxhighlight lang=&quot;python&quot;&gt;def gf_div(x,y):
    if y==0:
        raise ZeroDivisionError()
    if x==0:
        return 0
    return gf_exp[(gf_log[x] + 255 - gf_log[y])]&lt;/syntaxhighlight&gt;
''Python note:'' The &lt;kbd&gt;raise&lt;/kbd&gt; statement throws an exception and aborts execution of the &lt;kbd&gt;gf_div&lt;/kbd&gt; function.

With this definition of division, &lt;kbd&gt;gf_div(gf_mul(x,y),y)==x&lt;/kbd&gt; for any &lt;kbd&gt;x&lt;/kbd&gt; and any nonzero &lt;kbd&gt;y&lt;/kbd&gt;.

Readers who are more advanced programmers may find it interesting to write a class encapsulating Galois field arithmetic. [[w:Operator overloading|Operator overloading]] can be used to replace calls to &lt;kbd&gt;gf_mul&lt;/kbd&gt; and &lt;kbd&gt;gf_div&lt;/kbd&gt; with the familiar operators &lt;kbd&gt;*&lt;/kbd&gt; and &lt;kbd&gt;/&lt;/kbd&gt;, but this can lead to confusion as to exactly what type of operation is being performed. Certain details can be generalized in ways that would make the class more widely useful. For example, [[w:Aztec Code|Aztec codes]] use five different Galois fields with element sizes ranging from 4 to 12 bits.

===Power and Inverse===
The logarithm table approach will once again simplify and speed up our calculations when computing the power and the inverse:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_pow(x, power):
    return gf_exp[(gf_log[x] * power) % 255]

def gf_inverse(x):
    return gf_exp[255 - gf_log[x]] # gf_inverse(x) == gf_div(1, x)
&lt;/syntaxhighlight&gt;

===Polynomials===
Before moving on to Reed&amp;ndash;Solomon codes, we need to define several operations on polynomials whose coefficients are Galois field elements. This is a potential source of confusion, since the elements themselves are described as polynomials; my advice is not to think about it too much. Adding to the confusion is the fact that ''x'' is still used as the placeholder. This ''x'' has nothing to do with the ''x'' mentioned previously, so don't mix them up.

The binary notation used previously for Galois field elements starts to become inconveniently bulky at this point, so I will switch to hexadecimal instead.
:00000001 ''x''&lt;sup&gt;4&lt;/sup&gt; + 00001111 ''x''&lt;sup&gt;3&lt;/sup&gt; + 00110110 ''x''&lt;sup&gt;2&lt;/sup&gt; + 01111000 ''x'' + 01000000 = &lt;kbd&gt;01&lt;/kbd&gt; ''x''&lt;sup&gt;4&lt;/sup&gt; + &lt;kbd&gt;0f&lt;/kbd&gt; ''x''&lt;sup&gt;3&lt;/sup&gt; + &lt;kbd&gt;36&lt;/kbd&gt; ''x''&lt;sup&gt;2&lt;/sup&gt; + &lt;kbd&gt;78&lt;/kbd&gt; ''x'' + &lt;kbd&gt;40&lt;/kbd&gt;
In Python, polynomials will be represented by a list of numbers in descending order of powers of ''x'', so the polynomial above becomes &lt;kbd&gt;[ 0x01, 0x0f, 0x36, 0x78, 0x40 ]&lt;/kbd&gt;. (The reverse order could have been used instead; both choices have their advantages and disadvantages.)

The first function multiplies a polynomial by a scalar.
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_poly_scale(p,x):
    r = [0] * len(p)
    for i in range(0, len(p)):
        r[i] = gf_mul(p[i], x)
    return r
&lt;/syntaxhighlight&gt;
''Note to Python programmers:'' This function is not written in a &quot;pythonic&quot; style. It could be expressed quite elegantly as a [[w:List comprehension|list comprehension]], but I have limited myself to language features that are easier to translate to other programming languages.

This function &quot;adds&quot; two polynomials (using exclusive-or, as usual).
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_poly_add(p,q):
    r = [0] * max(len(p),len(q))
    for i in range(0,len(p)):
        r[i+len(r)-len(p)] = p[i]
    for i in range(0,len(q)):
        r[i+len(r)-len(q)] ^= q[i]
    return r
&lt;/syntaxhighlight&gt;

The next function multiplies two polynomials.
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_poly_mul(p,q):
    '''Multiply two polynomials, inside Galois Field'''
    # Pre-allocate the result array
    r = [0] * (len(p)+len(q)-1)
    # Compute the polynomial multiplication (just like the outer product of two vectors,
    # we multiply each coefficients of p with all coefficients of q)
    for j in range(0, len(q)):
        for i in range(0, len(p)):
            r[i+j] ^= gf_mul(p[i], q[j]) # equivalent to: r[i + j] = gf_add(r[i+j], gf_mul(p[i], q[j]))
                                                         # -- you can see it's your usual polynomial multiplication
    return r
&lt;/syntaxhighlight&gt;

Finally, we need a function to evaluate a polynomial at a particular value of ''x'', producing a scalar result. [[w:Horner's method|Horner's method]] is used to avoid explicitly calculating powers of ''x''. Horner's method works by factorizing consecutively the terms, so that we always deal with ''x^1'', iteratively, avoiding the computation of higher degree terms:
:&lt;kbd&gt;01&lt;/kbd&gt; ''x''&lt;sup&gt;4&lt;/sup&gt; + &lt;kbd&gt;0f&lt;/kbd&gt; ''x''&lt;sup&gt;3&lt;/sup&gt; + &lt;kbd&gt;36&lt;/kbd&gt; ''x''&lt;sup&gt;2&lt;/sup&gt; + &lt;kbd&gt;78&lt;/kbd&gt; ''x'' + &lt;kbd&gt;40&lt;/kbd&gt; = (((&lt;kbd&gt;01&lt;/kbd&gt; ''x'' + &lt;kbd&gt;0f&lt;/kbd&gt;) ''x'' + &lt;kbd&gt;36&lt;/kbd&gt;) ''x'' + &lt;kbd&gt;78&lt;/kbd&gt;) ''x'' + &lt;kbd&gt;40&lt;/kbd&gt;
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_poly_eval(poly, x):
    '''Evaluates a polynomial in GF(2^p) given the value for x. This is based on Horner's scheme for maximum efficiency.'''
    y = poly[0]
    for i in range(1, len(poly)):
        y = gf_mul(y, x) ^ poly[i]
    return y
&lt;/syntaxhighlight&gt;

There's still one missing polynomial operation that we will need: polynomial division. This is more complicated than the other operations on polynomial, so we will study it in the next chapter, along with Reed&amp;ndash;Solomon encoding.

==Reed&amp;ndash;Solomon codes==
Now that the preliminaries are out of the way, we are ready to begin looking at Reed&amp;ndash;Solomon codes.

===Insight of the coding theory===

But first, why did we have to learn about finite fields and polynomials? Because this is the main insight of error-correcting codes like Reed&amp;ndash;Solomon: instead of just seeing a message as a series of (ASCII) numbers, we see it as '''a polynomial''' following the very well-defined '''rules of finite field arithmetic'''. In other words, by representing the data using polynomials and finite fields arithmetic, '''we added a structure to the data'''. The values of the message are still the same, but this conceptual structure now allows us to operate on the message, on its characters values, using well defined mathematical rules. This structure, that we always know because it's outside and independent of the data, is what allows us to repair a corrupted message.

Thus, even if in your code implementation you may choose to not explicitly represent the polynomials and the finite field arithmetic, these notions are essential for the error-correcting codes to work, and you will find these notions to underlie (even if implicitly) any implementation.

And now we will put these notions into practice!

===RS encoding===

====Encoding outline====
Like BCH codes, Reed&amp;ndash;Solomon codes are encoded by dividing the polynomial representing the message by an irreducible generator polynomial, and then the remainder is the RS code, which we will just append to the original message.

Why? We previously said that the principle behind BCH codes, and most other error correcting codes, is to use a reduced dictionary with very different words as to maximize the distance between words, and that longer words have greater distance: here it's the same principle, first because we lengthen the original message with additional symbols (the remainder) which raises the distance, and secondly because the remainder is almost unique (thanks to the carefully designed irreducible generator polynomial), so that it can be exploited by clever algorithms to deduce parts of the original message.

To summarize, with an approximated analogy to encryption: our '''generator polynomial''' is our encoding '''dictionary''', and '''polynomial division''' is the operator to '''convert''' our message using the dictionary (the generator polynomial) into a RS code.

====Exception management====
To manage errors and cases where we can't correct a message, we will display a meaningful error message, by raising an exception. We will make our own custom exception so that users can easily catch and manage them:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
class ReedSolomonError(Exception):
    pass
&lt;/syntaxhighlight&gt;

To display an error by raising our custom exception, we can then simply do the following:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
raise ReedSolomonError(&quot;Some error message&quot;)
&lt;/syntaxhighlight&gt;

And you can easily catch this exception to manage it by using a try/except block:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
try:
    raise ReedSolomonError(&quot;Some error message&quot;)
except ReedSolomonError as e:
    pass # do something here
&lt;/syntaxhighlight&gt;

====RS generator polynomial====
Reed&amp;ndash;Solomon codes use a '''generator polynomial''' similar to BCH codes (not to be confused with the generator number alpha). The generator is the product of factors (''x'' - &amp;alpha;&lt;sup&gt;''n''&lt;/sup&gt;), starting with ''n''=0 for QR codes.

For example:
''g''&lt;sub&gt;4&lt;/sub&gt;(''x'') = (''x'' - &amp;alpha;&lt;sup&gt;0&lt;/sup&gt;) (''x'' - &amp;alpha;&lt;sup&gt;1&lt;/sup&gt;) (''x'' - &amp;alpha;&lt;sup&gt;2&lt;/sup&gt;) (''x'' - &amp;alpha;&lt;sup&gt;3&lt;/sup&gt;) 

The same as (x + a&lt;sup&gt;''i''&lt;/sup&gt;) because of GF(2^8). 

''g''&lt;sub&gt;4&lt;/sub&gt;(''x'') = ''x''&lt;sup&gt;4&lt;/sup&gt; - (&amp;alpha;&lt;sup&gt;3&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;2&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;1&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;0&lt;/sup&gt;) ''x''&lt;sup&gt;3&lt;/sup&gt; + ((&amp;alpha;&lt;sup&gt;0&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;1&lt;/sup&gt;) (&amp;alpha;&lt;sup&gt;2&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;3&lt;/sup&gt;)+(&amp;alpha;&lt;sup&gt;5&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;1&lt;/sup&gt;)) ''x''&lt;sup&gt;2&lt;/sup&gt; + (&amp;alpha;&lt;sup&gt;6&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;5&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;4&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;3&lt;/sup&gt;) ''x'' +&amp;alpha;&lt;sup&gt;6&lt;/sup&gt;

''g''&lt;sub&gt;4&lt;/sub&gt;(''x'') = ''x''&lt;sup&gt;4&lt;/sup&gt; - (&amp;alpha;&lt;sup&gt;3&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;2&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;1&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;0&lt;/sup&gt;) ''x''&lt;sup&gt;3&lt;/sup&gt; + (&amp;alpha;&lt;sup&gt;2&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;3&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;3&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;4&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;5&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;1&lt;/sup&gt;) ''x''&lt;sup&gt;2&lt;/sup&gt; + (&amp;alpha;&lt;sup&gt;6&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;5&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;4&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;3&lt;/sup&gt;) ''x'' +&amp;alpha;&lt;sup&gt;6&lt;/sup&gt; 

''g''&lt;sub&gt;4&lt;/sub&gt;(''x'') = ''x''&lt;sup&gt;4&lt;/sup&gt; - (&amp;alpha;&lt;sup&gt;3&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;2&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;1&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;0&lt;/sup&gt;) ''x''&lt;sup&gt;3&lt;/sup&gt; + (&amp;alpha;&lt;sup&gt;5&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;4&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;2&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;1&lt;/sup&gt;) ''x''&lt;sup&gt;2&lt;/sup&gt; + (&amp;alpha;&lt;sup&gt;6&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;5&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;4&lt;/sup&gt;+&amp;alpha;&lt;sup&gt;3&lt;/sup&gt;) ''x'' +&amp;alpha;&lt;sup&gt;6&lt;/sup&gt; 

''g''&lt;sub&gt;4&lt;/sub&gt;(''x'') = &lt;kbd&gt;01&lt;/kbd&gt; ''x''&lt;sup&gt;4&lt;/sup&gt; + &lt;kbd&gt;0f&lt;/kbd&gt; ''x''&lt;sup&gt;3&lt;/sup&gt; + &lt;kbd&gt;36&lt;/kbd&gt; ''x''&lt;sup&gt;2&lt;/sup&gt; + &lt;kbd&gt;78&lt;/kbd&gt; ''x'' + &lt;kbd&gt;40&lt;/kbd&gt;

Here is a function that computes the generator polynomial for a given number of error correction symbols.
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_generator_poly(nsym):
    '''Generate an irreducible generator polynomial (necessary to encode a message into Reed-Solomon)'''
    g = [1]
    for i in range(0, nsym):
        g = gf_poly_mul(g, [1, gf_pow(2, i)])
    return g
&lt;/syntaxhighlight&gt;
This function is somewhat inefficient in that it allocates successively larger arrays for &lt;kbd&gt;g&lt;/kbd&gt;. While this is unlikely to be a performance problem in practice, readers who are inveterate optimizers may find it interesting to rewrite it so that &lt;kbd&gt;g&lt;/kbd&gt; is only allocated once, or you can compute once and memorize g since it is fixed for a given nsym, so you can reuse g.

====Polynomial division====
Several algorithms for polynomial division exist, the simplest one that is often taught in elementary school is [[w:Polynomial_long_division|long division]]. This example shows the calculation for the message &lt;kbd&gt;12 34 56&lt;/kbd&gt;.
                &lt;u&gt;              12 da df&lt;/u&gt;
 01 0f 36 78 40 ) 12 34 56 00 00 00 00
                ^ &lt;u&gt;12 ee 2b 23 f4&lt;/u&gt;
                     da 7d 23 f4 00
                   ^ &lt;u&gt;da a2 85 79 84&lt;/u&gt;
                        df a6 8d 84 00
                      ^ &lt;u&gt;df 91 6b fc d9&lt;/u&gt;
                           37 e6 78 d9

Note: The concepts of polynomial long division apply, but there are a few important differences: When computing the resulting terms/coefficients that will be Galois Field subtracted from the divisor, bitwise carryless multiplication is performed and the result &quot;bitstream&quot; is XORed from the first encountered MSB with the chosen primitive polynomial until the answer is less than the Galois Field value, in this case, 256. The XOR &quot;subtractions&quot; are then performed as usual.

To illustrate the method for one operation (0x12 * 0x36):
 
     00010010 ( 12 )
   x &lt;u&gt;00110110&lt;/u&gt; ( 36 )
    00110110
 &lt;u&gt;00110110    &lt;/u&gt;
 001100001100
  ^100011101   &lt;-- XOR with primitive polynomial value (11D)...
 000100110110
   ^100011101  &lt;-- ...until answer is less than 256.
     00101011
        2   b

The remainder is concatenated with the message, so the encoded message is &lt;kbd&gt;12 34 56 37 e6 78 d9&lt;/kbd&gt;.

However, long division is quite slow as it requires a lot of recursive iterations to terminate. More efficient strategies can be devised, such as using [[synthetic division]] (also called Horner's method, a good tutorial video can be found on [https://www.khanacademy.org/math/algebra2/polynomial_and_rational/synthetic-division/v/synthetic-division Khan Academy]). Here is a function that implements [[w:Synthetic_division#Expanded_synthetic_division|extended synthetic division]] of GF(2^p) polynomials (extended because the divisor is a polynomial instead of a monomial):

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def gf_poly_div(dividend, divisor):
    '''Fast polynomial division by using Extended Synthetic Division and optimized for GF(2^p) computations
    (doesn't work with standard polynomials outside of this galois field, see the Wikipedia article for generic algorithm).'''
    # CAUTION: this function expects polynomials to follow the opposite convention at decoding:
    # the terms must go from the biggest to lowest degree (while most other functions here expect
    # a list from lowest to biggest degree). eg: 1 + 2x + 5x^2 = [5, 2, 1], NOT [1, 2, 5]

    msg_out = list(dividend) # Copy the dividend
    #normalizer = divisor[0] # precomputing for performance
    for i in range(0, len(dividend) - (len(divisor)-1)):
        #msg_out[i] /= normalizer # for general polynomial division (when polynomials are non-monic), the usual way of using
                                  # synthetic division is to divide the divisor g(x) with its leading coefficient, but not needed here.
        coef = msg_out[i] # precaching
        if coef != 0: # log(0) is undefined, so we need to avoid that case explicitly (and it's also a good optimization).
            for j in range(1, len(divisor)): # in synthetic division, we always skip the first coefficient of the divisior,
                                              # because it's only used to normalize the dividend coefficient
                if divisor[j] != 0: # log(0) is undefined
                    msg_out[i + j] ^= gf_mul(divisor[j], coef) # equivalent to the more mathematically correct
                                                               # (but xoring directly is faster): msg_out[i + j] += -divisor[j] * coef

    # The resulting msg_out contains both the quotient and the remainder, the remainder being the size of the divisor
    # (the remainder has necessarily the same degree as the divisor -- not length but degree == length-1 -- since it's
    # what we couldn't divide from the dividend), so we compute the index where this separation is, and return the quotient and remainder.
    separator = -(len(divisor)-1)
    return msg_out[:separator], msg_out[separator:] # return quotient, remainder.
&lt;/syntaxhighlight&gt;

====Encoding main function====

And now, here's how to encode a message to get its RS code:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_encode_msg(msg_in, nsym):
    '''Reed-Solomon main encoding function'''
    gen = rs_generator_poly(nsym)

    # Pad the message, then divide it by the irreducible generator polynomial
    _, remainder = gf_poly_div(msg_in + [0] * (len(gen)-1), gen)
    # The remainder is our RS code! Just append it to our original message to get our full codeword (this represents a polynomial of max 256 terms)
    msg_out = msg_in + remainder
    # Return the codeword
    return msg_out
&lt;/syntaxhighlight&gt;

Simple, isn't it? Encoding is in fact the easiest part in Reed&amp;ndash;Solomon, and it's always the same approach (polynomial division). Decoding is the tough part of Reed&amp;ndash;Solomon, and you will find a lot of different algorithms depending on your needs, but we will touch on that later on.

This function is quite fast, but since encoding is quite critical, here is an enhanced encoding function that inlines the polynomial synthetic division, which is the form that you will most often find in Reed&amp;ndash;Solomon software libraries:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_encode_msg(msg_in, nsym):
    '''Reed-Solomon main encoding function, using polynomial division (algorithm Extended Synthetic Division)'''
    if (len(msg_in) + nsym) &gt; 255: raise ValueError(&quot;Message is too long (%i when max is 255)&quot; % (len(msg_in)+nsym))
    gen = rs_generator_poly(nsym)
    # Init msg_out with the values inside msg_in and pad with len(gen)-1 bytes (which is the number of ecc symbols).
    msg_out = [0] * (len(msg_in) + len(gen)-1)
    # Initializing the Synthetic Division with the dividend (= input message polynomial)
    msg_out[:len(msg_in)] = msg_in

    # Synthetic division main loop
    for i in range(len(msg_in)):
        # Note that it's msg_out here, not msg_in. Thus, we reuse the updated value at each iteration
        # (this is how Synthetic Division works: instead of storing in a temporary register the intermediate values,
        # we directly commit them to the output).
        coef = msg_out[i]

        # log(0) is undefined, so we need to manually check for this case. There's no need to check
        # the divisor here because we know it can't be 0 since we generated it.
        if coef != 0:
            # in synthetic division, we always skip the first coefficient of the divisior, because it's only used to normalize the dividend coefficient (which is here useless since the divisor, the generator polynomial, is always monic)
            for j in range(1, len(gen)):
                msg_out[i+j] ^= gf_mul(gen[j], coef) # equivalent to msg_out[i+j] += gf_mul(gen[j], coef)

    # At this point, the Extended Synthetic Divison is done, msg_out contains the quotient in msg_out[:len(msg_in)]
    # and the remainder in msg_out[len(msg_in):]. Here for RS encoding, we don't need the quotient but only the remainder
    # (which represents the RS code), so we can just overwrite the quotient with the input message, so that we get
    # our complete codeword composed of the message + code.
    msg_out[:len(msg_in)] = msg_in

    return msg_out
&lt;/syntaxhighlight&gt;
This algorithm is faster, but it's still quite slow for practical use, particularly in Python. There are some ways to optimize the speed by using various tricks, such as inlining (instead of gf_mul, replace by the operation to avoid a call), by precomputing (the logarithm of gen and of coef, or even by generating a multiplication table &amp;ndash; but it seems the latter does not work well in Python), by using statically typed constructs (assign gf_log and gf_exp to &lt;kbd&gt;array.array('i', [...])&lt;/kbd&gt;), by using memoryviews (like by changing all your lists to bytearrays), by running it with PyPy, or by converting the algorithm into a Cython or a C extension&lt;ref&gt;Optimizing a reed-solomon encoder, question on StackOverflow.com http://stackoverflow.com/questions/30363903/optimizing-a-reed-solomon-encoder-polynomial-division&lt;/ref&gt;.

This example shows the encode function applied to the message in the sample QR code introduced earlier. The calculated error correction symbols (on the second line) match the values decoded from the QR code.
&lt;pre&gt;
&gt;&gt;&gt; msg_in = [ 0x40, 0xd2, 0x75, 0x47, 0x76, 0x17, 0x32, 0x06,
...            0x27, 0x26, 0x96, 0xc6, 0xc6, 0x96, 0x70, 0xec ]
&gt;&gt;&gt; msg = rs_encode_msg(msg_in, 10)
&gt;&gt;&gt; for i in range(0,len(msg)):
...    print(hex(msg[i]), end=' ')
... 
0x40 0xd2 0x75 0x47 0x76 0x17 0x32 0x6 0x27 0x26 0x96 0xc6 0xc6 0x96 0x70 0xec
0xbc 0x2a 0x90 0x13 0x6b 0xaf 0xef 0xfd 0x4b 0xe0
&lt;/pre&gt;
''Python version note:'' The syntax for the &lt;kbd&gt;print&lt;/kbd&gt; function has changed, and this example uses the Python 3.0+ version. In previous versions of Python (particularly Python 2.x), replace the &lt;kbd&gt;print&lt;/kbd&gt; line with &lt;kbd&gt;print hex(msg[i]),&lt;/kbd&gt; (including the final comma) and &lt;kbd&gt;range&lt;/kbd&gt; by &lt;kbd&gt;xrange&lt;/kbd&gt;.

===RS decoding===

====Decoding outline====

Reed&amp;ndash;Solomon decoding is the process that, from a potentially corrupted message and a RS code, returns a corrected message. In other words, decoding is the process of repairing your message using the previously computed RS code.

Although there is only one way to encode a message with Reed&amp;ndash;Solomon, there are lots of different ways to decode them, and thus there are a lot of different decoding algorithms.

However, we can generally outline the decoding process in 5 steps&lt;ref&gt;Tilavat, V., &amp; Shukla, Y. (2014). Simplification of procedure for decoding Reed&amp;ndash;Solomon codes using various algorithms: an introductory survey. International Journal of Engineering Development and Research, 2(1), 279-283.&lt;/ref&gt;&lt;ref&gt;Sarwate, D. V., &amp; Morrison, R. D. (1990). Decoder malfunction in BCH decoders. Information Theory, IEEE Transactions on, 36(4), 884-889.&lt;/ref&gt;:

# Compute the '''syndromes polynomial'''. This allows us to analyze what characters are in error using Berlekamp-Massey (or another algorithm), and also to quickly check if the input message is corrupted at all.
# Compute the erasure/error '''locator polynomial''' (from the syndromes). This is computed by Berlekamp-Massey, and is a detector that will tell us exactly what characters are corrupted.
# Compute the erasure/error '''evaluator polynomial''' (from the syndromes and erasure/error locator polynomial). Necessary to evaluate how much the characters were tampered (ie, helps to compute the magnitude).
# Compute the erasure/error '''magnitude polynomial''' (from all 3 polynomials above): this polynomial can also be called the corruption polynomial, since in fact it exactly stores the values that need to be subtracted from the received message to get the original, correct message (i.e., with correct values for erased characters). In other words, at this point, we extracted the noise and stored it in this polynomial, and we just have to remove this noise from the input message to repair it.
# '''Repair the input message''' simply by subtracting the magnitude polynomial from the input message.

We will describe each of those five steps below.

In addition, decoders can also be classified by the type of error they can repair: erasures (we know the location of the corrupted characters but not the magnitude), errors (we ignore both the location and magnitude), or a mix of errors-and-erasures. We will describe how to support all of these.

====Syndrome calculation====
Decoding a Reed&amp;ndash;Solomon message involves several steps. The first step is to calculate the &quot;syndrome&quot; of the message. Treat the message as a polynomial and evaluate it at &amp;alpha;&lt;sup&gt;0&lt;/sup&gt;, &amp;alpha;&lt;sup&gt;1&lt;/sup&gt;, &amp;alpha;&lt;sup&gt;2&lt;/sup&gt;, ..., &amp;alpha;&lt;sup&gt;''n''&lt;/sup&gt;. Since these are the zeros of the generator polynomial, the result should be zero if the scanned message is undamaged (this can be used to check if the message is corrupted, and after correction of a corrupted message if the message was completely repaired). If not, the syndromes contain all the information necessary to determine the correction that should be made. It is simple to write a function to calculate the syndromes.
&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_calc_syndromes(msg, nsym):
    '''Given the received codeword msg and the number of error correcting symbols (nsym), computes the syndromes polynomial.
    Mathematically, it's essentially equivalent to a Fourrier Transform (Chien search being the inverse).
    '''
    # Note the &quot;[0] +&quot; : we add a 0 coefficient for the lowest degree (the constant). This effectively shifts the syndrome, and will shift every computations depending on the syndromes (such as the errors locator polynomial, errors evaluator polynomial, etc. but not the errors positions).
    # This is not necessary, you can adapt subsequent computations to start from 0 instead of skipping the first iteration (ie, the often seen range(1, n-k+1)),
    synd = [0] * nsym
    for i in range(0, nsym):
        synd[i] = gf_poly_eval(msg, gf_pow(2,i))
    return [0] + synd # pad with one 0 for mathematical precision (else we can end up with weird calculations sometimes)
&lt;/syntaxhighlight&gt;

Continuing the example, we see that the syndromes of the original codeword without any corruption are indeed zero. Introducing a corruption of at least one character into the message or its RS code gives nonzero syndromes.
&lt;pre&gt;
&gt;&gt;&gt; synd = rs_calc_syndromes(msg, 10)
&gt;&gt;&gt; print(synd)
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] # not corrupted message = all 0 syndromes
&gt;&gt;&gt; msg[0] = 0  # deliberately damage the message
&gt;&gt;&gt; synd = rs_calc_syndromes(msg, 10)
&gt;&gt;&gt; print(synd)
[0, 64, 192, 93, 231, 52, 92, 228, 49, 83, 245] # when corrupted, the syndromes will be non zero
&lt;/pre&gt;

Here is the code to automate this checking:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_check(msg, nsym):
    '''Returns true if the message + ecc has no error or false otherwise (may not always catch a wrong decoding or a wrong message, particularly if there are too many errors -- above the Singleton bound --, but it usually does)'''
    return ( max(rs_calc_syndromes(msg, nsym)) == 0 )
&lt;/syntaxhighlight&gt;

====Erasure correction====
It is simplest to correct mistakes in the code if the locations of the mistakes are already known. This is known as '''erasure correction'''. It is possible to correct one erased symbol (ie, character) for each error-correction symbol added to the code. If the error locations are not known, two EC symbols are needed for each symbol error (so you can correct twice less errors than erasures). This makes erasure correction useful in practice if part of the QR code being scanned is covered or physically torn away. It may be difficult for a scanner to determine that this has happened, though, so not all QR code scanners can perform erasure correction.

Now that we already have the syndromes, we need to compute the locator polynomial. This is easy:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_find_errata_locator(e_pos):
    '''Compute the erasures/errors/errata locator polynomial from the erasures/errors/errata positions
       (the positions must be relative to the x coefficient, eg: &quot;hello worldxxxxxxxxx&quot; is tampered to &quot;h_ll_ worldxxxxxxxxx&quot;
       with xxxxxxxxx being the ecc of length n-k=9, here the string positions are [1, 4], but the coefficients are reversed
       since the ecc characters are placed as the first coefficients of the polynomial, thus the coefficients of the
       erased characters are n-1 - [1, 4] = [18, 15] = erasures_loc to be specified as an argument.'''

    e_loc = [1] # just to init because we will multiply, so it must be 1 so that the multiplication starts correctly without nulling any term
    # erasures_loc = product(1 - x*alpha**i) for i in erasures_pos and where alpha is the alpha chosen to evaluate polynomials.
    for i in e_pos:
        e_loc = gf_poly_mul( e_loc, gf_poly_add([1], [gf_pow(2, i), 0]) )
    return e_loc
&lt;/syntaxhighlight&gt;

Next, computing the erasure/error evaluator polynomial from the locator polynomial is easy, it's simply a polynomial multiplication followed by a polynomial division (that you can replace by a list slicing because that's the effect we want in the end):

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_find_error_evaluator(synd, err_loc, nsym):
    '''Compute the error (or erasures if you supply sigma=erasures locator polynomial, or errata) evaluator polynomial Omega
       from the syndrome and the error/erasures/errata locator Sigma.'''

    # Omega(x) = [ Synd(x) * Error_loc(x) ] mod x^(n-k+1)
    _, remainder = gf_poly_div( gf_poly_mul(synd, err_loc), ([1] + [0]*(nsym+1)) ) # first multiply syndromes * errata_locator, then do a
                                                                                   # polynomial division to truncate the polynomial to the
                                                                                   # required length

    # Faster way that is equivalent
    #remainder = gf_poly_mul(synd, err_loc) # first multiply the syndromes with the errata locator polynomial
    #remainder = remainder[len(remainder)-(nsym+1):] # then slice the list to truncate it (which represents the polynomial), which
                                                                          # is equivalent to dividing by a polynomial of the length we want

    return remainder
&lt;/syntaxhighlight&gt;

Finally, the [[w:Forney algorithm|Forney algorithm]] is used to calculate the correction values (also called the error magnitude polynomial). It is implemented in the function below.

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_correct_errata(msg_in, synd, err_pos): # err_pos is a list of the positions of the errors/erasures/errata
    '''Forney algorithm, computes the values (error magnitude) to correct the input message.'''
    # calculate errata locator polynomial to correct both errors and erasures (by combining the errors positions given by the error locator polynomial found by BM with the erasures positions given by caller)
    coef_pos = [len(msg_in) - 1 - p for p in err_pos] # need to convert the positions to coefficients degrees for the errata locator algo to work (eg: instead of [0, 1, 2] it will become [len(msg)-1, len(msg)-2, len(msg) -3])
    err_loc = rs_find_errata_locator(coef_pos)
    # calculate errata evaluator polynomial (often called Omega or Gamma in academic papers)
    err_eval = rs_find_error_evaluator(synd[::-1], err_loc, len(err_loc)-1)[::-1]

    # Second part of Chien search to get the error location polynomial X from the error positions in err_pos (the roots of the error locator polynomial, ie, where it evaluates to 0)
    X = [] # will store the position of the errors
    for i in range(0, len(coef_pos)):
        l = 255 - coef_pos[i]
        X.append( gf_pow(2, -l) )

    # Forney algorithm: compute the magnitudes
    E = [0] * (len(msg_in)) # will store the values that need to be corrected (substracted) to the message containing errors. This is sometimes called the error magnitude polynomial.
    Xlength = len(X)
    for i, Xi in enumerate(X):

        Xi_inv = gf_inverse(Xi)

        # Compute the formal derivative of the error locator polynomial (see Blahut, Algebraic codes for data transmission, pp 196-197).
        # the formal derivative of the errata locator is used as the denominator of the Forney Algorithm, which simply says that the ith error value is given by error_evaluator(gf_inverse(Xi)) / error_locator_derivative(gf_inverse(Xi)). See Blahut, Algebraic codes for data transmission, pp 196-197.
        err_loc_prime_tmp = []
        for j in range(0, Xlength):
            if j != i:
                err_loc_prime_tmp.append( gf_sub(1, gf_mul(Xi_inv, X[j])) )
        # compute the product, which is the denominator of the Forney algorithm (errata locator derivative)
        err_loc_prime = 1
        for coef in err_loc_prime_tmp:
            err_loc_prime = gf_mul(err_loc_prime, coef)
        # equivalent to: err_loc_prime = functools.reduce(gf_mul, err_loc_prime_tmp, 1)

        # Compute y (evaluation of the errata evaluator polynomial)
        # This is a more faithful translation of the theoretical equation contrary to the old forney method. Here it is an exact reproduction:
        # Yl = omega(Xl.inverse()) / prod(1 - Xj*Xl.inverse()) for j in len(X)
        y = gf_poly_eval(err_eval[::-1], Xi_inv) # numerator of the Forney algorithm (errata evaluator evaluated)
        y = gf_mul(gf_pow(Xi, 1), y)
        
        # Check: err_loc_prime (the divisor) should not be zero.
        if err_loc_prime == 0:
            raise ReedSolomonError(&quot;Could not find error magnitude&quot;)    # Could not find error magnitude

        # Compute the magnitude
        magnitude = gf_div(y, err_loc_prime) # magnitude value of the error, calculated by the Forney algorithm (an equation in fact): dividing the errata evaluator with the errata locator derivative gives us the errata magnitude (ie, value to repair) the ith symbol
        E[err_pos[i]] = magnitude # store the magnitude for this error into the magnitude polynomial

    # Apply the correction of values to get our message corrected! (note that the ecc bytes also gets corrected!)
    # (this isn't the Forney algorithm, we just apply the result of decoding here)
    msg_in = gf_poly_add(msg_in, E) # equivalent to Ci = Ri - Ei where Ci is the correct message, Ri the received (senseword) message, and Ei the errata magnitudes (minus is replaced by XOR since it's equivalent in GF(2^p)). So in fact here we substract from the received message the errors magnitude, which logically corrects the value to what it should be.
    return msg_in
&lt;/syntaxhighlight&gt;

''Mathematics note:'' The denominator of the expression for the error value is the [[w:Formal derivative|formal derivative]] of the error locator polynomial &lt;kbd&gt;q&lt;/kbd&gt;. This is calculated by the usual procedure of replacing each term ''c''&lt;sub&gt;''n''&lt;/sub&gt;&amp;nbsp;''x''&lt;sup&gt;''n''&lt;/sup&gt; with ''n''&amp;nbsp;''c''&lt;sub&gt;''n''&lt;/sub&gt;&amp;nbsp;''x''&lt;sup&gt;''n''-1&lt;/sup&gt;. Since we're working in a field of [[w:Characteristic (algebra)|characteristic]] two, ''n''&amp;nbsp;''c''&lt;sub&gt;''n''&lt;/sub&gt; is equal to ''c''&lt;sub&gt;''n''&lt;/sub&gt; when ''n'' is odd, and 0 when ''n'' is even. Thus, we can simply remove the even coefficients (resulting in the polynomial &lt;kbd&gt;qprime&lt;/kbd&gt;) and evaluate &lt;kbd&gt;qprime(x&lt;sup&gt;2&lt;/sup&gt;)&lt;/kbd&gt;.

''Python note:'' This function uses [::-1] to inverse the order of the elements in a list. This is necessary because the functions do not all use the same ordering convention (ie, some use the least item first, others use the biggest item first). It also use a [[w:List_comprehension#Python|list comprehension]], which is simply a concise way to write a for loop where items are appended in a list, but the Python interpreter can optimize this a bit more than a loop.

Continuing the example, here we use &lt;kbd&gt;rs_correct_errata&lt;/kbd&gt; to restore the first byte of the message.

&lt;pre&gt;
&gt;&gt;&gt; msg[0] = 0
&gt;&gt;&gt; synd = rs_calc_syndromes(msg, 10)
&gt;&gt;&gt; msg = rs_correct_errata(msg, synd, [0]) # [0] is the list of the erasures locations, here it's the first character, at position 0
&gt;&gt;&gt; print(hex(msg[0]))
0x40
&lt;/pre&gt;

====Error correction====
In the more likely situation where the error locations are unknown (what we usually call '''errors''', in opposition to '''erasures''' where the locations are known), we will use the same steps as for erasures, but we now need additional steps to find the location. The [[w:Berlekamp&amp;ndash;Massey algorithm|Berlekamp&amp;ndash;Massey algorithm]] is used to calculate the error '''locator polynomial''', which we can use later on to determine the errors locations:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_find_error_locator(synd, nsym, erase_loc=None, erase_count=0):
    '''Find error/errata locator and evaluator polynomials with Berlekamp-Massey algorithm'''
    # The idea is that BM will iteratively estimate the error locator polynomial.
    # To do this, it will compute a Discrepancy term called Delta, which will tell us if the error locator polynomial needs an update or not
    # (hence why it's called discrepancy: it tells us when we are getting off board from the correct value).

    # Init the polynomials
    if erase_loc: # if the erasure locator polynomial is supplied, we init with its value, so that we include erasures in the final locator polynomial
        err_loc = list(erase_loc)
        old_loc = list(erase_loc)
    else:
        err_loc = [1] # This is the main variable we want to fill, also called Sigma in other notations or more formally the errors/errata locator polynomial.
        old_loc = [1] # BM is an iterative algorithm, and we need the errata locator polynomial of the previous iteration in order to update other necessary variables.
    #L = 0 # update flag variable, not needed here because we use an alternative equivalent way of checking if update is needed (but using the flag could potentially be faster depending on if using length(list) is taking linear time in your language, here in Python it's constant so it's as fast.

    # Fix the syndrome shifting: when computing the syndrome, some implementations may prepend a 0 coefficient for the lowest degree term (the constant). This is a case of syndrome shifting, thus the syndrome will be bigger than the number of ecc symbols (I don't know what purpose serves this shifting). If that's the case, then we need to account for the syndrome shifting when we use the syndrome such as inside BM, by skipping those prepended coefficients.
    # Another way to detect the shifting is to detect the 0 coefficients: by definition, a syndrome does not contain any 0 coefficient (except if there are no errors/erasures, in this case they are all 0). This however doesn't work with the modified Forney syndrome, which set to 0 the coefficients corresponding to erasures, leaving only the coefficients corresponding to errors.
    synd_shift = len(synd) - nsym

    for i in range(0, nsym-erase_count): # generally: nsym-erase_count == len(synd), except when you input a partial erase_loc and using the full syndrome instead of the Forney syndrome, in which case nsym-erase_count is more correct (len(synd) will fail badly with IndexError).
        if erase_loc: # if an erasures locator polynomial was provided to init the errors locator polynomial, then we must skip the FIRST erase_count iterations (not the last iterations, this is very important!)
            K = erase_count+i+synd_shift
        else: # if erasures locator is not provided, then either there's no erasures to account or we use the Forney syndromes, so we don't need to use erase_count nor erase_loc (the erasures have been trimmed out of the Forney syndromes).
            K = i+synd_shift

        # Compute the discrepancy Delta
        # Here is the close-to-the-books operation to compute the discrepancy Delta: it's a simple polynomial multiplication of error locator with the syndromes, and then we get the Kth element.
        #delta = gf_poly_mul(err_loc[::-1], synd)[K] # theoretically it should be gf_poly_add(synd[::-1], [1])[::-1] instead of just synd, but it seems it's not absolutely necessary to correctly decode.
        # But this can be optimized: since we only need the Kth element, we don't need to compute the polynomial multiplication for any other element but the Kth. Thus to optimize, we compute the polymul only at the item we need, skipping the rest (avoiding a nested loop, thus we are linear time instead of quadratic).
        # This optimization is actually described in several figures of the book &quot;Algebraic codes for data transmission&quot;, Blahut, Richard E., 2003, Cambridge university press.
        delta = synd[K]
        for j in range(1, len(err_loc)):
            delta ^= gf_mul(err_loc[-(j+1)], synd[K - j]) # delta is also called discrepancy. Here we do a partial polynomial multiplication (ie, we compute the polynomial multiplication only for the term of degree K). Should be equivalent to brownanrs.polynomial.mul_at().
        #print &quot;delta&quot;, K, delta, list(gf_poly_mul(err_loc[::-1], synd)) # debugline

        # Shift polynomials to compute the next degree
        old_loc = old_loc + [0]

        # Iteratively estimate the errata locator and evaluator polynomials
        if delta != 0: # Update only if there's a discrepancy
            if len(old_loc) &gt; len(err_loc): # Rule B (rule A is implicitly defined because rule A just says that we skip any modification for this iteration)
            #if 2*L &lt;= K+erase_count: # equivalent to len(old_loc) &gt; len(err_loc), as long as L is correctly computed
                # Computing errata locator polynomial Sigma
                new_loc = gf_poly_scale(old_loc, delta)
                old_loc = gf_poly_scale(err_loc, gf_inverse(delta)) # effectively we are doing err_loc * 1/delta = err_loc // delta
                err_loc = new_loc
                # Update the update flag
                #L = K - L # the update flag L is tricky: in Blahut's schema, it's mandatory to use `L = K - L - erase_count` (and indeed in a previous draft of this function, if you forgot to do `- erase_count` it would lead to correcting only 2*(errors+erasures) &lt;= (n-k) instead of 2*errors+erasures &lt;= (n-k)), but in this latest draft, this will lead to a wrong decoding in some cases where it should correctly decode! Thus you should try with and without `- erase_count` to update L on your own implementation and see which one works OK without producing wrong decoding failures.

            # Update with the discrepancy
            err_loc = gf_poly_add(err_loc, gf_poly_scale(old_loc, delta))

    # Check if the result is correct, that there's not too many errors to correct
    while len(err_loc) and err_loc[0] == 0: del err_loc[0] # drop leading 0s, else errs will not be of the correct size
    errs = len(err_loc) - 1
    if (errs-erase_count) * 2 + erase_count &gt; nsym:
        raise ReedSolomonError(&quot;Too many errors to correct&quot;)    # too many errors to correct

    return err_loc
&lt;/syntaxhighlight&gt;

Then, using the error locator polynomial, we simply use a brute-force approach called trial substitution to find the zeros of this polynomial, which identifies the error locations (ie, the index of the characters that need to be corrected). A more efficient algorithm called Chien search exists, which avoids recomputing the whole evaluation at each iteration step, but this algorithm is left as an exercise to the reader.

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_find_errors(err_loc, nmess): # nmess is len(msg_in)
    '''Find the roots (ie, where evaluation = zero) of error polynomial by brute-force trial, this is a sort of Chien's search
    (but less efficient, Chien's search is a way to evaluate the polynomial such that each evaluation only takes constant time).'''
    errs = len(err_loc) - 1
    err_pos = []
    for i in range(nmess): # normally we should try all 2^8 possible values, but here we optimize to just check the interesting symbols
        if gf_poly_eval(err_loc, gf_pow(2, i)) == 0: # It's a 0? Bingo, it's a root of the error locator polynomial,
                                                                                   # in other terms this is the location of an error
            err_pos.append(nmess - 1 - i)
    # Sanity check: the number of errors/errata positions found should be exactly the same as the length of the errata locator polynomial
    if len(err_pos) != errs:
        # couldn't find error locations
        raise ReedSolomonError(&quot;Too many (or few) errors found by Chien Search for the errata locator polynomial!&quot;)
    return err_pos
&lt;/syntaxhighlight&gt;
''Mathematics note:'' When the error locator polynomial is linear (&lt;kbd&gt;err_poly&lt;/kbd&gt; has length 2), it can be solved easily without resorting to a brute-force approach. The function presented above does not take advantage of this fact, but the interested reader may wish to implement the more efficient solution. Similarly, when the error locator is quadratic, it can be solved by using a [[w:Quadratic equation#Generalization of quadratic equation|generalization of the quadratic formula]]. A more ambitious reader may wish to implement this procedure as well.

Here is an example where three errors in the message are corrected:

&lt;pre&gt;
&gt;&gt;&gt; print(hex(msg[10]))
0x96
&gt;&gt;&gt; msg[0] = 6
&gt;&gt;&gt; msg[10] = 7
&gt;&gt;&gt; msg[20] = 8
&gt;&gt;&gt; synd = rs_calc_syndromes(msg, 10)
&gt;&gt;&gt; err_loc = rs_find_error_locator(synd, nsym)
&gt;&gt;&gt; pos = rs_find_errors(err_loc[::-1], len(msg)) # find the errors locations
&gt;&gt;&gt; print(pos)
[20, 10, 0]
&gt;&gt;&gt; msg = rs_correct_errata(msg, synd, pos)
&gt;&gt;&gt; print(hex(msg[10]))
0x96
&lt;/pre&gt;

====Error and erasure correction====

It is possible for a Reed&amp;ndash;Solomon decoder to decode both erasures and errors at the same time, up to a limit (called the Singleton Bound) of &lt;kbd&gt;2*e+v &lt;= (n-k)&lt;/kbd&gt;, where &lt;kbd&gt;e&lt;/kbd&gt; is the number of errors, &lt;kbd&gt;v&lt;/kbd&gt; the number of erasures and &lt;kbd&gt;(n-k)&lt;/kbd&gt; the number of RS code characters (called &lt;kbd&gt;nsym&lt;/kbd&gt; in the code). Basically, it means that for every erasures, you just need one RS code character to repair it, while for every errors you need two RS code characters (because you need to find the position in addition of the value/magnitude to correct). Such a decoder is called an errors-and-erasures decoder, or an '''errata decoder'''.

In order to correct both errors and erasures, we must prevent the erasures from interfering with the error location process. This can be done by calculating the Forney syndromes, as follows.

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_forney_syndromes(synd, pos, nmess):
    # Compute Forney syndromes, which computes a modified syndromes to compute only errors (erasures are trimmed out). Do not confuse this with Forney algorithm, which allows to correct the message based on the location of errors.
    erase_pos_reversed = [nmess-1-p for p in pos] # prepare the coefficient degree positions (instead of the erasures positions)

    # Optimized method, all operations are inlined
    fsynd = list(synd[1:])      # make a copy and trim the first coefficient which is always 0 by definition
    for i in range(0, len(pos)):
        x = gf_pow(2, erase_pos_reversed[i])
        for j in range(0, len(fsynd) - 1):
            fsynd[j] = gf_mul(fsynd[j], x) ^ fsynd[j + 1]

    # Equivalent, theoretical way of computing the modified Forney syndromes: fsynd = (erase_loc * synd) % x^(n-k)
    # See Shao, H. M., Truong, T. K., Deutsch, L. J., &amp; Reed, I. S. (1986, April). A single chip VLSI Reed-Solomon decoder. In Acoustics, Speech, and Signal Processing, IEEE International Conference on ICASSP'86. (Vol. 11, pp. 2151-2154). IEEE.ISO 690
    #erase_loc = rs_find_errata_locator(erase_pos_reversed, generator=generator) # computing the erasures locator polynomial
    #fsynd = gf_poly_mul(erase_loc[::-1], synd[1:]) # then multiply with the syndrome to get the untrimmed forney syndrome
    #fsynd = fsynd[len(pos):] # then trim the first erase_pos coefficients which are useless. Seems to be not necessary, but this reduces the computation time later in BM (thus it's an optimization).

    return fsynd
&lt;/syntaxhighlight&gt;

The Forney syndromes can then be used in place of the regular syndromes in the error location process.

The function &lt;kbd&gt;rs_correct_msg&lt;/kbd&gt; below brings the complete procedure together. Erasures are indicated by providing &lt;kbd&gt;erase_pos&lt;/kbd&gt;, a list of erasures index positions in the message &lt;kbd&gt;msg_in&lt;/kbd&gt; (the full received message: original message + ecc).

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
def rs_correct_msg(msg_in, nsym, erase_pos=None):
    '''Reed-Solomon main decoding function'''
    if len(msg_in) &gt; 255: # can't decode, message is too big
        raise ValueError(&quot;Message is too long (%i when max is 255)&quot; % len(msg_in))

    msg_out = list(msg_in)     # copy of message
    # erasures: set them to null bytes for easier decoding (but this is not necessary, they will be corrected anyway, but debugging will be easier with null bytes because the error locator polynomial values will only depend on the errors locations, not their values)
    if erase_pos is None:
        erase_pos = []
    else:
        for e_pos in erase_pos:
            msg_out[e_pos] = 0
    # check if there are too many erasures to correct (beyond the Singleton bound)
    if len(erase_pos) &gt; nsym: raise ReedSolomonError(&quot;Too many erasures to correct&quot;)
    # prepare the syndrome polynomial using only errors (ie: errors = characters that were either replaced by null byte
    # or changed to another character, but we don't know their positions)
    synd = rs_calc_syndromes(msg_out, nsym)
    # check if there's any error/erasure in the input codeword. If not (all syndromes coefficients are 0), then just return the message as-is.
    if max(synd) == 0:
        return msg_out[:-nsym], msg_out[-nsym:]  # no errors

    # compute the Forney syndromes, which hide the erasures from the original syndrome (so that BM will just have to deal with errors, not erasures)
    fsynd = rs_forney_syndromes(synd, erase_pos, len(msg_out))
    # compute the error locator polynomial using Berlekamp-Massey
    err_loc = rs_find_error_locator(fsynd, nsym, erase_count=len(erase_pos))
    # locate the message errors using Chien search (or brute-force search)
    err_pos = rs_find_errors(err_loc[::-1] , len(msg_out))
    if err_pos is None:
        raise ReedSolomonError(&quot;Could not locate error&quot;)    # error location failed

    # Find errors values and apply them to correct the message
    # compute errata evaluator and errata magnitude polynomials, then correct errors and erasures
    msg_out = rs_correct_errata(msg_out, synd, (erase_pos + err_pos)) # note that we here use the original syndrome, not the forney syndrome
                                                                                                                                  # (because we will correct both errors and erasures, so we need the full syndrome)
    # check if the final message is fully repaired
    synd = rs_calc_syndromes(msg_out, nsym)
    if max(synd) &gt; 0:
        raise ReedSolomonError(&quot;Could not correct message&quot;)     # message could not be repaired
    # return the successfully decoded message
    return msg_out[:-nsym], msg_out[-nsym:] # also return the corrected ecc block so that the user can check()
&lt;/syntaxhighlight&gt;
''Python note:'' The lists &lt;kbd&gt;erase_pos&lt;/kbd&gt; and &lt;kbd&gt;err_pos&lt;/kbd&gt; are concatenated with the &lt;kbd&gt;+&lt;/kbd&gt; operator.

This is the last piece needed for a fully operational error-and-erasure correcting Reed&amp;ndash;Solomon decoder. If you want to delve more into the inner workings of errata (errors-and-erasures) decoders, you can read the excellent book &quot;Algebraic Codes for Data Transmission&quot; (2003) by Richard E. Blahut.

Mathematics note: in some software implementations, particularly the ones using a language optimized for linear algebra and matrix operations, you will find that the algorithms (encoding, Berlekamp-Massey, etc.) will seem totally different and use the Fourier Transform. This is because this is totally equivalent: when stated in the jargon of spectral estimation, decoding Reed&amp;ndash;Solomon consists of a Fourier transform (syndrome computer), followed by a spectral analysis (Berlekamp-Massey or Euclidian algorithm), followed by an inverse Fourier transform (Chien search). See the Blahut book for more info&lt;ref&gt;Richard E. Blahut, &quot;Algebraic Codes for Data Transmission&quot;, 2003, chapter 7.6 &quot;Decoding in Time Domain&quot;&lt;/ref&gt;. Indeed, if you are using a programming language optimized for linear algebra, or if you want to use fast linear algebra libraries, it can be a very good idea to use Fourier Transform since it's very fast nowadays (particularly the Fast Fourier Transform or Number Theoretic Transform&lt;ref name=&quot;ntt&quot;/&gt;).

===Wrapping up with an example===

Here's an example of how to use the functions you have just made, and how to decode both errors-and-erasures:

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Configuration of the parameters and input message
prim = 0x11d
n = 20 # set the size you want, it must be &gt; k, the remaining n-k symbols will be the ECC code (more is better)
k = 11 # k = len(message)
message = &quot;hello world&quot; # input message

# Initializing the log/antilog tables
init_tables(prim)

# Encoding the input message
mesecc = rs_encode_msg([ord(x) for x in message], n-k)
print(&quot;Original: %s&quot; % mesecc)

# Tampering 6 characters of the message (over 9 ecc symbols, so we are above the Singleton Bound)
mesecc[0] = 0
mesecc[1] = 2
mesecc[2] = 2
mesecc[3] = 2
mesecc[4] = 2
mesecc[5] = 2
print(&quot;Corrupted: %s&quot; % mesecc)

# Decoding/repairing the corrupted message, by providing the locations of a few erasures, we get below the Singleton Bound
# Remember that the Singleton Bound is: 2*e+v &lt;= (n-k)
corrected_message, corrected_ecc = rs_correct_msg(mesecc, n-k, erase_pos=[0, 1, 2])
print(&quot;Repaired: %s&quot; % (corrected_message+corrected_ecc))
print(''.join([chr(x) for x in corrected_message]))
&lt;/syntaxhighlight&gt;

This should output the following:

&lt;pre&gt;
Original:  [104, 101, 108, 108, 111,  32, 119, 111, 114, 108, 100, 145, 124, 96, 105, 94, 31, 179, 149, 163]
Corrupted: [  0,   2,   2,   2,   2,   2, 119, 111, 114, 108, 100, 145, 124, 96, 105, 94, 31, 179, 149, 163]
Repaired:  [104, 101, 108, 108, 111,  32, 119, 111, 114, 108, 100, 145, 124, 96, 105, 94, 31, 179, 149, 163]
hello world
&lt;/pre&gt;

==Conclusion and going further==

The basic principles of Reed–Solomon codes have been presented in this essay. Working Python code for a particular implementation (QR codes using a generic Reed&amp;ndash;Solomon codec to correct misreadings) has been included. The code presented here is quite generic and can be used for any purpose beyond QR codes where you need to correct errors/erasures, such as file protection, networking, etc. Many variations and refinements of these ideas are possible, since coding theory is a very rich field of study.

If your code is just intended for your own data (eg, you want to be able to generate and read your own QR codes), then you're fine, but if you intend to work with data provided by others (eg, you want to read and decode QR codes of other apps), then this decoder probably won't be enough, because there are some hidden parameters that were here fixed for simplicity (namely: the generator/alpha number and the first consecutive root). If you want to decode Reed&amp;ndash;Solomon codes generated by other libraries, you will need to use a '''universal''' Reed&amp;ndash;Solomon codec, which will allow you to specify your own parameters, and even go beyond the field 2^8.

[[Reed–Solomon codes for coders/Additional information#Universal_Reed-Solomon_Codec|On the complementary resource page, you will find an extended, universal version]] of the code presented here that you can use to decode almost any Reed&amp;ndash;Solomon code, with also a function to generate the list of prime polynomials, and [[Reed–Solomon codes for coders/Additional information#Autodetecting_the_Reed-Solomon_parameters|an algorithm to detect the parameters of an unknown RS code]]. Note that whatever the parameters you use, the repairing capabilities will always be the same: the generated values for the log/antilog tables and for the generator polynomial do not change the structure of Reed&amp;ndash;Solomon code, so that you always get the same functionality whatever the parameters. Indeed, modifying any of the available parameter will not change the theoretical Singleton bound which defines the maximal repairing capacity of Reed-Solomon (and in theory of any error correction code).

One immediate issue that you may have noticed is that we can only encode messages of up to 256 characters. This limit can be circumvented by several ways, the three most common being:

* using a higher Galois Field, for example 2&lt;sup&gt;16&lt;/sup&gt; which would allow for 65536 characters, or 2&lt;sup&gt;32&lt;/sup&gt;, 2&lt;sup&gt;64&lt;/sup&gt;, 2&lt;sup&gt;128&lt;/sup&gt;, etc. The issue here is that polynomial computations required to encode and decode Reed&amp;ndash;Solomon become very costly with big polynomials (most algorithms being in quadratic time, the most efficient being in ''n'' log ''n'' such as with number theoretic transform&lt;ref name=&quot;ntt&quot;&gt;Lin, S. J., Chung, W. H., &amp; Han, Y. S. (2014, October). Novel polynomial basis and its application to reed-solomon erasure codes. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on (pp. 316-325). IEEE.&lt;/ref&gt;).
* by &quot;chunking&quot;, which means that you simply encode your big data stream by chunks of 256 characters.
* using a variant algorithm that includes a packet size such as Cauchy Reed&amp;ndash;Solomon (see below).

If you want to go further, there are a lot of books and scientific articles on Reed&amp;ndash;Solomon codes, a good starting point is the author Richard Blahut who is notable in the domain. Also, there are a lot of different ways that Reed&amp;ndash;Solomon codes can be encoded and decoded, and thus you will find many different algorithms, in particular for decoding (Berlekamp-Massey, Berlekamp-Welch, Euclidian algorithm, etc.).

If you are looking for more performance, you will find in the literature several variants of the algorithms presented here, such as Cauchy&amp;ndash;Reed&amp;ndash;Solomon. The programming implementation also plays a big role in the performance of your Reed&amp;ndash;Solomon codec, which can lead into a 1000x speed difference. For more information, please read the [[Reed–Solomon codes for coders/Additional information#Optimizing performances|&quot;Optimizing performances&quot; section of the additional resources]].

Even if near-optimal forward error correction algorithms are all the rage nowadays (such as LDPC codes, Turbo codes, etc.) because of their great speed, Reed&amp;ndash;Solomon is an optimal FEC, which means that it can attain the theoretical limit known as the [[w:Singleton_bound|Singleton bound]]. In practice, this means that RS can correct up to &lt;kbd&gt;2*e+v &lt;= (n-k)&lt;/kbd&gt; errors and erasures at the same time, where e is the number of errors, v the number of erasures, k the message size, n the message+code size and &lt;kbd&gt;(n-k)&lt;/kbd&gt; the [[w:Minimum_distance|minimum distance]]. This is not to say that near-optimal FEC are useless: they are unimaginably faster than Reed&amp;ndash;Solomon could ever be, and they may suffer less from the [[w:Forward_error_correction#Averaging_noise_to_reduce_errors|cliff effect]] (which means they may still partially decode parts of the message even if there are too many errors to correct all errors, contrary to RS which will surely fail and even silently by decoding wrong messages without any detection&lt;ref&gt;Sofair, Isaac. &quot;Probability of miscorrection for Reed-Solomon codes.&quot; Information Technology: Coding and Computing, 2000. Proceedings. International Conference on. IEEE, 2000.&lt;/ref&gt;), but they surely can't correct as many errors as Reed&amp;ndash;Solomon. Choosing between a near-optimal and an optimal FEC is mainly a concern of speed.

Lately, the research field on Reed&amp;ndash;Solomon has regained some vitality since the discovery of [[w:List_decoding]] (not to confuse with soft decoding), which allows to decode/repair more symbols than the theoretical optimal limit. The core idea is that, instead of standard Reed&amp;ndash;Solomon which only do a unique decoding (meaning that it always results in a single solution, if it cannot because it's above the theoretical limit the decoder will return an error or a wrong result), Reed&amp;ndash;Solomon with list decoding will still try to decode beyond the limit and get several possible results, but by a clever examination of the different results, it's often possible to discriminate only one polynomial that is probably the correct one.

A few list decoding algorithms are already available that allows to repair up to &lt;kbd&gt;n - sqrt(n*k)&lt;/kbd&gt;&lt;ref&gt;&quot;Reed-Solomon Error-correcting Codes - The Deep Hole Problem&quot;, by Matt Keti, Nov 2012&lt;/ref&gt; instead of &lt;kbd&gt;2*e+v &lt;= (n-k)&lt;/kbd&gt;, and other list decoding algorithms (more efficient or decoding more symbols) are currently being investigated.

==Third-party implementations==

Here are a few implementations of Reed&amp;ndash;Solomon if you want to see practical examples:

* [https://github.com/tomerfiliba/reedsolomon Purely functional pure-Python Reedsolomon library] by Tomer Filiba and LRQ3000, inspired and expanding on this tutorial by supporting more features.
* [https://github.com/lrq3000/unireedsolomon Object-oriented Reed Solomon library in pure-Python] by Andrew Brown and LRQ3000 (same features as Tomer Filiba's lib, but object-oriented so closer to mathematical nomenclatura).
* [http://lxr.free-electrons.com/source/lib/reed_solomon/ Reed-Solomon in the Linux Kernel] (with a [https://github.com/tierney/reed-solomon userspace port here], initially ported from Phil Karn's library [http://www.ka9q.net/code/fec libfec] and [https://github.com/quiet/libfec libfec clone]).
* [https://github.com/zxing/zxing/ ZXing (Zebra Crossing)], a full-blown library to generate and decode QR codes.
* [https://github.com/catid/wirehair/blob/master/wirehair-mobile/wirehair_codec_8.cpp Speed-optimized Reed-Solomon] and [https://github.com/catid/longhair Cauchy-Reed-Solomon] with lots of comments and an associated [http://catid.mechafetus.com/news/news.php blog] for more details.
* [https://github.com/klauspost/reedsolomon Another high speed-optimized Reed-Solomon] in Go language.
* [https://github.com/mersinvald/reed-solomon-rs Port of code in the article] in Rust language.
* [https://github.com/mersinvald/Reed-Solomon C++ Reed Solomon implementation] with on-stack memory allocation and compile-time changable msg\ecc sizes for embedded, inspired by this tutorial.
* [https://github.com/NinjaDevelper/ReedSolomon Interleaved Reed Solomon implementation in C++] by NinjaDevelper.
* [https://github.com/Bulat-Ziganshin/FastECC FastECC, C++ Reed Solomon implementation in O(n log n) using Number Theoretic Transforms (NTT)] (open source, Apache License 2.0). Claims to have fast encoding rates even for large data.
* [https://github.com/catid/leopard Leopard-RS], another library in C++ for fast large data encoding, with a similar (but a bit different) algorithm as FastECC.
* [https://github.com/colin-davis/reedSolomon Pure Go Implementation] by Colin Davis (open source, GLPv3 License).
* [https://github.com/catid/shorthair Shorthair], an implementation of error correction code combined with UDP for fast reliable networking to replace the TCP stack or UDP duplication technique (which can be seen as a low efficiency redundancy scheme). [https://github.com/catid/shorthair/blob/master/docs/ErasureCodesInSoftware.pdf Slides] are provided, describing this approach for realtime game networking.
*[https://github.com/jackchouchani/reedsolomon Pure C Implementation] optimised using uint8_t and very efficient. 
*[https://github.com/hqm/rscode hqm rscode] ANSI C implementation, for 8-bit symbols 

==External links==
* [[w:Reed–Solomon_error_correction]]
* [[w:Finite_field_arithmetic]]
* [http://research.swtch.com/field Short tutorial on Reed-Solomon encoding with an introduction to finite fields]
* [https://www.academia.edu/31243287/Reed_Solomon_Encoding_Simplified_Explanation_for_Programmers A practical tutorial article to implement the core mathematical (galois field) operators].

==References==

[[Category:Essays]]
[[Category:Applied mathematics]]
[[Category:Algorithms]]</text>
      <sha1>f44jorsn2gkz857rnazg6cg4n9h4fgo</sha1>
    </revision>
  </page>
  <page>
    <title>The necessities in Digital Design</title>
    <ns>0</ns>
    <id>119422</id>
    <revision>
      <id>2815570</id>
      <parentid>2812607</parentid>
      <timestamp>2026-06-13T19:32:47Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/*  Timing Analysis  */</comment>
      <origin>2815570</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9835" sha1="t2u35e09h68x68n94ln37awcqblxdrl" xml:space="preserve">
== ''' Number Systems '''== 

=== ''' Binary Representation '''=== 
* Binary Numbers ([[Media:DD1.1.A.BinaryNum.20130918.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD1.2.A.HexaNum.20130918.pdf|A.pdf]])
* Other Codes ([[Media:DD1.3A.Code.20250329.pdf|A.pdf]])

=== ''' Binary Arithmetic '''=== 
* Binary Arithmetic ([[Media:DD1.4.A.BinaryArith.20150425.pdf|A.pdf]])
* BCD Arithmetic ([[Media:DD1.5.A.BCDArith.20130918.pdf|A.pdf]])

=== ''' C Program Examples '''=== 
* Binary Numbers in C programs ([[Media:DD1.6.A.BNumInC.20140103.pdf|A.pdf]])
* Binary Addition in C programs ([[Media:DD1.7.A.BArithInC.20140103.pdf|A.pdf]])
&lt;/br&gt;

* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|C.pdf]])
&lt;/br&gt;

=== ''' Floating Point Numbers '''=== 

* Floating Point Representations  ([[Media:CDesign.5.A.FPoint.20140121.pdf|5A.pdf]])&lt;/br&gt;
:: See [http://www.iro.umontreal.ca/~aboulham/F1214/Session%206Arithm/Floating_Point_Numbers.pdf Floating Point Overview]
:: See [http://www.cs.auckland.ac.nz/~patrice/210-2006/210%20LN04_2.pdf Offset Binary Overview]
:: See [http://www.intersil.com/content/dam/Intersil/documents/an96/an9657.pdf Offset Binary &amp; Sin / Cosine]
:: See [http://www.ee.ic.ac.uk/hp/staff/dmb/courses/dig2/4_Analog.pdf Offset Binary &amp; ADC / DAC]
&lt;/br&gt;

=== ''' Interfacing Digital and Analog Signals '''=== 
* Sampling and Quantization   ([[Media:DD1.10.A.SampleQuant.20150425.pdf|A.pdf]])
* Digital-to-Analog Conversion  ([[Media:DD1.8.A.DAC.20140208.pdf|A.pdf]])
* Analog-to-Digital Conversion ([[Media:DD1.9.A.DAC.20140208.pdf|A.pdf]])
&lt;/br&gt;

== '''Combinational Circuits'''==

=== ''' Design '''=== 
* Boolean Algebra ([[Media:DD2.A1.BAlgebra.20250503.pdf|A1.pdf]])
* Truth Tables ([[Media:DD2.A2.TTable.20250424.pdf|A2.pdf]])
* K-Map ([[Media:DD2.A3.KMap.20250424.pdf|A3.pdf]])
* Design Examples ([[Media:DD2.A4.CombEx.20250414.pdf|A4.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Decoder ([[Media:DD2.B.1.Decoder.20130928.pdf|B1.pdf]])
* Encoder ([[Media:DD2.B.2.Encoder.20130917.pdf|B2.pdf]])
* Multiplexer ([[Media:DD2.B.3.Multiplexer.20130928.pdf|B3.pdf]])
* Adder ([[Media:DD2.B.4..Adder.20131007.pdf|B4.pdf]], [[Media:Fa.sch.20131002.pdf|fa.sch.pdf]], [[Media:Adder4.sch.20131002.pdf|adder4.sch.pdf]])
&lt;/br&gt;

=== ''' Design Metric '''=== 
* Noise Margin ([[Media:DD2.C1.NoiseMargin.20250415.pdf|C1.pdf]])
&lt;/br&gt;

== '''Sequential Circuits'''==

=== ''' Design '''=== 
* Types of Flip-Flops ([[Media:CDesign.1.A.FF.20130412.pdf |1A.pdf]])&lt;/br&gt;
* Latches and Flipflops ([[Media:DD3.A.1.LatchFF.20160308.pdf|A1.pdf]])
* State Transition Table ([[Media:DD3.A.2.pdf|A2.pdf]])
* FSM (Finite State Machine) ([[Media:DD3.A.3.FSM.20131030.pdf|A3.pdf]])
&lt;/br&gt;

* The Classic FF Design ([[Media:DD3.A.6.ClassicFF.20131126.pdf|A7.pdf]])
* The Modern FF Design ([[Media:DD3.A.6.ClassicFF.20131204.2.pdf|A8.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Latches and Flip-flops ([[Media:DD3.B.1.LatchFF.20131008.pdf|B1.pdf]])
* Registers ([[Media:DD3.B.2.Register.20150326.pdf|B2.pdf]], [[Media:Register.20131118.pdf|register.pdf]])
* Counters ([[Media:DD3.B.2.Counter.20150420.pdf|B3.pdf]])
&lt;/br&gt;

=== ''' Timing Analysis '''=== 
* Metastability ([[Media:DD3.A.4.MetaState.20131030.pdf|A4.pdf]])
* Flip-flop Timing ([[Media:DD3.A5.FFTiming.20260608.pdf|A5.pdf]])
* SR Latch Forbidden State ([[Media:DD3.A.5.ForbiddenState.20131030.pdf|A6.pdf]])
&lt;/br&gt;

* FF Min Max Timing Constraints ([[Media:CArch.MinMaxTiming.20131121.pdf |pdf]])
* FF Clock Skew Timing Constraints ([[Media:CArch.ClockSkew.20131121.pdf |pdf]])

* Synchronizer ([[Media:CArch.Synchronizer.20131216.pdf |pdf]])
* Resolution Time Analysis  ([[Media:CArch.Resolution.20131216.pdf |pdf]])
&lt;/br&gt;

== '''Finite State Machine'''==

* FSM State Encoding 
* FSM Types : Mealy and Moore Machines
* FSM Example ([[Media:CArch.2.A.FSMExample.20141018.pdf |pdf]])
&lt;/br&gt;

== '''Array  Devices''' ==

=== ''' Memory Arrays '''=== 
* RAM 
** RAM Structure ([[Media:DD4.A.1.RAM.20131111.pdf|A.pdf]])
** RAM Timing ([[Media:DD4.B.1.RAMTiming.20131130.pdf|B.pdf]])
** FPGA RAM ([[Media:DD4.C.1.FPGARAM.20160513.pdf|C.pdf]])

* ROM
&lt;/br&gt;

=== ''' Logic Arrays '''=== 
* PLA 
* PAL
* PLD
* FPGA
** FPGA Structure
** FPGA Configuration ([[Media:DD4.C.1.FPGAConf.20131130.pdf|B.pdf]])
&lt;/br&gt;
&lt;/br&gt; 
[http://www.ece.cmu.edu/~ece548/localcpy/sramop.pdf  Synchronous SRAM Timing] &lt;/br&gt;
[http://www.micron.com/~/media/Documents/Products/Technical%20Note/DRAM/tn4529.pdf Asynchronous SRAM Timing]&lt;/br&gt;
[http://www.ece.cmu.edu/~ece548/localcpy/dramop.pdf DRAM Timing] &lt;/br&gt;

[http://www.ece.unm.edu/~jimp/415/slides/fpga_arch1.pdf FPGA Architectures] &lt;/br&gt;
[http://www.engr.siu.edu/~haibo/ece428/notes/ece428_fpgaarch.pdf CPLD &amp; FPGA] &lt;/br&gt;
&lt;/br&gt;

== ''' RTL Design Techniques''' ==
&lt;/br&gt;

''' Design Methodology '''
&lt;/br&gt;


''' Synthesis '''
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

== '''Logic Families and IOs''' ==
* BJT Based 
:: DTL (Diode-Transistor Logic)
:: TTL (Transistor-Transistor Logic)
:: ECL (Emitter-Coupled Logic)
 
* MOS Based
:: CMOS (Complementary MOS)
:: Pseudo-nMOS 
:: Transmission Gate
:: BiCMOS (Bipolr + CMOS)
 
* Dynamic CMOS 
:: Domino
:: Clocked-CMOS (C&lt;sup&gt;2&lt;/sup&gt;MOS)
&lt;/br&gt; 

* Modern I/O Standards
:: TTL and LVTTL (Low Voltage TTL)
:: CMOS and LVCMOS (Low Voltage CMOS)
:: SSTL (Stub Series Terminated Logic)
:: HSTL (High Speed Tranceiver Logic)
:: LVDS (Low Voltage Differential Signaling) 

&lt;/br&gt;

* Wikipedia Pages for Logic Families ([[Media:Logic Families.wiki.20140812.pdf|A.pdf]])
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&lt;[[The necessities in Computer Design]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Architecture]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Organization]]&gt;  &lt;/br&gt;


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

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

== '''Old''' ==
'''Until 2011.12''' 
'''Chapter 1. Binary Numbers'''
* 1.1 Binary Numbers([[Media:BinaryNumbers.1.A.pdf|pdf]])

''' Minterm, Maxterm, HW '''
* 1.1 Lecture01([[Media:DigitalDesign.20110922.pdf|pdf]])

''' Overflow HW '''
* Overflow Table([[Media:Overflow table.20110924.pdf|pdf]])

''' K-Map '''
* K-Map([[Media:DigitalDesign.20110926.pdf|pdf]])

''' Binary Adder '''
* Binary Adder (C, S) ([[Media:DigitalDesign.20110929.pdf|pdf]])
* Overflow detection circuit (V) ([[Media:HW Overflow20111001.pdf|pdf]])

''' BCD to Ex3 Code Coversion, Dont' Care '''
* BCD to Ex3 Code Conversion ([[Media:DigitalDesign.20111006.pdf|pdf]])

''' Prime Implicant, Dont' Care '''
* Prime Implicant, Don't Care ([[Media:DigitalDesign.20111010.pdf|pdf]])
* HW 3.6 - explain the method of combining 0's and X's

''' Multiplexer / Demultiplexer '''
* Multiplexer ([[Media:DigitalDesign.20111024.pdf|pdf]])
* HW (TBD)

''' Flip Flop / Latch '''
* FF &amp; Latch ([[Media:DigitalDesign.20111027.pdf|pdf]])
* FF &amp; Latch HW ([[Media:DigitalDesign (HW).20111027.pdf|pdf]])
* Gated D Latch &amp; Master-Slave D FlipFlop ([[Media:DigitalDesign.20111031.pdf|pdf]])
* HW (Forbidden state and Indeterminate state) ([[Media:DigitalDesign (HW).20111102.pdf|pdf]]) (note in #2, S' R' instead of S R)
* Classical Edge Triggered D FlipFlop ([[Media:DigitalDesign.20111112.pdf|pdf]])
* HW (addition in SW and HW) ([[Media:DigitalDesign (HW).20111112.pdf|pdf]]) 
* FSM1  ([[Media:DigitalDesign.FSM1.20111117.pdf|pdf]]) 
* FSM2  ([[Media:DigitalDesign.FSM2.20111117.pdf|pdf]]) 
* HW (FSM Waveforms) ([[Media:DigitalDesign (HW).20111118.pdf|pdf]]) 
 
 
''' Counter '''
* Sychronous Counter ([[Media:DigitalDesign.20111121.pdf|pdf]])
* Ripple Counter, Multiplexer, Tri-state buffer([[Media:DigitalDesign.20111124.pdf|pdf]])
* Register ([[Media:DigitalDesign.register.20111201.pdf|pdf]])
* Timing ([[Media:DigitalDesign.timing.20111201.pdf|pdf]])
* HW (Multiplexer, Shift Register) ([[Media:DigitalDesign (HW).20111201.pdf|pdf]])
* Universal Shift Register, Memory Cell ([[Media:DigitalDesign.20111206.pdf|pdf]])
* HW (Bit Serial Adder) ([[Media:DigitalDesign (HW).20111206.pdf|pdf]])
 
 
''' Memory '''
* Memory ([[Media:DigitalDesign.20111208.pdf|pdf]])

''' Comparator, Multiplier '''
* Comparator, Multiplier ([[Media:DigitalDesign.20111219.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111219.draw.pdf|2.pdf]])
 
 
'''Multiplexer based design method '''
* Multiplexer Design Method ([[Media:DigitalDesign.20111221.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111221.draw.pdf|2.pdf]])


midterm result ([[Media:MidReult.20111027.pdf|pdf]])

 
* Edge Triggered Flip Flop ([[Media:EdgeTrigFF.20111224.pdf|pdf]])
* FF Timing ([[Media:FFTiming.20111203.pdf|pdf]])


&lt;/br&gt; &lt;/br&gt;
'''Until 2013.07''' 

''' Number Systems '''
* Binary Numbers ([[Media:DD.1.A.BinNum.20130309.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD.1.B.HexaNum.20130417.pdf|B.pdf]])
* Numbers in C programs ([[Media:DD.1.C.CNum.20130309.pdf|C.pdf]])
* Codes ([[Media:DD.1.D.Coding.20130319.pdf|pdf]])
&lt;/br&gt;

&lt;/br&gt;
* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|pdf]])
&lt;/br&gt;

''' Combinational Circuits '''
* Truth Tables and Boolean Functions ([[Media:DD.2.A.TTable.20130325.pdf|2A.pdf]])&lt;/br&gt;
* K-Map ([[Media:DD.2.A.KMap.20130329.pdf|2B.pdf]])&lt;/br&gt;
* Binary Addition in C ([[Media:DD.2.C.BAinC.20130329.pdf|2.C.pdf]])&lt;/br&gt;
* Binary Arithmetic ([[Media:DD.2.D.BAri.2013.pdf|2.D.pdf]])&lt;/br&gt;
* Boolean Algebra ([[Media:DD.2.E.BAlgebra.20130419.pdf|2.E.pdf]])&lt;/br&gt;
&lt;/br&gt;

''' Sequential Circuits '''
* Latches and Flip-flops ([[Media:DD.3.A.LatchFF.20130413.pdf|3A.pdf]])&lt;/br&gt;
* FSM (Finite State Machine) ([[Media:DD.3.B.FSM.20130417.pdf|3B.pdf]])&lt;/br&gt;
* SR Latch Forbidden State ([[Media:DD.3.C.FState.20130413.pdf|3C.pdf]])&lt;/br&gt;
* Flip-flop Timing ([[Media:DD.3.D.Timing.20130413.pdf|3D.pdf]])&lt;/br&gt;
* Metastability ([[Media:DD.3.E.MetaState.20130628.pdf|3E.pdf]])&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&quot;[[The necessities in Computer Design]]&quot;  &lt;/br&gt;
&quot;[[The necessities in Computer Architecture]]&quot;  &lt;/br&gt;

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
      <sha1>t2u35e09h68x68n94ln37awcqblxdrl</sha1>
    </revision>
    <revision>
      <id>2815573</id>
      <parentid>2815570</parentid>
      <timestamp>2026-06-13T19:33:49Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/*  Timing Analysis  */</comment>
      <origin>2815573</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9835" sha1="d7tuftc9svlonhett4h9a25qga2hxzz" xml:space="preserve">
== ''' Number Systems '''== 

=== ''' Binary Representation '''=== 
* Binary Numbers ([[Media:DD1.1.A.BinaryNum.20130918.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD1.2.A.HexaNum.20130918.pdf|A.pdf]])
* Other Codes ([[Media:DD1.3A.Code.20250329.pdf|A.pdf]])

=== ''' Binary Arithmetic '''=== 
* Binary Arithmetic ([[Media:DD1.4.A.BinaryArith.20150425.pdf|A.pdf]])
* BCD Arithmetic ([[Media:DD1.5.A.BCDArith.20130918.pdf|A.pdf]])

=== ''' C Program Examples '''=== 
* Binary Numbers in C programs ([[Media:DD1.6.A.BNumInC.20140103.pdf|A.pdf]])
* Binary Addition in C programs ([[Media:DD1.7.A.BArithInC.20140103.pdf|A.pdf]])
&lt;/br&gt;

* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|C.pdf]])
&lt;/br&gt;

=== ''' Floating Point Numbers '''=== 

* Floating Point Representations  ([[Media:CDesign.5.A.FPoint.20140121.pdf|5A.pdf]])&lt;/br&gt;
:: See [http://www.iro.umontreal.ca/~aboulham/F1214/Session%206Arithm/Floating_Point_Numbers.pdf Floating Point Overview]
:: See [http://www.cs.auckland.ac.nz/~patrice/210-2006/210%20LN04_2.pdf Offset Binary Overview]
:: See [http://www.intersil.com/content/dam/Intersil/documents/an96/an9657.pdf Offset Binary &amp; Sin / Cosine]
:: See [http://www.ee.ic.ac.uk/hp/staff/dmb/courses/dig2/4_Analog.pdf Offset Binary &amp; ADC / DAC]
&lt;/br&gt;

=== ''' Interfacing Digital and Analog Signals '''=== 
* Sampling and Quantization   ([[Media:DD1.10.A.SampleQuant.20150425.pdf|A.pdf]])
* Digital-to-Analog Conversion  ([[Media:DD1.8.A.DAC.20140208.pdf|A.pdf]])
* Analog-to-Digital Conversion ([[Media:DD1.9.A.DAC.20140208.pdf|A.pdf]])
&lt;/br&gt;

== '''Combinational Circuits'''==

=== ''' Design '''=== 
* Boolean Algebra ([[Media:DD2.A1.BAlgebra.20250503.pdf|A1.pdf]])
* Truth Tables ([[Media:DD2.A2.TTable.20250424.pdf|A2.pdf]])
* K-Map ([[Media:DD2.A3.KMap.20250424.pdf|A3.pdf]])
* Design Examples ([[Media:DD2.A4.CombEx.20250414.pdf|A4.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Decoder ([[Media:DD2.B.1.Decoder.20130928.pdf|B1.pdf]])
* Encoder ([[Media:DD2.B.2.Encoder.20130917.pdf|B2.pdf]])
* Multiplexer ([[Media:DD2.B.3.Multiplexer.20130928.pdf|B3.pdf]])
* Adder ([[Media:DD2.B.4..Adder.20131007.pdf|B4.pdf]], [[Media:Fa.sch.20131002.pdf|fa.sch.pdf]], [[Media:Adder4.sch.20131002.pdf|adder4.sch.pdf]])
&lt;/br&gt;

=== ''' Design Metric '''=== 
* Noise Margin ([[Media:DD2.C1.NoiseMargin.20250415.pdf|C1.pdf]])
&lt;/br&gt;

== '''Sequential Circuits'''==

=== ''' Design '''=== 
* Types of Flip-Flops ([[Media:CDesign.1.A.FF.20130412.pdf |1A.pdf]])&lt;/br&gt;
* Latches and Flipflops ([[Media:DD3.A.1.LatchFF.20160308.pdf|A1.pdf]])
* State Transition Table ([[Media:DD3.A.2.pdf|A2.pdf]])
* FSM (Finite State Machine) ([[Media:DD3.A.3.FSM.20131030.pdf|A3.pdf]])
&lt;/br&gt;

* The Classic FF Design ([[Media:DD3.A.6.ClassicFF.20131126.pdf|A7.pdf]])
* The Modern FF Design ([[Media:DD3.A.6.ClassicFF.20131204.2.pdf|A8.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Latches and Flip-flops ([[Media:DD3.B.1.LatchFF.20131008.pdf|B1.pdf]])
* Registers ([[Media:DD3.B.2.Register.20150326.pdf|B2.pdf]], [[Media:Register.20131118.pdf|register.pdf]])
* Counters ([[Media:DD3.B.2.Counter.20150420.pdf|B3.pdf]])
&lt;/br&gt;

=== ''' Timing Analysis '''=== 
* Metastability ([[Media:DD3.A.4.MetaState.20131030.pdf|A4.pdf]])
* Flip-flop Timing ([[Media:DD3.A5.FFTiming.20260609.pdf|A5.pdf]])
* SR Latch Forbidden State ([[Media:DD3.A.5.ForbiddenState.20131030.pdf|A6.pdf]])
&lt;/br&gt;

* FF Min Max Timing Constraints ([[Media:CArch.MinMaxTiming.20131121.pdf |pdf]])
* FF Clock Skew Timing Constraints ([[Media:CArch.ClockSkew.20131121.pdf |pdf]])

* Synchronizer ([[Media:CArch.Synchronizer.20131216.pdf |pdf]])
* Resolution Time Analysis  ([[Media:CArch.Resolution.20131216.pdf |pdf]])
&lt;/br&gt;

== '''Finite State Machine'''==

* FSM State Encoding 
* FSM Types : Mealy and Moore Machines
* FSM Example ([[Media:CArch.2.A.FSMExample.20141018.pdf |pdf]])
&lt;/br&gt;

== '''Array  Devices''' ==

=== ''' Memory Arrays '''=== 
* RAM 
** RAM Structure ([[Media:DD4.A.1.RAM.20131111.pdf|A.pdf]])
** RAM Timing ([[Media:DD4.B.1.RAMTiming.20131130.pdf|B.pdf]])
** FPGA RAM ([[Media:DD4.C.1.FPGARAM.20160513.pdf|C.pdf]])

* ROM
&lt;/br&gt;

=== ''' Logic Arrays '''=== 
* PLA 
* PAL
* PLD
* FPGA
** FPGA Structure
** FPGA Configuration ([[Media:DD4.C.1.FPGAConf.20131130.pdf|B.pdf]])
&lt;/br&gt;
&lt;/br&gt; 
[http://www.ece.cmu.edu/~ece548/localcpy/sramop.pdf  Synchronous SRAM Timing] &lt;/br&gt;
[http://www.micron.com/~/media/Documents/Products/Technical%20Note/DRAM/tn4529.pdf Asynchronous SRAM Timing]&lt;/br&gt;
[http://www.ece.cmu.edu/~ece548/localcpy/dramop.pdf DRAM Timing] &lt;/br&gt;

[http://www.ece.unm.edu/~jimp/415/slides/fpga_arch1.pdf FPGA Architectures] &lt;/br&gt;
[http://www.engr.siu.edu/~haibo/ece428/notes/ece428_fpgaarch.pdf CPLD &amp; FPGA] &lt;/br&gt;
&lt;/br&gt;

== ''' RTL Design Techniques''' ==
&lt;/br&gt;

''' Design Methodology '''
&lt;/br&gt;


''' Synthesis '''
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

== '''Logic Families and IOs''' ==
* BJT Based 
:: DTL (Diode-Transistor Logic)
:: TTL (Transistor-Transistor Logic)
:: ECL (Emitter-Coupled Logic)
 
* MOS Based
:: CMOS (Complementary MOS)
:: Pseudo-nMOS 
:: Transmission Gate
:: BiCMOS (Bipolr + CMOS)
 
* Dynamic CMOS 
:: Domino
:: Clocked-CMOS (C&lt;sup&gt;2&lt;/sup&gt;MOS)
&lt;/br&gt; 

* Modern I/O Standards
:: TTL and LVTTL (Low Voltage TTL)
:: CMOS and LVCMOS (Low Voltage CMOS)
:: SSTL (Stub Series Terminated Logic)
:: HSTL (High Speed Tranceiver Logic)
:: LVDS (Low Voltage Differential Signaling) 

&lt;/br&gt;

* Wikipedia Pages for Logic Families ([[Media:Logic Families.wiki.20140812.pdf|A.pdf]])
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&lt;[[The necessities in Computer Design]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Architecture]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Organization]]&gt;  &lt;/br&gt;


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

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

== '''Old''' ==
'''Until 2011.12''' 
'''Chapter 1. Binary Numbers'''
* 1.1 Binary Numbers([[Media:BinaryNumbers.1.A.pdf|pdf]])

''' Minterm, Maxterm, HW '''
* 1.1 Lecture01([[Media:DigitalDesign.20110922.pdf|pdf]])

''' Overflow HW '''
* Overflow Table([[Media:Overflow table.20110924.pdf|pdf]])

''' K-Map '''
* K-Map([[Media:DigitalDesign.20110926.pdf|pdf]])

''' Binary Adder '''
* Binary Adder (C, S) ([[Media:DigitalDesign.20110929.pdf|pdf]])
* Overflow detection circuit (V) ([[Media:HW Overflow20111001.pdf|pdf]])

''' BCD to Ex3 Code Coversion, Dont' Care '''
* BCD to Ex3 Code Conversion ([[Media:DigitalDesign.20111006.pdf|pdf]])

''' Prime Implicant, Dont' Care '''
* Prime Implicant, Don't Care ([[Media:DigitalDesign.20111010.pdf|pdf]])
* HW 3.6 - explain the method of combining 0's and X's

''' Multiplexer / Demultiplexer '''
* Multiplexer ([[Media:DigitalDesign.20111024.pdf|pdf]])
* HW (TBD)

''' Flip Flop / Latch '''
* FF &amp; Latch ([[Media:DigitalDesign.20111027.pdf|pdf]])
* FF &amp; Latch HW ([[Media:DigitalDesign (HW).20111027.pdf|pdf]])
* Gated D Latch &amp; Master-Slave D FlipFlop ([[Media:DigitalDesign.20111031.pdf|pdf]])
* HW (Forbidden state and Indeterminate state) ([[Media:DigitalDesign (HW).20111102.pdf|pdf]]) (note in #2, S' R' instead of S R)
* Classical Edge Triggered D FlipFlop ([[Media:DigitalDesign.20111112.pdf|pdf]])
* HW (addition in SW and HW) ([[Media:DigitalDesign (HW).20111112.pdf|pdf]]) 
* FSM1  ([[Media:DigitalDesign.FSM1.20111117.pdf|pdf]]) 
* FSM2  ([[Media:DigitalDesign.FSM2.20111117.pdf|pdf]]) 
* HW (FSM Waveforms) ([[Media:DigitalDesign (HW).20111118.pdf|pdf]]) 
 
 
''' Counter '''
* Sychronous Counter ([[Media:DigitalDesign.20111121.pdf|pdf]])
* Ripple Counter, Multiplexer, Tri-state buffer([[Media:DigitalDesign.20111124.pdf|pdf]])
* Register ([[Media:DigitalDesign.register.20111201.pdf|pdf]])
* Timing ([[Media:DigitalDesign.timing.20111201.pdf|pdf]])
* HW (Multiplexer, Shift Register) ([[Media:DigitalDesign (HW).20111201.pdf|pdf]])
* Universal Shift Register, Memory Cell ([[Media:DigitalDesign.20111206.pdf|pdf]])
* HW (Bit Serial Adder) ([[Media:DigitalDesign (HW).20111206.pdf|pdf]])
 
 
''' Memory '''
* Memory ([[Media:DigitalDesign.20111208.pdf|pdf]])

''' Comparator, Multiplier '''
* Comparator, Multiplier ([[Media:DigitalDesign.20111219.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111219.draw.pdf|2.pdf]])
 
 
'''Multiplexer based design method '''
* Multiplexer Design Method ([[Media:DigitalDesign.20111221.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111221.draw.pdf|2.pdf]])


midterm result ([[Media:MidReult.20111027.pdf|pdf]])

 
* Edge Triggered Flip Flop ([[Media:EdgeTrigFF.20111224.pdf|pdf]])
* FF Timing ([[Media:FFTiming.20111203.pdf|pdf]])


&lt;/br&gt; &lt;/br&gt;
'''Until 2013.07''' 

''' Number Systems '''
* Binary Numbers ([[Media:DD.1.A.BinNum.20130309.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD.1.B.HexaNum.20130417.pdf|B.pdf]])
* Numbers in C programs ([[Media:DD.1.C.CNum.20130309.pdf|C.pdf]])
* Codes ([[Media:DD.1.D.Coding.20130319.pdf|pdf]])
&lt;/br&gt;

&lt;/br&gt;
* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|pdf]])
&lt;/br&gt;

''' Combinational Circuits '''
* Truth Tables and Boolean Functions ([[Media:DD.2.A.TTable.20130325.pdf|2A.pdf]])&lt;/br&gt;
* K-Map ([[Media:DD.2.A.KMap.20130329.pdf|2B.pdf]])&lt;/br&gt;
* Binary Addition in C ([[Media:DD.2.C.BAinC.20130329.pdf|2.C.pdf]])&lt;/br&gt;
* Binary Arithmetic ([[Media:DD.2.D.BAri.2013.pdf|2.D.pdf]])&lt;/br&gt;
* Boolean Algebra ([[Media:DD.2.E.BAlgebra.20130419.pdf|2.E.pdf]])&lt;/br&gt;
&lt;/br&gt;

''' Sequential Circuits '''
* Latches and Flip-flops ([[Media:DD.3.A.LatchFF.20130413.pdf|3A.pdf]])&lt;/br&gt;
* FSM (Finite State Machine) ([[Media:DD.3.B.FSM.20130417.pdf|3B.pdf]])&lt;/br&gt;
* SR Latch Forbidden State ([[Media:DD.3.C.FState.20130413.pdf|3C.pdf]])&lt;/br&gt;
* Flip-flop Timing ([[Media:DD.3.D.Timing.20130413.pdf|3D.pdf]])&lt;/br&gt;
* Metastability ([[Media:DD.3.E.MetaState.20130628.pdf|3E.pdf]])&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&quot;[[The necessities in Computer Design]]&quot;  &lt;/br&gt;
&quot;[[The necessities in Computer Architecture]]&quot;  &lt;/br&gt;

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
      <sha1>d7tuftc9svlonhett4h9a25qga2hxzz</sha1>
    </revision>
    <revision>
      <id>2815575</id>
      <parentid>2815573</parentid>
      <timestamp>2026-06-13T19:34:42Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/*  Timing Analysis  */</comment>
      <origin>2815575</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9835" sha1="f38zl8f0eep7xrnv5ad2ll116oytxj5" xml:space="preserve">
== ''' Number Systems '''== 

=== ''' Binary Representation '''=== 
* Binary Numbers ([[Media:DD1.1.A.BinaryNum.20130918.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD1.2.A.HexaNum.20130918.pdf|A.pdf]])
* Other Codes ([[Media:DD1.3A.Code.20250329.pdf|A.pdf]])

=== ''' Binary Arithmetic '''=== 
* Binary Arithmetic ([[Media:DD1.4.A.BinaryArith.20150425.pdf|A.pdf]])
* BCD Arithmetic ([[Media:DD1.5.A.BCDArith.20130918.pdf|A.pdf]])

=== ''' C Program Examples '''=== 
* Binary Numbers in C programs ([[Media:DD1.6.A.BNumInC.20140103.pdf|A.pdf]])
* Binary Addition in C programs ([[Media:DD1.7.A.BArithInC.20140103.pdf|A.pdf]])
&lt;/br&gt;

* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|C.pdf]])
&lt;/br&gt;

=== ''' Floating Point Numbers '''=== 

* Floating Point Representations  ([[Media:CDesign.5.A.FPoint.20140121.pdf|5A.pdf]])&lt;/br&gt;
:: See [http://www.iro.umontreal.ca/~aboulham/F1214/Session%206Arithm/Floating_Point_Numbers.pdf Floating Point Overview]
:: See [http://www.cs.auckland.ac.nz/~patrice/210-2006/210%20LN04_2.pdf Offset Binary Overview]
:: See [http://www.intersil.com/content/dam/Intersil/documents/an96/an9657.pdf Offset Binary &amp; Sin / Cosine]
:: See [http://www.ee.ic.ac.uk/hp/staff/dmb/courses/dig2/4_Analog.pdf Offset Binary &amp; ADC / DAC]
&lt;/br&gt;

=== ''' Interfacing Digital and Analog Signals '''=== 
* Sampling and Quantization   ([[Media:DD1.10.A.SampleQuant.20150425.pdf|A.pdf]])
* Digital-to-Analog Conversion  ([[Media:DD1.8.A.DAC.20140208.pdf|A.pdf]])
* Analog-to-Digital Conversion ([[Media:DD1.9.A.DAC.20140208.pdf|A.pdf]])
&lt;/br&gt;

== '''Combinational Circuits'''==

=== ''' Design '''=== 
* Boolean Algebra ([[Media:DD2.A1.BAlgebra.20250503.pdf|A1.pdf]])
* Truth Tables ([[Media:DD2.A2.TTable.20250424.pdf|A2.pdf]])
* K-Map ([[Media:DD2.A3.KMap.20250424.pdf|A3.pdf]])
* Design Examples ([[Media:DD2.A4.CombEx.20250414.pdf|A4.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Decoder ([[Media:DD2.B.1.Decoder.20130928.pdf|B1.pdf]])
* Encoder ([[Media:DD2.B.2.Encoder.20130917.pdf|B2.pdf]])
* Multiplexer ([[Media:DD2.B.3.Multiplexer.20130928.pdf|B3.pdf]])
* Adder ([[Media:DD2.B.4..Adder.20131007.pdf|B4.pdf]], [[Media:Fa.sch.20131002.pdf|fa.sch.pdf]], [[Media:Adder4.sch.20131002.pdf|adder4.sch.pdf]])
&lt;/br&gt;

=== ''' Design Metric '''=== 
* Noise Margin ([[Media:DD2.C1.NoiseMargin.20250415.pdf|C1.pdf]])
&lt;/br&gt;

== '''Sequential Circuits'''==

=== ''' Design '''=== 
* Types of Flip-Flops ([[Media:CDesign.1.A.FF.20130412.pdf |1A.pdf]])&lt;/br&gt;
* Latches and Flipflops ([[Media:DD3.A.1.LatchFF.20160308.pdf|A1.pdf]])
* State Transition Table ([[Media:DD3.A.2.pdf|A2.pdf]])
* FSM (Finite State Machine) ([[Media:DD3.A.3.FSM.20131030.pdf|A3.pdf]])
&lt;/br&gt;

* The Classic FF Design ([[Media:DD3.A.6.ClassicFF.20131126.pdf|A7.pdf]])
* The Modern FF Design ([[Media:DD3.A.6.ClassicFF.20131204.2.pdf|A8.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Latches and Flip-flops ([[Media:DD3.B.1.LatchFF.20131008.pdf|B1.pdf]])
* Registers ([[Media:DD3.B.2.Register.20150326.pdf|B2.pdf]], [[Media:Register.20131118.pdf|register.pdf]])
* Counters ([[Media:DD3.B.2.Counter.20150420.pdf|B3.pdf]])
&lt;/br&gt;

=== ''' Timing Analysis '''=== 
* Metastability ([[Media:DD3.A.4.MetaState.20131030.pdf|A4.pdf]])
* Flip-flop Timing ([[Media:DD3.A5.FFTiming.20260610.pdf|A5.pdf]])
* SR Latch Forbidden State ([[Media:DD3.A.5.ForbiddenState.20131030.pdf|A6.pdf]])
&lt;/br&gt;

* FF Min Max Timing Constraints ([[Media:CArch.MinMaxTiming.20131121.pdf |pdf]])
* FF Clock Skew Timing Constraints ([[Media:CArch.ClockSkew.20131121.pdf |pdf]])

* Synchronizer ([[Media:CArch.Synchronizer.20131216.pdf |pdf]])
* Resolution Time Analysis  ([[Media:CArch.Resolution.20131216.pdf |pdf]])
&lt;/br&gt;

== '''Finite State Machine'''==

* FSM State Encoding 
* FSM Types : Mealy and Moore Machines
* FSM Example ([[Media:CArch.2.A.FSMExample.20141018.pdf |pdf]])
&lt;/br&gt;

== '''Array  Devices''' ==

=== ''' Memory Arrays '''=== 
* RAM 
** RAM Structure ([[Media:DD4.A.1.RAM.20131111.pdf|A.pdf]])
** RAM Timing ([[Media:DD4.B.1.RAMTiming.20131130.pdf|B.pdf]])
** FPGA RAM ([[Media:DD4.C.1.FPGARAM.20160513.pdf|C.pdf]])

* ROM
&lt;/br&gt;

=== ''' Logic Arrays '''=== 
* PLA 
* PAL
* PLD
* FPGA
** FPGA Structure
** FPGA Configuration ([[Media:DD4.C.1.FPGAConf.20131130.pdf|B.pdf]])
&lt;/br&gt;
&lt;/br&gt; 
[http://www.ece.cmu.edu/~ece548/localcpy/sramop.pdf  Synchronous SRAM Timing] &lt;/br&gt;
[http://www.micron.com/~/media/Documents/Products/Technical%20Note/DRAM/tn4529.pdf Asynchronous SRAM Timing]&lt;/br&gt;
[http://www.ece.cmu.edu/~ece548/localcpy/dramop.pdf DRAM Timing] &lt;/br&gt;

[http://www.ece.unm.edu/~jimp/415/slides/fpga_arch1.pdf FPGA Architectures] &lt;/br&gt;
[http://www.engr.siu.edu/~haibo/ece428/notes/ece428_fpgaarch.pdf CPLD &amp; FPGA] &lt;/br&gt;
&lt;/br&gt;

== ''' RTL Design Techniques''' ==
&lt;/br&gt;

''' Design Methodology '''
&lt;/br&gt;


''' Synthesis '''
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

== '''Logic Families and IOs''' ==
* BJT Based 
:: DTL (Diode-Transistor Logic)
:: TTL (Transistor-Transistor Logic)
:: ECL (Emitter-Coupled Logic)
 
* MOS Based
:: CMOS (Complementary MOS)
:: Pseudo-nMOS 
:: Transmission Gate
:: BiCMOS (Bipolr + CMOS)
 
* Dynamic CMOS 
:: Domino
:: Clocked-CMOS (C&lt;sup&gt;2&lt;/sup&gt;MOS)
&lt;/br&gt; 

* Modern I/O Standards
:: TTL and LVTTL (Low Voltage TTL)
:: CMOS and LVCMOS (Low Voltage CMOS)
:: SSTL (Stub Series Terminated Logic)
:: HSTL (High Speed Tranceiver Logic)
:: LVDS (Low Voltage Differential Signaling) 

&lt;/br&gt;

* Wikipedia Pages for Logic Families ([[Media:Logic Families.wiki.20140812.pdf|A.pdf]])
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&lt;[[The necessities in Computer Design]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Architecture]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Organization]]&gt;  &lt;/br&gt;


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

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

== '''Old''' ==
'''Until 2011.12''' 
'''Chapter 1. Binary Numbers'''
* 1.1 Binary Numbers([[Media:BinaryNumbers.1.A.pdf|pdf]])

''' Minterm, Maxterm, HW '''
* 1.1 Lecture01([[Media:DigitalDesign.20110922.pdf|pdf]])

''' Overflow HW '''
* Overflow Table([[Media:Overflow table.20110924.pdf|pdf]])

''' K-Map '''
* K-Map([[Media:DigitalDesign.20110926.pdf|pdf]])

''' Binary Adder '''
* Binary Adder (C, S) ([[Media:DigitalDesign.20110929.pdf|pdf]])
* Overflow detection circuit (V) ([[Media:HW Overflow20111001.pdf|pdf]])

''' BCD to Ex3 Code Coversion, Dont' Care '''
* BCD to Ex3 Code Conversion ([[Media:DigitalDesign.20111006.pdf|pdf]])

''' Prime Implicant, Dont' Care '''
* Prime Implicant, Don't Care ([[Media:DigitalDesign.20111010.pdf|pdf]])
* HW 3.6 - explain the method of combining 0's and X's

''' Multiplexer / Demultiplexer '''
* Multiplexer ([[Media:DigitalDesign.20111024.pdf|pdf]])
* HW (TBD)

''' Flip Flop / Latch '''
* FF &amp; Latch ([[Media:DigitalDesign.20111027.pdf|pdf]])
* FF &amp; Latch HW ([[Media:DigitalDesign (HW).20111027.pdf|pdf]])
* Gated D Latch &amp; Master-Slave D FlipFlop ([[Media:DigitalDesign.20111031.pdf|pdf]])
* HW (Forbidden state and Indeterminate state) ([[Media:DigitalDesign (HW).20111102.pdf|pdf]]) (note in #2, S' R' instead of S R)
* Classical Edge Triggered D FlipFlop ([[Media:DigitalDesign.20111112.pdf|pdf]])
* HW (addition in SW and HW) ([[Media:DigitalDesign (HW).20111112.pdf|pdf]]) 
* FSM1  ([[Media:DigitalDesign.FSM1.20111117.pdf|pdf]]) 
* FSM2  ([[Media:DigitalDesign.FSM2.20111117.pdf|pdf]]) 
* HW (FSM Waveforms) ([[Media:DigitalDesign (HW).20111118.pdf|pdf]]) 
 
 
''' Counter '''
* Sychronous Counter ([[Media:DigitalDesign.20111121.pdf|pdf]])
* Ripple Counter, Multiplexer, Tri-state buffer([[Media:DigitalDesign.20111124.pdf|pdf]])
* Register ([[Media:DigitalDesign.register.20111201.pdf|pdf]])
* Timing ([[Media:DigitalDesign.timing.20111201.pdf|pdf]])
* HW (Multiplexer, Shift Register) ([[Media:DigitalDesign (HW).20111201.pdf|pdf]])
* Universal Shift Register, Memory Cell ([[Media:DigitalDesign.20111206.pdf|pdf]])
* HW (Bit Serial Adder) ([[Media:DigitalDesign (HW).20111206.pdf|pdf]])
 
 
''' Memory '''
* Memory ([[Media:DigitalDesign.20111208.pdf|pdf]])

''' Comparator, Multiplier '''
* Comparator, Multiplier ([[Media:DigitalDesign.20111219.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111219.draw.pdf|2.pdf]])
 
 
'''Multiplexer based design method '''
* Multiplexer Design Method ([[Media:DigitalDesign.20111221.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111221.draw.pdf|2.pdf]])


midterm result ([[Media:MidReult.20111027.pdf|pdf]])

 
* Edge Triggered Flip Flop ([[Media:EdgeTrigFF.20111224.pdf|pdf]])
* FF Timing ([[Media:FFTiming.20111203.pdf|pdf]])


&lt;/br&gt; &lt;/br&gt;
'''Until 2013.07''' 

''' Number Systems '''
* Binary Numbers ([[Media:DD.1.A.BinNum.20130309.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD.1.B.HexaNum.20130417.pdf|B.pdf]])
* Numbers in C programs ([[Media:DD.1.C.CNum.20130309.pdf|C.pdf]])
* Codes ([[Media:DD.1.D.Coding.20130319.pdf|pdf]])
&lt;/br&gt;

&lt;/br&gt;
* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|pdf]])
&lt;/br&gt;

''' Combinational Circuits '''
* Truth Tables and Boolean Functions ([[Media:DD.2.A.TTable.20130325.pdf|2A.pdf]])&lt;/br&gt;
* K-Map ([[Media:DD.2.A.KMap.20130329.pdf|2B.pdf]])&lt;/br&gt;
* Binary Addition in C ([[Media:DD.2.C.BAinC.20130329.pdf|2.C.pdf]])&lt;/br&gt;
* Binary Arithmetic ([[Media:DD.2.D.BAri.2013.pdf|2.D.pdf]])&lt;/br&gt;
* Boolean Algebra ([[Media:DD.2.E.BAlgebra.20130419.pdf|2.E.pdf]])&lt;/br&gt;
&lt;/br&gt;

''' Sequential Circuits '''
* Latches and Flip-flops ([[Media:DD.3.A.LatchFF.20130413.pdf|3A.pdf]])&lt;/br&gt;
* FSM (Finite State Machine) ([[Media:DD.3.B.FSM.20130417.pdf|3B.pdf]])&lt;/br&gt;
* SR Latch Forbidden State ([[Media:DD.3.C.FState.20130413.pdf|3C.pdf]])&lt;/br&gt;
* Flip-flop Timing ([[Media:DD.3.D.Timing.20130413.pdf|3D.pdf]])&lt;/br&gt;
* Metastability ([[Media:DD.3.E.MetaState.20130628.pdf|3E.pdf]])&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&quot;[[The necessities in Computer Design]]&quot;  &lt;/br&gt;
&quot;[[The necessities in Computer Architecture]]&quot;  &lt;/br&gt;

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
      <sha1>f38zl8f0eep7xrnv5ad2ll116oytxj5</sha1>
    </revision>
    <revision>
      <id>2815577</id>
      <parentid>2815575</parentid>
      <timestamp>2026-06-13T19:35:36Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/*  Timing Analysis  */</comment>
      <origin>2815577</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9835" sha1="b3z4ew282ewlko7khi9fji7anahmytm" xml:space="preserve">
== ''' Number Systems '''== 

=== ''' Binary Representation '''=== 
* Binary Numbers ([[Media:DD1.1.A.BinaryNum.20130918.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD1.2.A.HexaNum.20130918.pdf|A.pdf]])
* Other Codes ([[Media:DD1.3A.Code.20250329.pdf|A.pdf]])

=== ''' Binary Arithmetic '''=== 
* Binary Arithmetic ([[Media:DD1.4.A.BinaryArith.20150425.pdf|A.pdf]])
* BCD Arithmetic ([[Media:DD1.5.A.BCDArith.20130918.pdf|A.pdf]])

=== ''' C Program Examples '''=== 
* Binary Numbers in C programs ([[Media:DD1.6.A.BNumInC.20140103.pdf|A.pdf]])
* Binary Addition in C programs ([[Media:DD1.7.A.BArithInC.20140103.pdf|A.pdf]])
&lt;/br&gt;

* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|C.pdf]])
&lt;/br&gt;

=== ''' Floating Point Numbers '''=== 

* Floating Point Representations  ([[Media:CDesign.5.A.FPoint.20140121.pdf|5A.pdf]])&lt;/br&gt;
:: See [http://www.iro.umontreal.ca/~aboulham/F1214/Session%206Arithm/Floating_Point_Numbers.pdf Floating Point Overview]
:: See [http://www.cs.auckland.ac.nz/~patrice/210-2006/210%20LN04_2.pdf Offset Binary Overview]
:: See [http://www.intersil.com/content/dam/Intersil/documents/an96/an9657.pdf Offset Binary &amp; Sin / Cosine]
:: See [http://www.ee.ic.ac.uk/hp/staff/dmb/courses/dig2/4_Analog.pdf Offset Binary &amp; ADC / DAC]
&lt;/br&gt;

=== ''' Interfacing Digital and Analog Signals '''=== 
* Sampling and Quantization   ([[Media:DD1.10.A.SampleQuant.20150425.pdf|A.pdf]])
* Digital-to-Analog Conversion  ([[Media:DD1.8.A.DAC.20140208.pdf|A.pdf]])
* Analog-to-Digital Conversion ([[Media:DD1.9.A.DAC.20140208.pdf|A.pdf]])
&lt;/br&gt;

== '''Combinational Circuits'''==

=== ''' Design '''=== 
* Boolean Algebra ([[Media:DD2.A1.BAlgebra.20250503.pdf|A1.pdf]])
* Truth Tables ([[Media:DD2.A2.TTable.20250424.pdf|A2.pdf]])
* K-Map ([[Media:DD2.A3.KMap.20250424.pdf|A3.pdf]])
* Design Examples ([[Media:DD2.A4.CombEx.20250414.pdf|A4.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Decoder ([[Media:DD2.B.1.Decoder.20130928.pdf|B1.pdf]])
* Encoder ([[Media:DD2.B.2.Encoder.20130917.pdf|B2.pdf]])
* Multiplexer ([[Media:DD2.B.3.Multiplexer.20130928.pdf|B3.pdf]])
* Adder ([[Media:DD2.B.4..Adder.20131007.pdf|B4.pdf]], [[Media:Fa.sch.20131002.pdf|fa.sch.pdf]], [[Media:Adder4.sch.20131002.pdf|adder4.sch.pdf]])
&lt;/br&gt;

=== ''' Design Metric '''=== 
* Noise Margin ([[Media:DD2.C1.NoiseMargin.20250415.pdf|C1.pdf]])
&lt;/br&gt;

== '''Sequential Circuits'''==

=== ''' Design '''=== 
* Types of Flip-Flops ([[Media:CDesign.1.A.FF.20130412.pdf |1A.pdf]])&lt;/br&gt;
* Latches and Flipflops ([[Media:DD3.A.1.LatchFF.20160308.pdf|A1.pdf]])
* State Transition Table ([[Media:DD3.A.2.pdf|A2.pdf]])
* FSM (Finite State Machine) ([[Media:DD3.A.3.FSM.20131030.pdf|A3.pdf]])
&lt;/br&gt;

* The Classic FF Design ([[Media:DD3.A.6.ClassicFF.20131126.pdf|A7.pdf]])
* The Modern FF Design ([[Media:DD3.A.6.ClassicFF.20131204.2.pdf|A8.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Latches and Flip-flops ([[Media:DD3.B.1.LatchFF.20131008.pdf|B1.pdf]])
* Registers ([[Media:DD3.B.2.Register.20150326.pdf|B2.pdf]], [[Media:Register.20131118.pdf|register.pdf]])
* Counters ([[Media:DD3.B.2.Counter.20150420.pdf|B3.pdf]])
&lt;/br&gt;

=== ''' Timing Analysis '''=== 
* Metastability ([[Media:DD3.A.4.MetaState.20131030.pdf|A4.pdf]])
* Flip-flop Timing ([[Media:DD3.A5.FFTiming.20260611.pdf|A5.pdf]])
* SR Latch Forbidden State ([[Media:DD3.A.5.ForbiddenState.20131030.pdf|A6.pdf]])
&lt;/br&gt;

* FF Min Max Timing Constraints ([[Media:CArch.MinMaxTiming.20131121.pdf |pdf]])
* FF Clock Skew Timing Constraints ([[Media:CArch.ClockSkew.20131121.pdf |pdf]])

* Synchronizer ([[Media:CArch.Synchronizer.20131216.pdf |pdf]])
* Resolution Time Analysis  ([[Media:CArch.Resolution.20131216.pdf |pdf]])
&lt;/br&gt;

== '''Finite State Machine'''==

* FSM State Encoding 
* FSM Types : Mealy and Moore Machines
* FSM Example ([[Media:CArch.2.A.FSMExample.20141018.pdf |pdf]])
&lt;/br&gt;

== '''Array  Devices''' ==

=== ''' Memory Arrays '''=== 
* RAM 
** RAM Structure ([[Media:DD4.A.1.RAM.20131111.pdf|A.pdf]])
** RAM Timing ([[Media:DD4.B.1.RAMTiming.20131130.pdf|B.pdf]])
** FPGA RAM ([[Media:DD4.C.1.FPGARAM.20160513.pdf|C.pdf]])

* ROM
&lt;/br&gt;

=== ''' Logic Arrays '''=== 
* PLA 
* PAL
* PLD
* FPGA
** FPGA Structure
** FPGA Configuration ([[Media:DD4.C.1.FPGAConf.20131130.pdf|B.pdf]])
&lt;/br&gt;
&lt;/br&gt; 
[http://www.ece.cmu.edu/~ece548/localcpy/sramop.pdf  Synchronous SRAM Timing] &lt;/br&gt;
[http://www.micron.com/~/media/Documents/Products/Technical%20Note/DRAM/tn4529.pdf Asynchronous SRAM Timing]&lt;/br&gt;
[http://www.ece.cmu.edu/~ece548/localcpy/dramop.pdf DRAM Timing] &lt;/br&gt;

[http://www.ece.unm.edu/~jimp/415/slides/fpga_arch1.pdf FPGA Architectures] &lt;/br&gt;
[http://www.engr.siu.edu/~haibo/ece428/notes/ece428_fpgaarch.pdf CPLD &amp; FPGA] &lt;/br&gt;
&lt;/br&gt;

== ''' RTL Design Techniques''' ==
&lt;/br&gt;

''' Design Methodology '''
&lt;/br&gt;


''' Synthesis '''
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

== '''Logic Families and IOs''' ==
* BJT Based 
:: DTL (Diode-Transistor Logic)
:: TTL (Transistor-Transistor Logic)
:: ECL (Emitter-Coupled Logic)
 
* MOS Based
:: CMOS (Complementary MOS)
:: Pseudo-nMOS 
:: Transmission Gate
:: BiCMOS (Bipolr + CMOS)
 
* Dynamic CMOS 
:: Domino
:: Clocked-CMOS (C&lt;sup&gt;2&lt;/sup&gt;MOS)
&lt;/br&gt; 

* Modern I/O Standards
:: TTL and LVTTL (Low Voltage TTL)
:: CMOS and LVCMOS (Low Voltage CMOS)
:: SSTL (Stub Series Terminated Logic)
:: HSTL (High Speed Tranceiver Logic)
:: LVDS (Low Voltage Differential Signaling) 

&lt;/br&gt;

* Wikipedia Pages for Logic Families ([[Media:Logic Families.wiki.20140812.pdf|A.pdf]])
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&lt;[[The necessities in Computer Design]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Architecture]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Organization]]&gt;  &lt;/br&gt;


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

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

== '''Old''' ==
'''Until 2011.12''' 
'''Chapter 1. Binary Numbers'''
* 1.1 Binary Numbers([[Media:BinaryNumbers.1.A.pdf|pdf]])

''' Minterm, Maxterm, HW '''
* 1.1 Lecture01([[Media:DigitalDesign.20110922.pdf|pdf]])

''' Overflow HW '''
* Overflow Table([[Media:Overflow table.20110924.pdf|pdf]])

''' K-Map '''
* K-Map([[Media:DigitalDesign.20110926.pdf|pdf]])

''' Binary Adder '''
* Binary Adder (C, S) ([[Media:DigitalDesign.20110929.pdf|pdf]])
* Overflow detection circuit (V) ([[Media:HW Overflow20111001.pdf|pdf]])

''' BCD to Ex3 Code Coversion, Dont' Care '''
* BCD to Ex3 Code Conversion ([[Media:DigitalDesign.20111006.pdf|pdf]])

''' Prime Implicant, Dont' Care '''
* Prime Implicant, Don't Care ([[Media:DigitalDesign.20111010.pdf|pdf]])
* HW 3.6 - explain the method of combining 0's and X's

''' Multiplexer / Demultiplexer '''
* Multiplexer ([[Media:DigitalDesign.20111024.pdf|pdf]])
* HW (TBD)

''' Flip Flop / Latch '''
* FF &amp; Latch ([[Media:DigitalDesign.20111027.pdf|pdf]])
* FF &amp; Latch HW ([[Media:DigitalDesign (HW).20111027.pdf|pdf]])
* Gated D Latch &amp; Master-Slave D FlipFlop ([[Media:DigitalDesign.20111031.pdf|pdf]])
* HW (Forbidden state and Indeterminate state) ([[Media:DigitalDesign (HW).20111102.pdf|pdf]]) (note in #2, S' R' instead of S R)
* Classical Edge Triggered D FlipFlop ([[Media:DigitalDesign.20111112.pdf|pdf]])
* HW (addition in SW and HW) ([[Media:DigitalDesign (HW).20111112.pdf|pdf]]) 
* FSM1  ([[Media:DigitalDesign.FSM1.20111117.pdf|pdf]]) 
* FSM2  ([[Media:DigitalDesign.FSM2.20111117.pdf|pdf]]) 
* HW (FSM Waveforms) ([[Media:DigitalDesign (HW).20111118.pdf|pdf]]) 
 
 
''' Counter '''
* Sychronous Counter ([[Media:DigitalDesign.20111121.pdf|pdf]])
* Ripple Counter, Multiplexer, Tri-state buffer([[Media:DigitalDesign.20111124.pdf|pdf]])
* Register ([[Media:DigitalDesign.register.20111201.pdf|pdf]])
* Timing ([[Media:DigitalDesign.timing.20111201.pdf|pdf]])
* HW (Multiplexer, Shift Register) ([[Media:DigitalDesign (HW).20111201.pdf|pdf]])
* Universal Shift Register, Memory Cell ([[Media:DigitalDesign.20111206.pdf|pdf]])
* HW (Bit Serial Adder) ([[Media:DigitalDesign (HW).20111206.pdf|pdf]])
 
 
''' Memory '''
* Memory ([[Media:DigitalDesign.20111208.pdf|pdf]])

''' Comparator, Multiplier '''
* Comparator, Multiplier ([[Media:DigitalDesign.20111219.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111219.draw.pdf|2.pdf]])
 
 
'''Multiplexer based design method '''
* Multiplexer Design Method ([[Media:DigitalDesign.20111221.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111221.draw.pdf|2.pdf]])


midterm result ([[Media:MidReult.20111027.pdf|pdf]])

 
* Edge Triggered Flip Flop ([[Media:EdgeTrigFF.20111224.pdf|pdf]])
* FF Timing ([[Media:FFTiming.20111203.pdf|pdf]])


&lt;/br&gt; &lt;/br&gt;
'''Until 2013.07''' 

''' Number Systems '''
* Binary Numbers ([[Media:DD.1.A.BinNum.20130309.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD.1.B.HexaNum.20130417.pdf|B.pdf]])
* Numbers in C programs ([[Media:DD.1.C.CNum.20130309.pdf|C.pdf]])
* Codes ([[Media:DD.1.D.Coding.20130319.pdf|pdf]])
&lt;/br&gt;

&lt;/br&gt;
* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|pdf]])
&lt;/br&gt;

''' Combinational Circuits '''
* Truth Tables and Boolean Functions ([[Media:DD.2.A.TTable.20130325.pdf|2A.pdf]])&lt;/br&gt;
* K-Map ([[Media:DD.2.A.KMap.20130329.pdf|2B.pdf]])&lt;/br&gt;
* Binary Addition in C ([[Media:DD.2.C.BAinC.20130329.pdf|2.C.pdf]])&lt;/br&gt;
* Binary Arithmetic ([[Media:DD.2.D.BAri.2013.pdf|2.D.pdf]])&lt;/br&gt;
* Boolean Algebra ([[Media:DD.2.E.BAlgebra.20130419.pdf|2.E.pdf]])&lt;/br&gt;
&lt;/br&gt;

''' Sequential Circuits '''
* Latches and Flip-flops ([[Media:DD.3.A.LatchFF.20130413.pdf|3A.pdf]])&lt;/br&gt;
* FSM (Finite State Machine) ([[Media:DD.3.B.FSM.20130417.pdf|3B.pdf]])&lt;/br&gt;
* SR Latch Forbidden State ([[Media:DD.3.C.FState.20130413.pdf|3C.pdf]])&lt;/br&gt;
* Flip-flop Timing ([[Media:DD.3.D.Timing.20130413.pdf|3D.pdf]])&lt;/br&gt;
* Metastability ([[Media:DD.3.E.MetaState.20130628.pdf|3E.pdf]])&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&quot;[[The necessities in Computer Design]]&quot;  &lt;/br&gt;
&quot;[[The necessities in Computer Architecture]]&quot;  &lt;/br&gt;

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
      <sha1>b3z4ew282ewlko7khi9fji7anahmytm</sha1>
    </revision>
  </page>
  <page>
    <title>Linux System programming in plain view</title>
    <ns>0</ns>
    <id>136794</id>
    <revision>
      <id>2815527</id>
      <parentid>2812585</parentid>
      <timestamp>2026-06-13T18:08:21Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* File System */</comment>
      <origin>2815527</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3197" sha1="fqbx555i0swjfjhf4f0r9wspsmj3f9y" xml:space="preserve">This course belongs to the [[Electrical &amp; Computer Engineering Studies]]

== Introduction ==
* Introduction ([[Media:SysP.Intro.20161128.pdf|pdf]])

== File System ==
* File System ([[Media:SysP.FileSystem.20251023.pdf|pdf]])
* File Pointer ([[Media:SysP..FilePointer.20161103.pdf|pdf]])
* System Calls ([[Media:SysP.File.SysCall.20161128.pdf|pdf]])
* File IO ([[Media:SysP.FileIO.20251023.pdf|pdf]])


* Copilot: File System ([[Media:glibcFileSystem.20251029-2.pdf|pdf]])
* Copilot: File Buffer ([[Media:glibcFileBuffer.20251025-2.pdf|pdf]])
* Copilot: File IO ([[Media:glibcFileIO.20251025-2.pdf|pdf]])
* Copilot: File Permission ([[Media:glibcFilePerm.20260121.pdf|pdf]])
* Copilot: File Control ([[Media:CP.FileCntl.20260428.pdf|pdf]], [[Media:CP.FileCntl.A.20260608.pdf|A]], [[Media:CP.FileCntl.B.20260504.pdf|B]], [[Media:CP.FileCntl.C.20260501.pdf|C]])

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

== Process ==
* Process  ([[Media:SysP.Process.20251120.pdf|pdf]])
* Fork  ([[Media:SysP.Fork.20251126.pdf|pdf]])


* Copilot: Process Information ([[Media:glibc.Process.1Info.20251101.pdf|pdf]])
* Copilot: Process Control ([[Media:glibc.Process.2Control.20251103.pdf|pdf]])
* Copilot: Process Execution ([[Media:glibc.Proc.3Exec.20251105.pdf|pdf]])
* Copilot: Process Fork ([[Media:glibc.Proc.4Fork.20251106.pdf|pdf]])
* Copilot: Process Context Switching ([[Media:glibc.Proc.5Context.20251107.pdf|pdf]])
* Copilot: Process Exec family of functions ([[Media:glibc.Proc.6ExecCall.20251112.pdf|pdf]])
* Copilot: Process Wait family of functions ([[Media:glibc.Proc.7WaitCall.20251112.pdf|pdf]])
* Copilot: Process Exit ([[Media:glibc.Proc.8Exit.20251113.pdf|pdf]])

&lt;/br&gt;

== Inter Process Communication==

=== Signal ===
* Signal  ([[Media:SysP.7.A.Signal.20121206.pdf|pdf]])


* Copilot: Signal 1. Alarm ([[Media:glibc.Signal.Alarm.20251201.pdf|pdf]])
* Copilot: Signal 2. Other Functions ([[Media:glibc.Signal.2Other.20251205.pdf|pdf]])

&lt;/br&gt;

=== Pipe ===

* Pipe ([[Media:SysP.3.A.IPC.20121115.pdf|pdf]])


* Copilot: Pipe 1. A Special File ([[Media:glibc.Pipe.File.20260307.pdf|pdf]])

&lt;/br&gt;

=== System V IPC ===

* Message Queue  ([[Media:SysP.5.A.MessageQ.20121213.pdf|pdf]])

* Shared Memory  ([[Media:SysP.8.A.SharedMem.20121227.pdf|pdf]])

* Semaphore  ([[Media:SysP.6.A.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

* Copilot: Message Queue ([[Media:glibc.MessageQ.20251202.pdf|pdf]])
* Copilot: Shared Memory ([[Media:glibc.SharedMem.20251203.pdf|pdf]])
* Copilot: Semaphore ([[Media:glibc.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

=== Socket ===

* Socket  ([[Media:SysP.4.A.Socket.20121122.pdf|pdf]])

&lt;/br&gt;

== Thread ==
* POSIX thread (pthread)  ([[Media:SysP.9.A.Pthread.20130225.pdf|pdf]])

==External links==
* [http://www.tldp.org/LDP/tlk/tlk.html The Linux Kernel]
* [http://www.tldp.org/LDP/lpg/lpg.html The Linux Programmer's Guide]
* [http://www.cs.cf.ac.uk/Dave/C/ Programming in C - UNIX System Calls and Subroutines using C.]
* [http://www.cs.cmu.edu/afs/cs/academic/class/15492-f07/www/pthreads.html POSIX thread (pthread) libraries]
* [https://computing.llnl.gov/tutorials/pthreads/#Thread POSIX Threads Programming]

[[Category:Linux]]
[[Category:Computer programming]]
[[Category:C programming language]]</text>
      <sha1>fqbx555i0swjfjhf4f0r9wspsmj3f9y</sha1>
    </revision>
    <revision>
      <id>2815529</id>
      <parentid>2815527</parentid>
      <timestamp>2026-06-13T18:17:18Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* File System */</comment>
      <origin>2815529</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3197" sha1="gtnd28v1vlrd7z1tia1vmaoqw8i1fnq" xml:space="preserve">This course belongs to the [[Electrical &amp; Computer Engineering Studies]]

== Introduction ==
* Introduction ([[Media:SysP.Intro.20161128.pdf|pdf]])

== File System ==
* File System ([[Media:SysP.FileSystem.20251023.pdf|pdf]])
* File Pointer ([[Media:SysP..FilePointer.20161103.pdf|pdf]])
* System Calls ([[Media:SysP.File.SysCall.20161128.pdf|pdf]])
* File IO ([[Media:SysP.FileIO.20251023.pdf|pdf]])


* Copilot: File System ([[Media:glibcFileSystem.20251029-2.pdf|pdf]])
* Copilot: File Buffer ([[Media:glibcFileBuffer.20251025-2.pdf|pdf]])
* Copilot: File IO ([[Media:glibcFileIO.20251025-2.pdf|pdf]])
* Copilot: File Permission ([[Media:glibcFilePerm.20260121.pdf|pdf]])
* Copilot: File Control ([[Media:CP.FileCntl.20260428.pdf|pdf]], [[Media:CP.FileCntl.A.20260609.pdf|A]], [[Media:CP.FileCntl.B.20260504.pdf|B]], [[Media:CP.FileCntl.C.20260501.pdf|C]])

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

== Process ==
* Process  ([[Media:SysP.Process.20251120.pdf|pdf]])
* Fork  ([[Media:SysP.Fork.20251126.pdf|pdf]])


* Copilot: Process Information ([[Media:glibc.Process.1Info.20251101.pdf|pdf]])
* Copilot: Process Control ([[Media:glibc.Process.2Control.20251103.pdf|pdf]])
* Copilot: Process Execution ([[Media:glibc.Proc.3Exec.20251105.pdf|pdf]])
* Copilot: Process Fork ([[Media:glibc.Proc.4Fork.20251106.pdf|pdf]])
* Copilot: Process Context Switching ([[Media:glibc.Proc.5Context.20251107.pdf|pdf]])
* Copilot: Process Exec family of functions ([[Media:glibc.Proc.6ExecCall.20251112.pdf|pdf]])
* Copilot: Process Wait family of functions ([[Media:glibc.Proc.7WaitCall.20251112.pdf|pdf]])
* Copilot: Process Exit ([[Media:glibc.Proc.8Exit.20251113.pdf|pdf]])

&lt;/br&gt;

== Inter Process Communication==

=== Signal ===
* Signal  ([[Media:SysP.7.A.Signal.20121206.pdf|pdf]])


* Copilot: Signal 1. Alarm ([[Media:glibc.Signal.Alarm.20251201.pdf|pdf]])
* Copilot: Signal 2. Other Functions ([[Media:glibc.Signal.2Other.20251205.pdf|pdf]])

&lt;/br&gt;

=== Pipe ===

* Pipe ([[Media:SysP.3.A.IPC.20121115.pdf|pdf]])


* Copilot: Pipe 1. A Special File ([[Media:glibc.Pipe.File.20260307.pdf|pdf]])

&lt;/br&gt;

=== System V IPC ===

* Message Queue  ([[Media:SysP.5.A.MessageQ.20121213.pdf|pdf]])

* Shared Memory  ([[Media:SysP.8.A.SharedMem.20121227.pdf|pdf]])

* Semaphore  ([[Media:SysP.6.A.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

* Copilot: Message Queue ([[Media:glibc.MessageQ.20251202.pdf|pdf]])
* Copilot: Shared Memory ([[Media:glibc.SharedMem.20251203.pdf|pdf]])
* Copilot: Semaphore ([[Media:glibc.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

=== Socket ===

* Socket  ([[Media:SysP.4.A.Socket.20121122.pdf|pdf]])

&lt;/br&gt;

== Thread ==
* POSIX thread (pthread)  ([[Media:SysP.9.A.Pthread.20130225.pdf|pdf]])

==External links==
* [http://www.tldp.org/LDP/tlk/tlk.html The Linux Kernel]
* [http://www.tldp.org/LDP/lpg/lpg.html The Linux Programmer's Guide]
* [http://www.cs.cf.ac.uk/Dave/C/ Programming in C - UNIX System Calls and Subroutines using C.]
* [http://www.cs.cmu.edu/afs/cs/academic/class/15492-f07/www/pthreads.html POSIX thread (pthread) libraries]
* [https://computing.llnl.gov/tutorials/pthreads/#Thread POSIX Threads Programming]

[[Category:Linux]]
[[Category:Computer programming]]
[[Category:C programming language]]</text>
      <sha1>gtnd28v1vlrd7z1tia1vmaoqw8i1fnq</sha1>
    </revision>
    <revision>
      <id>2815532</id>
      <parentid>2815529</parentid>
      <timestamp>2026-06-13T18:18:36Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* File System */</comment>
      <origin>2815532</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3197" sha1="6k5rczhqgudny953a4r0l38b6qapqhl" xml:space="preserve">This course belongs to the [[Electrical &amp; Computer Engineering Studies]]

== Introduction ==
* Introduction ([[Media:SysP.Intro.20161128.pdf|pdf]])

== File System ==
* File System ([[Media:SysP.FileSystem.20251023.pdf|pdf]])
* File Pointer ([[Media:SysP..FilePointer.20161103.pdf|pdf]])
* System Calls ([[Media:SysP.File.SysCall.20161128.pdf|pdf]])
* File IO ([[Media:SysP.FileIO.20251023.pdf|pdf]])


* Copilot: File System ([[Media:glibcFileSystem.20251029-2.pdf|pdf]])
* Copilot: File Buffer ([[Media:glibcFileBuffer.20251025-2.pdf|pdf]])
* Copilot: File IO ([[Media:glibcFileIO.20251025-2.pdf|pdf]])
* Copilot: File Permission ([[Media:glibcFilePerm.20260121.pdf|pdf]])
* Copilot: File Control ([[Media:CP.FileCntl.20260428.pdf|pdf]], [[Media:CP.FileCntl.A.20260610.pdf|A]], [[Media:CP.FileCntl.B.20260504.pdf|B]], [[Media:CP.FileCntl.C.20260501.pdf|C]])

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

== Process ==
* Process  ([[Media:SysP.Process.20251120.pdf|pdf]])
* Fork  ([[Media:SysP.Fork.20251126.pdf|pdf]])


* Copilot: Process Information ([[Media:glibc.Process.1Info.20251101.pdf|pdf]])
* Copilot: Process Control ([[Media:glibc.Process.2Control.20251103.pdf|pdf]])
* Copilot: Process Execution ([[Media:glibc.Proc.3Exec.20251105.pdf|pdf]])
* Copilot: Process Fork ([[Media:glibc.Proc.4Fork.20251106.pdf|pdf]])
* Copilot: Process Context Switching ([[Media:glibc.Proc.5Context.20251107.pdf|pdf]])
* Copilot: Process Exec family of functions ([[Media:glibc.Proc.6ExecCall.20251112.pdf|pdf]])
* Copilot: Process Wait family of functions ([[Media:glibc.Proc.7WaitCall.20251112.pdf|pdf]])
* Copilot: Process Exit ([[Media:glibc.Proc.8Exit.20251113.pdf|pdf]])

&lt;/br&gt;

== Inter Process Communication==

=== Signal ===
* Signal  ([[Media:SysP.7.A.Signal.20121206.pdf|pdf]])


* Copilot: Signal 1. Alarm ([[Media:glibc.Signal.Alarm.20251201.pdf|pdf]])
* Copilot: Signal 2. Other Functions ([[Media:glibc.Signal.2Other.20251205.pdf|pdf]])

&lt;/br&gt;

=== Pipe ===

* Pipe ([[Media:SysP.3.A.IPC.20121115.pdf|pdf]])


* Copilot: Pipe 1. A Special File ([[Media:glibc.Pipe.File.20260307.pdf|pdf]])

&lt;/br&gt;

=== System V IPC ===

* Message Queue  ([[Media:SysP.5.A.MessageQ.20121213.pdf|pdf]])

* Shared Memory  ([[Media:SysP.8.A.SharedMem.20121227.pdf|pdf]])

* Semaphore  ([[Media:SysP.6.A.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

* Copilot: Message Queue ([[Media:glibc.MessageQ.20251202.pdf|pdf]])
* Copilot: Shared Memory ([[Media:glibc.SharedMem.20251203.pdf|pdf]])
* Copilot: Semaphore ([[Media:glibc.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

=== Socket ===

* Socket  ([[Media:SysP.4.A.Socket.20121122.pdf|pdf]])

&lt;/br&gt;

== Thread ==
* POSIX thread (pthread)  ([[Media:SysP.9.A.Pthread.20130225.pdf|pdf]])

==External links==
* [http://www.tldp.org/LDP/tlk/tlk.html The Linux Kernel]
* [http://www.tldp.org/LDP/lpg/lpg.html The Linux Programmer's Guide]
* [http://www.cs.cf.ac.uk/Dave/C/ Programming in C - UNIX System Calls and Subroutines using C.]
* [http://www.cs.cmu.edu/afs/cs/academic/class/15492-f07/www/pthreads.html POSIX thread (pthread) libraries]
* [https://computing.llnl.gov/tutorials/pthreads/#Thread POSIX Threads Programming]

[[Category:Linux]]
[[Category:Computer programming]]
[[Category:C programming language]]</text>
      <sha1>6k5rczhqgudny953a4r0l38b6qapqhl</sha1>
    </revision>
    <revision>
      <id>2815534</id>
      <parentid>2815532</parentid>
      <timestamp>2026-06-13T18:19:46Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* File System */</comment>
      <origin>2815534</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3197" sha1="0xdg7vk3um6lw91ldwrm71dyuibt8hb" xml:space="preserve">This course belongs to the [[Electrical &amp; Computer Engineering Studies]]

== Introduction ==
* Introduction ([[Media:SysP.Intro.20161128.pdf|pdf]])

== File System ==
* File System ([[Media:SysP.FileSystem.20251023.pdf|pdf]])
* File Pointer ([[Media:SysP..FilePointer.20161103.pdf|pdf]])
* System Calls ([[Media:SysP.File.SysCall.20161128.pdf|pdf]])
* File IO ([[Media:SysP.FileIO.20251023.pdf|pdf]])


* Copilot: File System ([[Media:glibcFileSystem.20251029-2.pdf|pdf]])
* Copilot: File Buffer ([[Media:glibcFileBuffer.20251025-2.pdf|pdf]])
* Copilot: File IO ([[Media:glibcFileIO.20251025-2.pdf|pdf]])
* Copilot: File Permission ([[Media:glibcFilePerm.20260121.pdf|pdf]])
* Copilot: File Control ([[Media:CP.FileCntl.20260428.pdf|pdf]], [[Media:CP.FileCntl.A.20260611.pdf|A]], [[Media:CP.FileCntl.B.20260504.pdf|B]], [[Media:CP.FileCntl.C.20260501.pdf|C]])

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

== Process ==
* Process  ([[Media:SysP.Process.20251120.pdf|pdf]])
* Fork  ([[Media:SysP.Fork.20251126.pdf|pdf]])


* Copilot: Process Information ([[Media:glibc.Process.1Info.20251101.pdf|pdf]])
* Copilot: Process Control ([[Media:glibc.Process.2Control.20251103.pdf|pdf]])
* Copilot: Process Execution ([[Media:glibc.Proc.3Exec.20251105.pdf|pdf]])
* Copilot: Process Fork ([[Media:glibc.Proc.4Fork.20251106.pdf|pdf]])
* Copilot: Process Context Switching ([[Media:glibc.Proc.5Context.20251107.pdf|pdf]])
* Copilot: Process Exec family of functions ([[Media:glibc.Proc.6ExecCall.20251112.pdf|pdf]])
* Copilot: Process Wait family of functions ([[Media:glibc.Proc.7WaitCall.20251112.pdf|pdf]])
* Copilot: Process Exit ([[Media:glibc.Proc.8Exit.20251113.pdf|pdf]])

&lt;/br&gt;

== Inter Process Communication==

=== Signal ===
* Signal  ([[Media:SysP.7.A.Signal.20121206.pdf|pdf]])


* Copilot: Signal 1. Alarm ([[Media:glibc.Signal.Alarm.20251201.pdf|pdf]])
* Copilot: Signal 2. Other Functions ([[Media:glibc.Signal.2Other.20251205.pdf|pdf]])

&lt;/br&gt;

=== Pipe ===

* Pipe ([[Media:SysP.3.A.IPC.20121115.pdf|pdf]])


* Copilot: Pipe 1. A Special File ([[Media:glibc.Pipe.File.20260307.pdf|pdf]])

&lt;/br&gt;

=== System V IPC ===

* Message Queue  ([[Media:SysP.5.A.MessageQ.20121213.pdf|pdf]])

* Shared Memory  ([[Media:SysP.8.A.SharedMem.20121227.pdf|pdf]])

* Semaphore  ([[Media:SysP.6.A.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

* Copilot: Message Queue ([[Media:glibc.MessageQ.20251202.pdf|pdf]])
* Copilot: Shared Memory ([[Media:glibc.SharedMem.20251203.pdf|pdf]])
* Copilot: Semaphore ([[Media:glibc.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

=== Socket ===

* Socket  ([[Media:SysP.4.A.Socket.20121122.pdf|pdf]])

&lt;/br&gt;

== Thread ==
* POSIX thread (pthread)  ([[Media:SysP.9.A.Pthread.20130225.pdf|pdf]])

==External links==
* [http://www.tldp.org/LDP/tlk/tlk.html The Linux Kernel]
* [http://www.tldp.org/LDP/lpg/lpg.html The Linux Programmer's Guide]
* [http://www.cs.cf.ac.uk/Dave/C/ Programming in C - UNIX System Calls and Subroutines using C.]
* [http://www.cs.cmu.edu/afs/cs/academic/class/15492-f07/www/pthreads.html POSIX thread (pthread) libraries]
* [https://computing.llnl.gov/tutorials/pthreads/#Thread POSIX Threads Programming]

[[Category:Linux]]
[[Category:Computer programming]]
[[Category:C programming language]]</text>
      <sha1>0xdg7vk3um6lw91ldwrm71dyuibt8hb</sha1>
    </revision>
  </page>
  <page>
    <title>Understanding Arithmetic Circuits</title>
    <ns>0</ns>
    <id>139384</id>
    <revision>
      <id>2815492</id>
      <parentid>2815339</parentid>
      <timestamp>2026-06-13T13:27:18Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
      <origin>2815492</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3088" sha1="ppfdieozupvjkkrqnn5n79i21jet6di" xml:space="preserve">
== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  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.20260613.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260613.pdf|B]] ||
|| [[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]]||
|| || 
|}
&lt;/br&gt;
 
=== 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
&lt;/br&gt;

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

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

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

=== 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]])
&lt;/br&gt;

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

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

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

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
      <sha1>ppfdieozupvjkkrqnn5n79i21jet6di</sha1>
    </revision>
  </page>
  <page>
    <title>User:Dl1gcr</title>
    <ns>2</ns>
    <id>170612</id>
    <revision>
      <id>2815594</id>
      <parentid>1285698</parentid>
      <timestamp>2026-06-14T02:08:26Z</timestamp>
      <contributor>
        <username>PieWriter</username>
        <id>3039865</id>
      </contributor>
      <minor />
      <comment>Remove template/category (via JWB)</comment>
      <origin>2815594</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="628" sha1="cooqsu9gi9gr2b88c1lspu9pw6mjuaf" xml:space="preserve">Dear colleagues,

I am about to write my first article in Wikipedia as a &quot;test case&quot; for further contribution. Convinced of the value of the idea working on a free encyclopedia it is my special interest to support this project. But also to be aware about the fact that free commitment sometimes implicates well-meant participation which is not always expedient. Being aware of this, I put my ideas on a platform which has been hopefully granted increasingly reliable.

P.S. User dl1gcr is also registered on fr.wikipedia.org after being invited. It is my particular pleasure and challenge to contribute to the francophone world.</text>
      <sha1>cooqsu9gi9gr2b88c1lspu9pw6mjuaf</sha1>
    </revision>
  </page>
  <page>
    <title>The necessities in Filter Theory</title>
    <ns>0</ns>
    <id>199550</id>
    <revision>
      <id>2815560</id>
      <parentid>2812595</parentid>
      <timestamp>2026-06-13T19:14:29Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Sample Processing Methods */</comment>
      <origin>2815560</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1524" sha1="gh3mtjsoh6uf00bxe5oeguduye80i4u" xml:space="preserve">==''' Background '''==
=== Bode plot ===

See [http://lpsa.swarthmore.edu/Bode/Bode.html swarthmore]
&lt;/br&gt;

=== OP Amp ===
Overview ([[Media:OPAmp.A.1.20151203.pdf  |pdf]])
 
See [http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/opampcon.html#c1 Hyperphysics]
&lt;/br&gt;

==''' Analog Filter Analysis (Continuous Time) '''==
=== First Order Filters ===
&lt;/br&gt;

=== Second Order Filters ===
&lt;/br&gt;

==''' Digital Filter Analysis (Discrete Time) '''==

=== Sample Processing Methods  ===
* Tapped Delays ([[Media:Sample.TappedDelay.20260608.pdf  |A.pdf]])
* Programming Considerations
* Circular Buffers

=== FIR Filter Realizations ===
* Direct Form FIR Filter 
* Canonical Form FIR Filter
* Cascade Form FIR Filter

=== IIR Filter Realizations ===
* Direct Form IIR Filter ([[Media:IIR.DirectForm.20231209.pdf  |A.pdf]])
* Canonical Form IIR Filter
* Cascade Form IIR Filter 
&lt;/br&gt;

=== FIR (Finite Impulse Response) Filters ===
* Block Processing Methods
* Sample Processing Methods
* Window Method
* Kaiser Window
* Frequency Sampling Method
&lt;/br&gt;

=== IIR (Infinite Impulse Response) Filters ===
* Bilinear Transform
* 1st Order Lowpass and Highpass Filters
* 2nd Order Lowpass and Highpass Filters
* Parametric Equalizer Filters
* Comb Filters
* High Order Filters
&lt;/br&gt;


=== Example Octave Codes for Digital Filters === 

==== Octave Functions for Filters ====
* Octave Functions for Filters ([[Media:Octave.1.Function.1.A.20180219.pdf  |A.pdf]]) 


&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>gh3mtjsoh6uf00bxe5oeguduye80i4u</sha1>
    </revision>
    <revision>
      <id>2815562</id>
      <parentid>2815560</parentid>
      <timestamp>2026-06-13T19:15:41Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Sample Processing Methods */</comment>
      <origin>2815562</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1524" sha1="16cswutq14elkx0xisibktzigmsnfzc" xml:space="preserve">==''' Background '''==
=== Bode plot ===

See [http://lpsa.swarthmore.edu/Bode/Bode.html swarthmore]
&lt;/br&gt;

=== OP Amp ===
Overview ([[Media:OPAmp.A.1.20151203.pdf  |pdf]])
 
See [http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/opampcon.html#c1 Hyperphysics]
&lt;/br&gt;

==''' Analog Filter Analysis (Continuous Time) '''==
=== First Order Filters ===
&lt;/br&gt;

=== Second Order Filters ===
&lt;/br&gt;

==''' Digital Filter Analysis (Discrete Time) '''==

=== Sample Processing Methods  ===
* Tapped Delays ([[Media:Sample.TappedDelay.20260609.pdf  |A.pdf]])
* Programming Considerations
* Circular Buffers

=== FIR Filter Realizations ===
* Direct Form FIR Filter 
* Canonical Form FIR Filter
* Cascade Form FIR Filter

=== IIR Filter Realizations ===
* Direct Form IIR Filter ([[Media:IIR.DirectForm.20231209.pdf  |A.pdf]])
* Canonical Form IIR Filter
* Cascade Form IIR Filter 
&lt;/br&gt;

=== FIR (Finite Impulse Response) Filters ===
* Block Processing Methods
* Sample Processing Methods
* Window Method
* Kaiser Window
* Frequency Sampling Method
&lt;/br&gt;

=== IIR (Infinite Impulse Response) Filters ===
* Bilinear Transform
* 1st Order Lowpass and Highpass Filters
* 2nd Order Lowpass and Highpass Filters
* Parametric Equalizer Filters
* Comb Filters
* High Order Filters
&lt;/br&gt;


=== Example Octave Codes for Digital Filters === 

==== Octave Functions for Filters ====
* Octave Functions for Filters ([[Media:Octave.1.Function.1.A.20180219.pdf  |A.pdf]]) 


&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>16cswutq14elkx0xisibktzigmsnfzc</sha1>
    </revision>
    <revision>
      <id>2815564</id>
      <parentid>2815562</parentid>
      <timestamp>2026-06-13T19:16:39Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/*  Digital Filter Analysis (Discrete Time)  */</comment>
      <origin>2815564</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1524" sha1="apd4gjrpoz9jmdszj385f8dguta910f" xml:space="preserve">==''' Background '''==
=== Bode plot ===

See [http://lpsa.swarthmore.edu/Bode/Bode.html swarthmore]
&lt;/br&gt;

=== OP Amp ===
Overview ([[Media:OPAmp.A.1.20151203.pdf  |pdf]])
 
See [http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/opampcon.html#c1 Hyperphysics]
&lt;/br&gt;

==''' Analog Filter Analysis (Continuous Time) '''==
=== First Order Filters ===
&lt;/br&gt;

=== Second Order Filters ===
&lt;/br&gt;

==''' Digital Filter Analysis (Discrete Time) '''==

=== Sample Processing Methods  ===
* Tapped Delays ([[Media:Sample.TappedDelay.20260610.pdf  |A.pdf]])
* Programming Considerations
* Circular Buffers

=== FIR Filter Realizations ===
* Direct Form FIR Filter 
* Canonical Form FIR Filter
* Cascade Form FIR Filter

=== IIR Filter Realizations ===
* Direct Form IIR Filter ([[Media:IIR.DirectForm.20231209.pdf  |A.pdf]])
* Canonical Form IIR Filter
* Cascade Form IIR Filter 
&lt;/br&gt;

=== FIR (Finite Impulse Response) Filters ===
* Block Processing Methods
* Sample Processing Methods
* Window Method
* Kaiser Window
* Frequency Sampling Method
&lt;/br&gt;

=== IIR (Infinite Impulse Response) Filters ===
* Bilinear Transform
* 1st Order Lowpass and Highpass Filters
* 2nd Order Lowpass and Highpass Filters
* Parametric Equalizer Filters
* Comb Filters
* High Order Filters
&lt;/br&gt;


=== Example Octave Codes for Digital Filters === 

==== Octave Functions for Filters ====
* Octave Functions for Filters ([[Media:Octave.1.Function.1.A.20180219.pdf  |A.pdf]]) 


&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>apd4gjrpoz9jmdszj385f8dguta910f</sha1>
    </revision>
    <revision>
      <id>2815566</id>
      <parentid>2815564</parentid>
      <timestamp>2026-06-13T19:17:48Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Sample Processing Methods */</comment>
      <origin>2815566</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1524" sha1="sf3q6655fmq9j1nbkzd8cx9i3v8lnlb" xml:space="preserve">==''' Background '''==
=== Bode plot ===

See [http://lpsa.swarthmore.edu/Bode/Bode.html swarthmore]
&lt;/br&gt;

=== OP Amp ===
Overview ([[Media:OPAmp.A.1.20151203.pdf  |pdf]])
 
See [http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/opampcon.html#c1 Hyperphysics]
&lt;/br&gt;

==''' Analog Filter Analysis (Continuous Time) '''==
=== First Order Filters ===
&lt;/br&gt;

=== Second Order Filters ===
&lt;/br&gt;

==''' Digital Filter Analysis (Discrete Time) '''==

=== Sample Processing Methods  ===
* Tapped Delays ([[Media:Sample.TappedDelay.20260611.pdf  |A.pdf]])
* Programming Considerations
* Circular Buffers

=== FIR Filter Realizations ===
* Direct Form FIR Filter 
* Canonical Form FIR Filter
* Cascade Form FIR Filter

=== IIR Filter Realizations ===
* Direct Form IIR Filter ([[Media:IIR.DirectForm.20231209.pdf  |A.pdf]])
* Canonical Form IIR Filter
* Cascade Form IIR Filter 
&lt;/br&gt;

=== FIR (Finite Impulse Response) Filters ===
* Block Processing Methods
* Sample Processing Methods
* Window Method
* Kaiser Window
* Frequency Sampling Method
&lt;/br&gt;

=== IIR (Infinite Impulse Response) Filters ===
* Bilinear Transform
* 1st Order Lowpass and Highpass Filters
* 2nd Order Lowpass and Highpass Filters
* Parametric Equalizer Filters
* Comb Filters
* High Order Filters
&lt;/br&gt;


=== Example Octave Codes for Digital Filters === 

==== Octave Functions for Filters ====
* Octave Functions for Filters ([[Media:Octave.1.Function.1.A.20180219.pdf  |A.pdf]]) 


&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>sf3q6655fmq9j1nbkzd8cx9i3v8lnlb</sha1>
    </revision>
  </page>
  <page>
    <title>Python programming in plain view</title>
    <ns>0</ns>
    <id>212733</id>
    <revision>
      <id>2815509</id>
      <parentid>2812437</parentid>
      <timestamp>2026-06-13T16:56:33Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
      <origin>2815509</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2647" sha1="k72vcldxlvnypczyb4ohupwujwa4ugk" xml:space="preserve">==''' Part I '''==
&lt;!----------------------------------------------------------------------&gt;
===  Introduction ===

* Overview 
* Memory 
* Number 

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


=== Python for C programmers ===

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

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


=== Using Libraries ===

* Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]])
* Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]])
* Packages ([[Media:Python.Work2.Package.1A.20241207.pdf |pdf]])
* Libraries ([[Media:Python.Work2.Library.1A.20260608.pdf |pdf]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


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

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

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

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

=== Handling a Big Work ===

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

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

=== Handling Series of Data ===

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

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

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


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

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

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

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

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
      <sha1>k72vcldxlvnypczyb4ohupwujwa4ugk</sha1>
    </revision>
    <revision>
      <id>2815511</id>
      <parentid>2815509</parentid>
      <timestamp>2026-06-13T16:58:25Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
      <origin>2815511</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2647" sha1="o8ar5r66jzhmd2bhzn3r7d4psk072x2" xml:space="preserve">==''' Part I '''==
&lt;!----------------------------------------------------------------------&gt;
===  Introduction ===

* Overview 
* Memory 
* Number 

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


=== Python for C programmers ===

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

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


=== Using Libraries ===

* Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]])
* Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]])
* Packages ([[Media:Python.Work2.Package.1A.20241207.pdf |pdf]])
* Libraries ([[Media:Python.Work2.Library.1A.20260609.pdf |pdf]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


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

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

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

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

=== Handling a Big Work ===

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

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

=== Handling Series of Data ===

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

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

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


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

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

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

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

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
      <sha1>o8ar5r66jzhmd2bhzn3r7d4psk072x2</sha1>
    </revision>
    <revision>
      <id>2815513</id>
      <parentid>2815511</parentid>
      <timestamp>2026-06-13T16:59:18Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
      <origin>2815513</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2647" sha1="46ctl0loa1sgoktdauh73bbxnk68v1s" xml:space="preserve">==''' Part I '''==
&lt;!----------------------------------------------------------------------&gt;
===  Introduction ===

* Overview 
* Memory 
* Number 

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


=== Python for C programmers ===

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

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


=== Using Libraries ===

* Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]])
* Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]])
* Packages ([[Media:Python.Work2.Package.1A.20241207.pdf |pdf]])
* Libraries ([[Media:Python.Work2.Library.1A.20260610.pdf |pdf]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


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

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

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

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

=== Handling a Big Work ===

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

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

=== Handling Series of Data ===

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

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

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


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

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

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

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

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
      <sha1>46ctl0loa1sgoktdauh73bbxnk68v1s</sha1>
    </revision>
    <revision>
      <id>2815515</id>
      <parentid>2815513</parentid>
      <timestamp>2026-06-13T17:00:05Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
      <origin>2815515</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2647" sha1="tt30c3kdlplra4laha5bcc6b9uxuzui" xml:space="preserve">==''' Part I '''==
&lt;!----------------------------------------------------------------------&gt;
===  Introduction ===

* Overview 
* Memory 
* Number 

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


=== Python for C programmers ===

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

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


=== Using Libraries ===

* Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]])
* Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]])
* Packages ([[Media:Python.Work2.Package.1A.20241207.pdf |pdf]])
* Libraries ([[Media:Python.Work2.Library.1A.20260611.pdf |pdf]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


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

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

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

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

=== Handling a Big Work ===

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

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

=== Handling Series of Data ===

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

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

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


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

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

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

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

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
      <sha1>tt30c3kdlplra4laha5bcc6b9uxuzui</sha1>
    </revision>
  </page>
  <page>
    <title>Vehicle identification number</title>
    <ns>0</ns>
    <id>223035</id>
    <revision>
      <id>2815610</id>
      <parentid>2815418</parentid>
      <timestamp>2026-06-14T02:49:00Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <comment>Restored revision 2607366 by [[Special:Contributions/Meatboat|Meatboat]] ([[en:w:User:BrandonXLF/Restorer|Restorer]])</comment>
      <origin>2815610</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="694" sha1="f533l8gd5n1khvfue6yd4i7g8e8wdxi" xml:space="preserve">[[File:Framenummer voorbeeld.jpg|thumb|Vehicle identification number]]

A '''vehicle identification number''' ('''VIN'''), also called a chassis number, is a unique code, including a serial number, used by the automotive industry to identify individual motor vehicles, towed vehicles, motorcycles, scooters and mopeds, as defined in ISO 3833.&lt;ref&gt;[[Wikipedia:Vehicle identification number]]&lt;/ref&gt;

== VIN Resources ==
* [https://www.edmunds.com/driving-tips/making-sense-of-your-vin.html Edmunds.com: Making sense of your VIN]
* [https://www.vindecoder.net/ VIN Decoder.net]
* [https://www.carfax.com/company/vehicle-identification-numbers-vins CARFAX VIN Decoder]

== References ==
{{Reflist}}</text>
      <sha1>f533l8gd5n1khvfue6yd4i7g8e8wdxi</sha1>
    </revision>
  </page>
  <page>
    <title>User:Marshallsumter/Radiation astronomy/Aerometeors</title>
    <ns>2</ns>
    <id>234478</id>
    <revision>
      <id>2815639</id>
      <parentid>2698115</parentid>
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      <text bytes="116074" sha1="0j9uxbb9amyflg6mlzq1cn9x61jh77g" xml:space="preserve">[[Image:April18 AR.gif|right|thumb|300px|An atmospheric river forms over Hawai'i then heads toward California 10-11 April 2017. Credit: UW-CIMSS.{{tlx|fairuse}}]]
&quot;Several times a year an atmospheric river [shown in the image on the right forming over Hawai'i]—a long, narrow conveyor belt of storms that stream in relentlessly from the Pacific Ocean—drops inches of rain or feet of snow on the U.S. west coast. Such a system triggered floods and mudslides in central and southern California this past weekend [2-3 February 2019].&quot;&lt;ref name=Fischetti&gt;{{ cite book
|author=Mark Fischetti
|title=Warning Scale Unveiled for Dangerous Rivers in the Sky
|publisher=Scientific American
|location=
|date=February 5, 2019
|url=https://www.scientificamerican.com/article/warning-scale-unveiled-for-dangerous-rivers-in-the-sky/
|accessdate=8 February 2019 }}&lt;/ref&gt;

&quot;Atmospheric rivers flow through the sky about a mile above the ocean surface, and may extend across a thousand miles of ocean to the coast. Some bring routine rain but the more intense systems can carry as much water as 15 Mississippi Rivers. The series of storms striking land can arrive for days or, occasionally, weeks on end. They hit west-facing coastlines worldwide, although the U.S. experiences more than most other national coasts.&quot;&lt;ref name=Fischetti/&gt;

The “atmospheric river scale” &quot;ranks severity and impacts, from category 1 (weak) to category 5 (exceptional).&quot;&lt;ref name=Fischetti/&gt;

&quot;Without a scale, we really had no way to objectively communicate what would be a strong storm or a weak one.&quot;&lt;ref name=Ralph&gt;{{ cite book
|author=Martin Ralph
|title=Warning Scale Unveiled for Dangerous Rivers in the Sky
|publisher=Scientific American
|location=
|date=February 5, 2019
|url=https://www.scientificamerican.com/article/warning-scale-unveiled-for-dangerous-rivers-in-the-sky/
|accessdate=8 February 2019 }}&lt;/ref&gt;

&quot;Scientists, the media and the public viewed atmospheric rivers as primarily a hazard, but the weaker ARs are quite beneficial. Water managers made it clear to us that a rating scale would be helpful.&quot;&lt;ref name=Ralph/&gt;

&quot;The scale, published Tuesday in the ''Bulletin of the American Meteorological Society'', ranks atmospheric rivers on five levels:&quot;&lt;ref name=Fischetti/&gt;
* Category 1: Weak—primarily beneficial
* Category 2: Moderate—mostly beneficial, but also somewhat hazardous
* Category 3: Strong—balance of beneficial and hazardous
* Category 4: Extreme—mostly hazardous, but also beneficial (if persistent drought)
* Category 5—Exceptional—primarily hazardous
{{clear}}

==Atmospheric rivers==
[[Image:Atmospheric River GOES WV 20101220.1200.goes11.vapor.x.pacus.x.jpg|thumb|right|250px|Water vapor imagery of the eastern Pacific Ocean from the GOES 11 satellite, shows a large atmospheric river aimed across California in December 2010. Credit: United States Naval Research Laboratory, Monterey.{{tlx|free media}}]]
[[Image:NASA_Atmospheric_river_AsiaNA2017_10_26.jpg|thumb|left|250px|NASA Image of the Day October 26, 2017, AR connects Asia to North America. Credit: NASA Earth Observatory.{{tlx|free media}}]]
[[Image:DesmondAtmosphericRiver.png|thumb|center|250px|Layered precipitable water imagery of particularly strong atmospheric rivers on 5 December 2015. Credit: NWS OPC.{{tlx|free media}}]]
The particularly intense storm system in the image on the right produced as much as {{convert|26|in|cm|abbr=on}} of precipitation in California and up to {{convert|17|ft|cm|abbr=on}} of snowfall in the Sierra Nevada during December 17–22, 2010.

Atmospheric rivers consist of narrow bands of enhanced water vapor transport, typically along the boundaries between large areas of divergent surface air flow, including some frontal zones in association with extratropical cyclones that form over the oceans.&lt;ref name=&quot;ar94&quot;&gt;{{ cite journal
|last=Zhu |first=Yong |author2=Reginald E. Newell
|title=Atmospheric rivers and bombs
|journal=Geophysical Research Letters
|year=1994
|volume=21
|issue=18
|pages=1999–2002
|doi=10.1029/94GL01710
|bibcode=1994GeoRL..21.1999Z
|url=https://web.archive.org/web/20100610063041/http://paos.colorado.edu/~dcn/ATOC6020/papers/AtmosphericRivers_94GL01710.pdf  }}&lt;/ref&gt;&lt;ref name=&quot;mwr98&quot;&gt;{{ cite journal
|last=Zhu|first=Yong|author2=Reginald E. Newell
|title=A Proposed Algorithm for Moisture Fluxes from Atmospheric Rivers
|journal=Monthly Weather Review
|year=1998
|volume=126
|issue=3
|pages= 725–735
|doi=10.1175/1520-0493(1998)126&lt;0725:APAFMF&gt;2.0.CO;2
|issn=1520-0493
|bibcode = 1998MWRv..126..725Z }}&lt;/ref&gt;&lt;ref name=&quot;sci06&quot;&gt;{{ cite journal
|last=Kerr|first=Richard A.
|title=Rivers in the Sky Are Flooding The World With Tropical Waters
|journal=Science
|date=28 July 2006
|volume=313
|issue=5786
|pages=435
|doi=10.1126/science.313.5786.435
|url=http://tenaya.ucsd.edu/~dettinge/atmos_rivers.science.pdf
|pmid=16873624 }}&lt;/ref&gt;&lt;ref name=&quot;ncaro&quot;&gt;{{ cite conference | first = Allen B. | last = White
| date = 2009-10-08
| title = The NOAA coastal atmospheric river observatory, In: ''34th Conference on Radar Meteorology''
| url = http://ams.confex.com/ams/34Radar/techprogram/paper_155601.htm
|display-authors=etal }}&lt;/ref&gt;

Pineapple Express storms are the most commonly represented and recognized type of atmospheric rivers; they are given the name due to the warm water vapor plumes originating over the Hawaiian tropics that follow a path towards California.&lt;ref name=Dettinger&gt;{{ cite journal
|last=Dettinger|first=Michael
|date=2011-06-01
|title=Climate Change, Atmospheric Rivers, and Floods in California – A Multimodel Analysis of Storm Frequency and Magnitude Changes1
|journal=JAWRA Journal of the American Water Resources Association
|volume=47
|issue=3
|pages=514–523
|doi=10.1111/j.1752-1688.2011.00546.x
|issn=1752-1688
|bibcode=2011JAWRA..47..514D }}&lt;/ref&gt;&lt;ref name=Dettinger2011&gt;{{ cite journal
|last=Dettinger|first=Michael D.|last2=Ralph|first2=Fred Martin|last3=Das|first3=Tapash|last4=Neiman|first4=Paul J.|last5=Cayan|first5=Daniel R.
|date=2011-03-24
|title=Atmospheric Rivers, Floods and the Water Resources of California
|url=http://www.mdpi.com/2073-4441/3/2/445
|journal=Water
|volume=3
|issue=2
|pages=445–478
|doi=10.3390/w3020445 }}&lt;/ref&gt;

Atmospheric rivers are typically several thousand kilometers long and only a few hundred kilometers wide, and a single one can carry a greater flux of water than the Earth's largest river, the Amazon River.&lt;ref name=&quot;mwr98&quot; /&gt;

The length and width factors in conjunction with an integrated water vapor depth greater than 2.0&amp;nbsp;cm are used as standards to categorize atmospheric river events.&lt;ref name=Dettinger2011/&gt;&lt;ref name=&quot;grl06&quot; /&gt;&lt;ref&gt;{{ cite journal
|last=Guan|first=Bin|last2=Waliser|first2=Duane E.|last3=Molotch|first3=Noah P.|last4=Fetzer|first4=Eric J.|last5=Neiman|first5=Paul J.
|date=2011-08-24
|title=Does the Madden–Julian Oscillation Influence Wintertime Atmospheric Rivers and Snowpack in the Sierra Nevada?
|journal=Monthly Weather Review
|volume=140
|issue=2
|pages=325–342
|doi=10.1175/MWR-D-11-00087.1
|issn=0027-0644
|bibcode=2012MWRv..140..325G }}&lt;/ref&gt;&lt;ref name=Guan&gt;{{ cite journal
|last=Guan|first=Bin|last2=Waliser|first2=Duane E.
|date=2015-12-27
|title=Detection of atmospheric rivers: Evaluation and application of an algorithm for global studies
|journal=Journal of Geophysical Research: Atmospheres
|volume=120
|issue=24
|pages=2015JD024257
|doi=10.1002/2015JD024257
|issn=2169-8996
|bibcode=2015JGRD..12012514G }}&lt;/ref&gt;

Integrated water vapor transport (IVT) is more directly attributed to orographic precipitation, a key factor in the production of intense rainfall and subsequent flooding.&lt;ref name=Guan/&gt;

On any given day, atmospheric rivers account for over 90% of the global meridional (north-south) water vapor transport, yet they cover less than 10% of the Earth's circumference.&lt;ref name=&quot;mwr98&quot; /&gt; Atmospheric rivers are also known to contribute to about 22% of total global runoff.&lt;ref name=Paltan&gt;{{ cite journal
|last=Paltan|first=Homero|last2=Waliser|first2=Duane|last3=Lim|first3=Wee Ho|last4=Guan|first4=Bin|last5=Yamazaki|first5=Dai|last6=Pant|first6=Raghav|last7=Dadson|first7=Simon
|date=2017-10-25
|title=Global Floods and Water Availability Driven by Atmospheric Rivers
|journal=Geophysical Research Letters
|volume=44
|issue=20
|pages=10,387–10,395
|doi=10.1002/2017gl074882
|issn=0094-8276
|bibcode=2017GeoRL..4410387P }}&lt;/ref&gt;

They also are the major cause of extreme precipitation events that cause severe flooding in many mid-latitude, westerly coastal regions of the world, including the West Coast of North America,&lt;ref name=&quot;neiman2010&quot;&gt;{{ cite conference
|first=Paul J. |last=Neiman
|date=2009-06-08
|title=Landfalling Impacts of Atmospheric Rivers: From Extreme Events to Long-term Consequences, In: ''The 2010 Mountain Climate Research Conference''
|url=http://www.fs.fed.us/psw/mtnclim/talks/pdf/Neiman_Talk2010.pdf
|display-authors=etal }}&lt;/ref&gt;&lt;ref name=&quot;mwr08&quot;&gt;{{ cite journal
|last=Neiman|first=Paul J.
|title=Diagnosis of an Intense Atmospheric River Impacting the Pacific Northwest: Storm Summary and Offshore Vertical Structure Observed with COSMIC Satellite Retrievals
|journal=Monthly Weather Review
|year=2008
|volume=136
|issue=11
|pages=4398–4420
|doi=10.1175/2008MWR2550.1
|url=http://tenaya.ucsd.edu/~dettinge/neiman_cosmic08.pdf
|bibcode = 2008MWRv..136.4398N
|display-authors=etal }}&lt;/ref&gt;&lt;ref name=&quot;jmet08&quot;&gt;{{ cite journal
|last=Neiman|first=Paul J.
|title=Meteorological Characteristics and Overland Precipitation Impacts of Atmospheric Rivers Affecting the West Coast of North America Based on Eight Years of SSM/I Satellite Observations
|journal=Journal of Hydrometeorology
|year=2008
|volume=9
|issue=1
|pages=22–47
|doi=10.1175/2007JHM855.1
|url=http://tenaya.ucsd.edu/~dettinge/Neiman_Ar-JHM08.pdf
|bibcode = 2008JHyMe...9...22N
|display-authors=etal }}&lt;/ref&gt;&lt;ref name=&quot;grl06&quot;&gt;{{ cite journal
|last=Ralph|first=F. Martin
|title=Flooding on California's Russian River: Role of atmospheric rivers
|journal=Geophys. Res. Lett.
|year=2006
|volume=33
|issue=13
|pages=L13801
|doi=10.1029/2006GL026689
|url=http://tenaya.ucsd.edu/~dettinge/atmos_rivers.pdf
|bibcode=2006GeoRL..3313801R
|display-authors=etal }}&lt;/ref&gt; Western Europe,&lt;ref&gt;{{ cite web
|title=Atmospheric river of moisture targets Britain and Ireland
|url=http://cimss.ssec.wisc.edu/goes/blog/archives/3838|work=CIMSS Satellite Blog
|date=November 19, 2009 }}&lt;/ref&gt;&lt;ref&gt;{{ cite journal
|last=Stohl|first=A. |author2=Forster, C. |author3=Sodermann, H.
|title=Remote sources of water vapor forming precipitation on the Norwegian west coast at 60°N–a tale of hurricanes and an atmospheric river
|journal=Journal of Geophysical Research
|date=March 2008
|volume=113
|issue=D5
|pages=n/a
|url=
| doi = 10.1029/2007jd009006
|bibcode=2008JGRD..113.5102S
|accessdate=10 July 2012 }}&lt;/ref&gt;&lt;ref&gt;{{ cite journal
|last=Lavers|first=David A|author2=R. P. Allan |author3=E. F. Wood |author4=G. Villarini |author5=D. J. Brayshaw |author6=A. J. Wade
|title=Winter floods in Britain are connected to atmospheric rivers
|journal=Geophysical Research Letters
|date=6 December 2011
|volume=38
|issue=23
|pages=n/a
|doi=10.1029/2011GL049783
|url=http://www.met.reading.ac.uk/~sgs02rpa/PAPERS/Lavers11GRL.pdf
|accessdate=12 August 2012
|bibcode = 2011GeoRL..3823803L
|citeseerx=10.1.1.722.4841 }}&lt;/ref&gt; the west coast of North Africa,&lt;ref name=&quot;sci06&quot; /&gt; the Iberian Peninsula, Iran and New Zealand.&lt;ref name=Guan/&gt; Equally, the absence of atmospheric rivers has been linked with the occurrence of droughts in several parts of the world including South Africa, Spain and Portugal.&lt;ref name=Paltan/&gt;

The inconsistency of California's rainfall is due to the variability in strength and quantity of these storms, which can produce strenuous effects on California's water budget, which make California a perfect case study to show the importance of proper water management and prediction of these storms.&lt;ref name=Dettinger2011/&gt; The significance atmospheric rivers have for the control of coastal water budgets juxtaposed against their creation of detrimental floods can be constructed and studied by looking at California and the surrounding coastal region of the western United States, where atmospheric rivers have contributed 30-50% of total annual rainfall.&lt;ref&gt;{{ cite journal
|last=Dettinger|first=Michael D.
|date=2013-06-28
|title=Atmospheric Rivers as Drought Busters on the U.S. West Coast
|journal=Journal of Hydrometeorology
|volume=14
|issue=6
|pages=1721–1732
|doi=10.1175/JHM-D-13-02.1
|issn=1525-755X 
|bibcode=2013JHyMe..14.1721D }}&lt;/ref&gt; The Fourth National Climate Assessment (NCA) report, released by the U.S. Global Change Research Program (USGCRP) on November 23, 2018&lt;ref name=&quot;CNN_Christensen_2018&quot;&gt;{{ cite news
|url=https://www.cnn.com/2018/11/23/health/climate-change-report-bn/index.html
|title=Climate change will shrink US economy and kill thousands, government report warns
|first1=Jen |last1=Christensen |first2=Michael |last2=Nedelman
|newspaper=CNN
|date=November 23, 2018
|accessdate=November 23, 2018 }}&lt;/ref&gt; confirmed that along the U.S. western coast, landfalling atmospheric rivers &quot;account for 30%–40% of precipitation and snowpack. These landfalling atmospheric rivers &quot;are associated with severe flooding events in California and other western states.&quot;&lt;ref name=Dettinger/&gt;&lt;ref name=&quot;grl06&quot; /&gt;&lt;ref name=&quot;NCA2018_Chap2&quot;&gt;{{ cite book
|series=National Climate Assessment (NCA)
|title=Chapter 2: Our Changing Climate
|date=November 23, 2018
|url=https://nca2018.globalchange.gov/chapter/2/
|publisher=USGCRP
|accessdate=November 23, 2018
|location=Washington, DC }}&lt;/ref&gt;

&quot;As the world warms, the &quot;landfalling atmospheric rivers on the West Coast are likely to increase&quot; in &quot;frequency and severity&quot; because of &quot;increasing evaporation and higher atmospheric water vapor levels in the atmosphere.&quot;&lt;ref name=&quot;CNN_Christensen_2018&quot;/&gt;&lt;ref name=&quot;NCA_IV_Wehner_2017&quot;&gt;{{ cite report
|last1=Wehner |first1=M. F. |first2=J. R. |last2=Arnold |first3=T. |last3=Knutson |first4=K. E. |last4=Kunkel |first5=A. N. |last5=LeGrande
|date=2017
|title=Droughts, Floods, and Wildfires
|series=Climate Science Special Report: Fourth National Climate Assessment
|volume=1
|editor-last1=Wuebbles  |editor-first1=D. J. |editor-first2=D. W. |editor-last2=Fahey |editor-first3=K. A. |editor-last3=Hibbard |editor-first4=D. J. |editor-last4=Dokken |editor-first5=B. C. |editor-last5=Stewart |editor-first6=T. K. |editor-last6=Maycock
|publisher=U.S. Global Change Research Program
|location=Washington, DC
|pages=231–256
|doi=10.7930/J0CJ8BNN }}&lt;/ref&gt;&lt;ref&gt;Dettinger, M., 2011: Climate change, atmospheric rivers, and floods in California–a multimodel analysis of storm frequency and magnitude changes. Journal of the American Water Resources Association, 47 (3), 514–523. doi:10.1111/j.1752-1688.2011.00546.x.&lt;/ref&gt;&lt;ref&gt;Warner, M. D., C. F. Mass, and E. P. Salathé Jr., 2015: Changes in winter atmospheric rivers along the North American West Coast in CMIP5 climate models. Journal of Hydrometeorology, 16 (1), 118–128. doi:10.1175/JHM-D-14-0080.1.&lt;/ref&gt;&lt;ref&gt;Gao, Y., J. Lu, L. R. Leung, Q. Yang, S. Hagos, and Y. Qian, 2015: Dynamical and thermodynamical modulations on future changes of landfalling atmospheric rivers over western North America. Geophysical Research Letters, 42 (17), 7179–7186. doi:10.1002/2015GL065435.&lt;/ref&gt;

Landfalling ARs were &quot;responsible for nearly all the annual peak daily flow (APDF)s in western Washington&quot; from 1998 through 2009.&lt;ref name=&quot;AMS_Neiman_2011&quot;&gt;{{ cite journal
|last1=Neiman |first1=Paul. J. |first2=L. J. |last2=Schick |first3=F. M. |last3=Ralph |first4=M. |last4=Hughes |first5=G. A. |last5=Wick
|date=December 2011
|title=Flooding in western Washington: The connection to atmospheric rivers
|journal=American Meteorological Society (AMS)
|volume=12
|issue=6
|pages=1337–1358
|doi=10.1175/2011JHM1358.1 }}&lt;/ref&gt;

This AR in the image on the left brought a &quot;stunning&quot; end to the American West's 5-year drought with &quot;some parts of California received nearly twice as much rain in a single deluge as normally falls in the preceding 5 months (October–February)&quot;.&lt;ref name=&quot;NCA4_Vol1_Wuebbles_2017_full&quot;&gt;{{ cite report
|date=October 2017
|title=Climate Science Special Report (CSSR)
|series=Fourth National Climate Assessment
|volume=1
|editor-last1=Wuebbles  |editor-first1=D. J. |editor-first2=D. W. |editor-last2=Fahey |editor-first3=K. A. |editor-last3=Hibbard |editor-first4=D. J. |editor-last4=Dokken |editor-first5=B. C. |editor-last5=Stewart |editor-first6=T. K. |editor-last6=Maycock
|publisher=U.S. Global Change Research Program
|location=Washington, DC
|url=https://science2017.globalchange.gov/downloads/CSSR2017_FullReport.pdf
|pages=470 |doi=10.7930/J0J964J6 }}&lt;/ref&gt; 
{{clear}}

==Anticyclones==
[[Image:High pressure Area Sep 08 2012.jpg|thumb|250px|right|True color satellite image of an unusual anticyclone off southern Australia in the Southern Hemisphere, on September 8, 2012, showing a counter-clockwise rotation around an oval area of clear skies. Credit: NASA, MODIS Rapid Response System.{{tlx|free media}}]]
[[Image:High Pressure.jpg|left|thumb|300px|Satellite image shows an unusual high-pressure area south of Australia. Credit: NASA, MODIS Rapid Response System.{{tlx|free media}}]]
[[Image:HadleyCross-sec.jpg|thumb|250px|right|Hadley cell circulation tends to create anticyclonic patterns in the Horse latitudes, depositing drier air and contributing to the world's great deserts. Credit: [[w:user:Dwindrim|Dwindrim]].{{tlx|free media}}]]
The Great Red Spot on Jupiter is, in fact, the inverse phenomenon, an anticyclone.&lt;ref name=&quot;HaydPlan&quot;&gt;{{ cite web
 |title = Jupiter's Great Red Spot
 |year = 2009
 |author = Ellen Cohen
 |publisher = Hayden Planetarium
 |accessdate = 2007-11-16
 |url = https://web.archive.org/web/20070808130633/http://haydenplanetarium.org/resources/ava/page/index.php?file=P0413jupispot }}&lt;/ref&gt;

An '''anticyclone''' is a weather phenomenon defined by the United States National Weather Service's glossary as &quot;a large-scale circulation of winds around a central region of high atmospheric pressure, clockwise in the Northern Hemisphere, counterclockwise in the Southern Hemisphere&quot;.&lt;ref&gt;{{ cite web
|author=
|title=Glossary: Anticyclone
|date=
|publisher=National Weather Service
|accessdate=January 19, 2010
|url=https://web.archive.org/web/20110629140523/http://www.nws.noaa.gov/glossary/index.php?word=anticyclone }}&lt;/ref&gt;

'''Def.''' &quot;a system of winds that spiral out from a centre of high pressure&quot;&lt;ref name=AnticycloneWikt&gt;{{ cite book
|author=[[wikt:User:SemperBlotto|SemperBlotto]]
|title=anticyclone
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=15 August 2005
|url=https://en.wiktionary.org/wiki/anticyclone
|accessdate=9 February 2019 }}&lt;/ref&gt; is called an '''anticyclone'''.

&quot;High-pressure weather systems often bring fair weather and relatively clear skies. In early June 2012, a high off the coast of Tasmania did just that...and in spectacular fashion.&quot;&lt;ref name=Voiland&gt;{{ cite book
|author=Adam Voiland
|author2=Patrick Minnis
|author3=Joanna Joiner
|author4=Steve Lang
|author5=Heather Hyre
|title=An Australian “Anti-storm”
|publisher=NASA
|location=Washington, DC USA
|date=June 5, 2012
|url=https://earthobservatory.nasa.gov/images/78208/an-australian-anti-storm
|accessdate=9 February 2019 }}&lt;/ref&gt;

&quot;The Moderate Resolution Imaging Spectroradiometer (MODIS) on NASA’s Aqua satellite acquired this view of a hole in a cloud formation [in the image on the left] at 3:00 p.m. local time (05:00 Universal Time) on June 5, 2012. The weather system over the Great Australian Bight cut out the oval-shaped hole from a blanket of marine stratocumulus clouds.&quot;&lt;ref name=Voiland/&gt;

&quot;The cloud hole, with a diameter that stretched as far as 1,000 kilometers (620 miles) across, was caused by sinking air associated with an area of high pressure near the surface. Globally, the average sea-level pressure is about 1013 millibars; at the center of this high, pressures topped 1,040 millibars.&quot;&lt;ref name=Voiland/&gt;

&quot;Sea-level pressure maps published by the Australian Bureau of Meteorology on June 5 showed that the shape of the cloud hole matched the shape of the high-pressure area. However, the center of high pressure and the cloud hole didn't match precisely; the center of the high was near the western edge of the clear area, about 100 kilometers from the cloud edge.&quot;&lt;ref name=Voiland/&gt;

&quot;In general, winds blow outward and away from areas of high pressure. As a result, areas of high pressure pull air downward. As the air sinks, it also warms, increasing the rate of evaporation and making it difficult for the air to sustain clouds. Areas of low pressure, by contrast, pull air upward and generate clouds and stormy weather.&quot;&lt;ref name=Voiland/&gt;

&quot;While low-pressure systems often produce circular cyclonic storms and clouds, high-pressure systems (which are sometimes called anticyclones) can yield large circular areas of clear skies.&quot;&lt;ref name=Voiland/&gt;

&quot;You could call it an anti-storm.&quot;&lt;ref name=Minnis&gt;{{ cite book
|author=Patrick Minnis
|title=An Australian “Anti-storm”
|publisher=NASA
|location=Washington, DC USA
|date=June 5, 2012
|url=https://earthobservatory.nasa.gov/images/78208/an-australian-anti-storm
|accessdate=9 February 2019 }}&lt;/ref&gt;

&quot;Weather models simulated the cloud formation quite accurately. We checked the Global Modeling and Assimilation Office (GMAO) forecast, and it really nailed the system.&quot;&lt;ref name=Joiner&gt;{{ cite book
|author=Joanna Joiner
|author2=Arlindo da Silva
|title=An Australian “Anti-storm”
|publisher=NASA
|location=Washington, DC USA
|date=June 5, 2012
|url=https://earthobservatory.nasa.gov/images/78208/an-australian-anti-storm
|accessdate=9 February 2019 }}&lt;/ref&gt;

The evolution of an anticyclone depends on a few variables such as its size, intensity, moist-convection, Coriolis force etc.&lt;ref&gt; Masoud Rostami &amp; Vladimir Zeitlin (2017) Influence of condensation and latent heat release upon barotropic and baroclinic instabilities of vortices in a rotating shallow water f-plane model, Geophysical &amp; Astrophysical Fluid Dynamics, 111:1, 1-31, DOI: 10.1080/03091929.2016.1269897  https://doi.org/10.1080/03091929.2016.1269897&lt;/ref&gt;

Surface anticyclones form due to downward motion through the troposphere, in areas within a synoptic flow pattern in higher levels of the troposphere, beneath the western side of troughs, on weather maps, these show converging winds (isotachs), also known as confluence, or converging height lines near or above the level of non-divergence, which is near the 500 hPa pressure surface about midway up the troposphere.&lt;ref&gt;{{ cite web
|title=Glossary of Meteorology
|date=2009
|url=http://archive.wikiwix.com/cache/20110628073639/http://amsglossary.allenpress.com/glossary/search?id=level-of-nondivergence1 |publisher=American Meteorological Society
|accessdate=2009-02-17 }}&lt;/ref&gt;&lt;ref name=Matchev&gt;{{ cite book
|author=Konstantin Matchev
|title=Middle-Latitude Cyclones - II
|url=https://web.archive.org/web/20090225025157/http://www.phys.ufl.edu/~matchev/MET1010/notes/Chapter12b.ppt
|date=2009-02-25
|publisher=University of Florida
|accessdate=2009-02-16 }}&lt;/ref&gt;
{{clear}}

==Cyclogenesis==
[[Image:StormCollage-GOES-13.jpg|thumb|upright=2|right|300px|This collage of GOES 13 satellite images shows the development of a nor'easter over several days. Credit: NASA GOES Project.{{tlx|free media}}]]
Cyclogenesis is the process of cyclone formation and intensification.&lt;ref name=&quot;Arc&quot;&gt;{{ cite web
|author=Nina A. Zaitseva
|year=2006
|publisher=National Snow and Ice Data Center
|title=Cyclogenesis
|accessdate=2006-12-04
|url=https://web.archive.org/web/20060830135741/http://www.nsidc.org/arcticmet/glossary/cyclogenesis.html }}&lt;/ref&gt;

Cyclogenesis is the development or strengthening of cyclonic circulation in the atmosphere (a low-pressure area).&lt;ref&gt;{{ cite web
|author=&lt;nowiki&gt;Arctic Climatology and Meteorology&lt;/nowiki&gt;
|title=Cyclogenesis
|publisher=National Snow and Ice Data Center
|accessdate=2006-12-04
|date=2006
|url=https://web.archive.org/web/20060830135741/http://www.nsidc.org/arcticmet/glossary/cyclogenesis.html }}&lt;/ref&gt;

The anticyclonic equivalent, the process of formation of high pressure systems, dealing with surface systems is anticyclogenesis.&lt;ref name=&quot;CyclogenesisDef&quot;&gt;{{ cite web
| publisher = American Meteorological Society
|work= Glossary of Meteorology
| title= Cyclogenesis
| date = 26 January 2012
| url = http://glossary.ametsoc.org/wiki/Cyclogenesis
| accessdate = 2016-07-23 }}&lt;/ref&gt;

Cyclogenesis is the opposite of cyclolysis, which concerns the weakening of surface cyclones. The term has an anticyclonic equivalent—Anticyclogenesis.&lt;ref name=&quot;CyclogenesisDef&quot;/&gt;
{{clear}}

==Cyclones==
[[Image:Cyclone Catarina from the ISS on March 26 2004.JPG|250px|thumb|right|Meteorology is represented by cyclone Catarina. Credit: NASA.{{tlx|free media}}]]
'''Def.''' a system of winds rotating around a center of low atmospheric pressure, the more or less violent small-scale circulations such as tornadoes, waterspouts, and dust devils is called a '''cyclone'''.

In [[meteorology]], a '''cyclone''' is a large scale air mass that rotates around a strong center of low atmospheric pressure.&lt;ref name=&quot;AMSCcDef&quot;&gt;{{ cite web
|title = Cyclonic  circulation
|author = Glossary of Meteorology
| date=June 2000
|publisher=American Meteorological Society
|url = http://glossary.ametsoc.org/wiki/Cyclonic_circulation|accessdate = 2008-09-17 }}&lt;/ref&gt;&lt;ref name=&quot;AMSCycDef&quot;&gt;{{ cite web
|title = Cyclone
|date=June 2000
|author= Glossary of Meteorology
|publisher= American Meteorological Society
|url = http://glossary.ametsoc.org/wiki/Cyclone
|accessdate = 2008-09-17 }}&lt;/ref&gt;

Cyclones are characterized by inward spiraling winds that rotate about a zone of low-pressure.&lt;ref name=&quot;BBCCycDef&quot;&gt;{{ cite web
|author = BBC Weather Glossary
|title = Cyclone
|date= July 2006
|publisher= British Broadcasting Corporation
|accessdate = 2006-10-24
|url = https://web.archive.org/web/20060829214837/http://www.bbc.co.uk/weather/weatherwise/glossary/c.shtml }}&lt;/ref&gt;&lt;ref name=&quot;UCARCycDef&quot;&gt;{{ cite web
|title = UCAR Glossary&amp;nbsp;— Cyclone
|publisher= University Corporation for Atmospheric Research
|url = http://meted.ucar.edu/satmet/goeschan/glossary.htm#c
|accessdate = 2006-10-24 }}&lt;/ref&gt;

The largest low-pressure systems are polar vortices and extratropical cyclones of the largest scale (the synoptic scale), which includes warm-core cyclones such as tropical cyclones and subtropical cyclones.&lt;ref&gt;National Hurricane Center (2012). [http://www.nhc.noaa.gov/aboutgloss.shtml Glossary of NHC terms.] Retrieved on 2012-08-13.&lt;/ref&gt;

Mesocyclones, tornadoes and dust devils lie within the smaller mesoscale.&lt;ref&gt;{{ cite journal
|author=I. Orlanski
|year=1975
|title=A rational subdivision of scales for atmospheric processes
|journal=Bulletin of the American Meteorological Society
|volume=56
|pages=527–530
|issue=5
|bibcode=1975BAMS...56..527
|doi=10.1175/1520-0477-56.5.527 }}&lt;/ref&gt;

Upper level cyclones can exist without the presence of a surface low, and can pinch off from the base of the tropical upper tropospheric trough during the summer months in the Northern Hemisphere.

Cyclones have also been seen on extraterrestrial planets, such as Mars and [[Neptune]].&lt;ref name=&quot;Brand&quot;&gt;{{ cite web
|author=David Brand
|title=Colossal cyclone swirling near Martian north pole is observed by Cornell-led team on Hubble telescope
|accessdate=2008-06-15
|date=1999-05-19
|publisher=Cornell University
|url = https://web.archive.org/web/20070613133949/http://www.news.cornell.edu/releases/May99/mars.cyclone.deb.html }}&lt;/ref&gt;&lt;ref name=&quot;WIZ&quot;&gt;{{ cite web
|publisher=NASA
|author=Samantha Harvey
|date=2006-10-02
|title=Historic Hurricanes
|accessdate=2008-06-14
|url=https://web.archive.org/web/20080415120400/http://solarsystem.nasa.gov/educ/themes/display.cfm?Item=hurricane }}&lt;/ref&gt;
{{clear}}

==Waterspouts==
[[Image:Trombe.jpg|thumb|right|250px|A waterspout near Florida has two flares with smoke trails near the bottom of the photograph for indicating wind direction and general speed. Credit: Dr. Joseph Golden, NOAA.{{tlx|free media}}]]
[[Image:Giant Waterspout Filmed by RAF Search and Rescue Crew MOD 45152038.jpg|thumb|left|250px|Waterspout filmed off Anglesey, Wales, on 12 November 2010 by an RAF Search and Rescue crew. Credit: RAF/MoD.{{tlx|free media}}]]
A waterspout is an intense columnar vortex (usually appearing as a funnel-shaped cloud) that occurs over a body of water.&lt;ref name=Burt&gt;{{Cite book
|title=Extreme weather : a guide &amp; record book
|last=Burt
|first=Christopher
|date=2004
|publisher=W.W. Norton
|others=Cartography by Stroud, Mark.
|isbn=978-0393326581
|edition=1st
|location=New York
|oclc=55671731
}}&lt;/ref&gt; Some are connected to a cumulus congestus cloud, some to a cumuliform cloud and some to a cumulonimbus cloud.&lt;ref name=Glossary&gt;{{cite web
|url= http://www.geographic.org/climate/w.html#waterspout
|title=Waterspout definition
|work=A Comprehensive Glossary Of Weather
|publisher=Geographic.org
|accessdate=2014-07-10
}}&lt;/ref&gt; In the common form, it is a non-supercell tornado over water.&lt;ref name=Glossary1&gt;{{cite web
|url= http://www.geographic.org/climate/w.html#waterspout
|title=Waterspout definition
|work=A Comprehensive Glossary Of Weather
|publisher=Geographic.org
|accessdate=2014-07-10
}}&lt;/ref&gt;&lt;ref&gt;[http://www.wcvb.com/video/the-weather-channel/what-is-a-waterspout/40714452 What Is a Waterspout?] (Weather Channel video)&lt;/ref&gt;&lt;ref&gt;[http://www.chron.com/neighborhood/bayarea/article/Waterspout-comes-ashore-in-Galveston-8382923.php Waterspout comes ashore in Galveston] by Jessica Hamilton, Houston Chronicle, July 17, 2016&lt;/ref&gt;

While it is often weaker than most of its land counterparts, stronger versions spawned by mesocyclones do occur.&lt;ref&gt;{{Cite web
|url        = http://www.answers.com/topic/waterspout
|title      = Waterspout
|editor     = Answer.com
|work       = McGraw-Hill Encyclopedia of Science and Technology
|accessdate = 6 December 2010}}&lt;/ref&gt;&lt;ref&gt;{{Cite web
|url        = http://www.islandnet.com/~see/weather/almanac/arc2002/alm02oct.htm
|title      = Water Twisters
|editor     = Islandnet.com
|author     = Keith C. Heidorn
|work       = The Weather Doctor Almanach
|accessdate = 6 December 2010}}&lt;/ref&gt; Most waterspouts do not suck up water; they are small and weak rotating columns of air over water.&lt;ref name=Glossary1/&gt;&lt;ref&gt;{{cite journal
|last=Schwiesow
|first=R.L.
|author2=Cupp, R.E.
|author3=Sinclair, P.C.
|author4=Abbey, R.F.
|title=Waterspout Velocity Measurements by Airborne Doppler Lidar
|journal=Journal of Applied Meteorology
|date=April 1981
|volume=20
|issue=4
|pages=341–348
|bibcode = 1981JApMe..20..341S
|doi = 10.1175/1520-0450(1981)020&lt;0341:WVMBAD&gt;2.0.CO;2
}}&lt;/ref&gt;

While waterspouts form mostly in the tropics and subtropical areas,&lt;ref name=Glossary1/&gt; other areas also report waterspouts, including Europe, Australia, New Zealand, the Great Lakes, Antarctica&lt;ref name=&quot;lake_michigan&quot;&gt;{{cite news
|publisher=BBC News
|title=Several waterspouts filmed on Lake Michigan in US
| url=https://www.bbc.co.uk/news/world-us-canada-19315824
| date=20 August 2012
|accessdate=20 August 2012
}}&lt;/ref&gt;&lt;ref&gt;{{cite web
|last=Taylor
|first=Stanley
|title=Antarctic Diary
|url=http://antarcticdiary.wordpress.com/part-4/
|accessdate=4 June 2013
|date=August 2011
}}&lt;/ref&gt; and on rare occasions, the Great Salt Lake.&lt;ref name=&quot;A Great Salt Lake Waterspout&quot;&gt;publisher=journals.ametsoc.org| url=journals.ametsoc.org/doi/pdf/10.1175/1520-0493-119-12-2740.1|author=J Simpson |date =1991&lt;/ref&gt; Some are also found on the East Coast of the United States, and the coast of California.&lt;ref name=Burt/&gt;
{{clear}}

==Dust devils==
[[Image:dust devil.jpg|right|thumb|300px|A dust devil occurs in Arizona. Credit: NASA.{{tlx|free media}}]]
[[Image:Dust devil krakow.jpg|250px|right|thumb|A dust devil occurs in Cracow, Poland. Credit: [[c:user:KHRoN|KHRoN]].{{tlx|free media}}]]
[[Image:Iraqi Dust Devil.jpg|thumb|250px|right|A dust devil occurs in Ramadi, Iraq. Credit: [[w:user:Ultratone85|Ultratone85]].{{tlx|free media}}]]
[[Image:Remolino (tourbillon de sable).ogv|thumb|left|250px|A large dust devil occurs in Colonia Omega, Saltillo, Coahuila, Mexico. Credit: [[c:user:Dupondt|Dupondt]].{{tlx|free media}}]]
A dust devil is a strong, well-formed, and relatively long-lived whirlwind, ranging from small (half a metre wide and a few metres tall) to large (more than 10&amp;nbsp;metres wide and more than 1000&amp;nbsp;metres tall). The primary vertical motion is upward. Dust devils are usually harmless, but can on rare occasions grow large enough to pose a threat to both people and property.&lt;ref name=&quot;Glossary2000&quot;&gt;{{cite book
|title= Glossary of Meteorology
|publisher=American Meteorological Society
|year= 2000
|url= http://amsglossary.allenpress.com/glossary/search?id=dust-devil1
|isbn= 978-1-878220-34-9
|archiveurl= https://web.archive.org/web/20090130003357/http://amsglossary.allenpress.com/glossary/search?id=dust-devil1
|archivedate= 2009-01-30
|df=
}}&lt;/ref&gt;

Dust devils form when a pocket of hot air near the surface rises quickly through  cooler air above it, forming an updraft. If conditions are just right, the updraft may begin to rotate. As the air rapidly rises, the column of hot air is stretched vertically, thereby moving mass closer to the axis of rotation, which causes intensification of the spinning effect by conservation of angular momentum. The secondary flow in the dust devil causes other hot air to speed horizontally inward to the bottom of the newly forming vortex. As more hot air rushes in toward the developing vortex to replace the air that is rising, the spinning effect becomes further intensified and self-sustaining. A dust devil, fully formed, is a funnel-like chimney through which hot air moves, both upwards and in a circle. As the hot air rises, it cools, loses its buoyancy and eventually ceases to rise.  As it rises, it displaces air which descends outside the core of the vortex. This cool air returning acts as a balance against the spinning hot-air outer wall and keeps the system stable.&lt;ref&gt;{{cite book
|author = Ludlum, David M.
|year = 1997
|title = National Audubon Society Field Guide to North American Weather
|publisher = Knopf
|isbn = 978-0-679-40851-2
}}&lt;/ref&gt;

As available hot air near the surface is channeled up the dust devil, eventually surrounding cooler air will be sucked in. Once this occurs, the effect is dramatic, and the dust devil dissipates in seconds. Usually this occurs when the dust devil is not moving fast enough (depletion) or begins to enter a terrain where the surface temperatures are cooler.&lt;ref name=death-valley&gt;[http://www.death-valley.us/article559.html http://www.death-valley.us/article559.html] {{dead link|date=September 2010|url=http://www.death-valley.us/article559.html}}&lt;/ref&gt;

On rare occasions, a dust devil can grow very large and intense, sometimes reaching a diameter of up to 300&amp;nbsp;feet (90&amp;nbsp;m) with winds in excess of 60&amp;nbsp;mph (100&amp;nbsp;km/h+) and can last for upwards of 20&amp;nbsp;minutes before dissipating.&lt;ref&gt;{{cite web | publisher=Arizona Vacation Planner | title=Dust Devils: Ephemeral Whirlwinds Can Stir Up Trouble | url=http://www.arizona-vacation-planner.com/dust-devils.html | archive-url=https://archive.is/20120718220124/http://www.arizona-vacation-planner.com/dust-devils.html | archive-date=2012-07-18 | accessdate=2007-10-05}} &lt;/ref&gt;

Dust devils typically do not cause injuries, but rare, severe dust devils have caused damage and even deaths in the past. One such dust devil struck the Coconino County, Arizona, Fairgrounds in Flagstaff, Arizona, on September 14, 2000, causing extensive damage to several temporary tents, stands and booths well as some permanent fairgrounds structures. Several injuries were reported, but there were no fatalities. Based on the degree of damage left behind, it is estimated that the dust devil produced winds as high as 75&amp;nbsp;mph (120&amp;nbsp;km/h), which is equivalent to an Enhanced Fujita Scale (EF-0) tornado.&lt;ref&gt;{{cite web
| publisher=National Weather Service-Flagstaff, AZ
| title=Damage From a Dust Devil at the Coconino County Fairgrounds - September 14, 2000
| url=http://www.wrh.noaa.gov/fgz/past/cocodust/coco_fair.php?wfo=fgz
| accessdate=2007-10-05
}}&lt;/ref&gt; On May 19, 2003, a dust devil lifted the roof off a two-story building in Lebanon, Maine, causing it to collapse and kill a man inside.&lt;ref&gt;{{ cite web
|url=https://web.archive.org/web/20090129192229/http://www4.ncdc.noaa.gov/cgi-win/wwcgi.dll?wwevent~ShowEvent~499035 |date=2009-01-29 |title=National Climatic Data Center |accessdate=2008-06-05.&lt;/ref&gt;&lt;ref&gt;{{cite news
| url=https://www.nytimes.com/2003/05/21/us/national-briefing-new-england-maine-man-dies-in-windstorm.html
| work=The New York Times
| title=Man Dies In Windstorm
| date=May 21, 2003
| accessdate=May 1, 2010
}}&lt;/ref&gt; In East El Paso, Texas in 2010, three children in an inflatable jump house were picked up by a dust devil and lifted over 10 feet (3&amp;nbsp;m), traveling over a fence and landing in a backyard three houses away.&lt;ref&gt;This rare weather incident was the subject of a United States Air Force Weather Squadron study: Clarence Giles, &quot;Air Force Weather Squadron forecasts, studies weather to keep servicemembers safe&quot;, http://fortblissbugle.com/air-force-weather-squadron-forecasts-studies-weather-to-keep-servicemembers-safe/ ''Fort Bliss Bugle'', Unit News p.1A (January 12, 2011)&lt;/ref&gt; In Commerce City, Colorado in 2018, a powerful dust devil hurtled two porta-potties into the air. No one was injured in the incident.&lt;ref&gt;{{cite web
|last1=Lane
|first1=Damon
|title=Colorado Dust Devil Tosses Porta-Potties
|url=http://texasstormwatch.com/2018/06/colorado-dust-devil-tosses-porta-potties.html
|website=Texas Storm Watch
|accessdate=16 June 2018
}}&lt;/ref&gt; In 2019 a large dust devil in Yucheng county, Henan province, China killed 2 children and injured 18 children and 2 adults when a bouncy castle was lifted into the air.&lt;ref&gt;[https://www.scmp.com/news/china/society/article/3004174/two-children-killed-when-bouncy-castle-swept-air-dust-devil Two children killed after bouncy castle is swept into air by ‘dust devil’ in central China], ''South China Morning Post'', April 1, 2019&lt;/ref&gt;

Dust devils have been implicated in around 100 aircraft accidents.&lt;ref&gt;{{cite journal
|last=Lorenz
|first=Ralph
|title=Dust Devil Hazard to Aviation: A Review of US Air Accident Reports,
|journal=Journal of Meteorology
|year=2005
|volume=28
|issue=298
|pages=178–184
|url=http://www.lpl.arizona.edu/~rlorenz/dustdevilaviation.pdf
|accessdate=17 September 2012
}}&lt;/ref&gt;  While many incidents have been simple taxiing problems, a few have had fatal consequences. Dust devils are also considered major hazards among skydivers and paragliding pilots as they can cause a parachute or a glider to collapse with little to no warning, at altitudes considered too low to cut away, and contribute to the serious injury or death of parachutists.&lt;ref&gt;{{cite web
| publisher=United States Parachute Association
| title=Dust Devils - July 9, 2012
| url=http://parachutistonline.com/safety_training/safety_check/dust-devils
| access-date=2014-08-12
| archive-url=https://web.archive.org/web/20170917124209/http://parachutistonline.com/safety_training/safety_check/dust-devils
| archive-date=2017-09-17 }}&lt;/ref&gt;&lt;ref&gt;{{cite news
|title=Skydiving instructor Tony Rokov killed in accident at Goulburn airport
|url=http://www.smh.com.au/nsw/skydiving-instructor-tony-rokov-killed-in-accident-at-goulburn-airport-20151121-gl4raf.html
|work=Sydney Morning Herald
|date=22 November 2015
|accessdate=22 November 2015}}&lt;/ref&gt;&lt;ref&gt;{{cite news
|url=https://www.smh.com.au/national/nsw/paraglider-landed-180km-away-after-being-thrown-off-cliff-by-dust-devil-20190103-p50pi9.html
|title=Paraglider landed 180km away after being thrown off cliff by dust devil
|work=Sydney Morning Herald
|date=3 January 2019
|accessdate=3 January 2019 }}&lt;/ref&gt;

Dust devils, even small ones (on Earth), can produce radio noise and electrical fields greater than 10,000 volts per meter.&lt;ref&gt;{{cite press release
| publisher=University of California, Berkeley
| date=29 May 2002
| title= Stalking Arizona dust devils helps scientists understand electrical, atmospheric effects of dust storms on Mars
| url=http://www.berkeley.edu/news/media/releases/2002/05/29_dust.html
| accessdate=2006-12-01 }}&lt;/ref&gt; A dust devil picks up small dirt and dust particles. As the particles whirl around, they bump and scrape into each other and become electrically charged. The whirling charged particles also create a magnetic field that fluctuates between 3 and 30 times each second.&lt;ref&gt;{{cite conference
|author1=Koch, J.
|author2=N.O. Renno
| title=Convective-radiative feedback mechanisms by dusty convective plumes and vortices, In: ''Fall meeting of the American Geophysical Union''
| date=December 5–9, 2005 }}&lt;/ref&gt;

A large dust devil measuring about 100&amp;nbsp;metres (330&amp;nbsp;ft) across at its base can lift about 15 metric tonnes (17 short tons) of dust into the air in 30&amp;nbsp;minutes. Giant dust storms that sweep across the world's deserts contribute 8% of the mineral dust in the atmosphere each year during the handful of storms that occur. In comparison, the significantly smaller dust devils that twist across the deserts during the summer lift about three times as much dust, thus having a greater combined impact on the dust content of the atmosphere. When this occurs, they are often called '''sand pillars'''.&lt;ref&gt;{{cite journal|last=Kok|first=J.F.|author2=Renno, N.O.|year=2006|title=Enhancement of the emission of mineral dust aerosols by electric forces|url=https://deepblue.lib.umich.edu/bitstream/2027.42/95661/1/grl21575.pdf|journal=Geophysical Research Letters|volume=33|issue=Aug. 28|pages=L19S10|bibcode=2006GeoRL..3319S10K|doi=10.1029/2006GL026284}}&lt;!--| accessdate = 2006-12-01 --&gt;&lt;/ref&gt;
{{clear}}

==Snow whirlwinds==
[[Image:Tourbillon de neige.png|thumb|right|250px|Snow whirlwind, similar to a dust devil, is seen on Mount Royal in Montreal, Quebec, Canada. Credit: [[c:user:Pierre cb|Pierre cb]].{{tlx|free media}}]]
[[Image:Winter waterspout.jpg|thumb|left|250px|A large winter waterspout over Lake Ontario, just off the shore of Whitby, Ontario on 26 January 1994. Credit: ERH at NOAA.{{tlx|fairuse}}]]
The same conditions can produce a snow whirlwind, although differential heating is more difficult in snow-covered areas.

A ''winter waterspout'', also known as a ''snow devil'', an ''icespout'', an ''ice devil'', a ''snownado'', or a ''snowspout'', is an extremely rare instance of a waterspout forming under the base of a snow squall.&lt;ref&gt;{{cite web
|url= http://www.acsu.buffalo.edu/~insrisg/nature/nw03/0414waterspouts.htm
|title= Waterspouts
|date= 14 April 2003
|author= The Buffalo News
|publisher= State University of New York in Buffalo
|accessdate=21 July 2008 }}&lt;/ref&gt;&lt;ref&gt;{{cite web
 |url         = http://www.weather.com/glossary/s.html
 |title       = Snow Devil
 |accessdate  = 21 July 2008
 |year        = 2008
 |work        = Glossary
 |author      = The Weather Channel
  |archiveurl  = https://web.archive.org/web/20080801101119/http://www.weather.com/glossary/s.html
 |archivedate = 1 August 2008
 |df          = dmy-all }}&lt;/ref&gt;

Like the more efficient lake-effect snow events, winds focusing down the axis of long lakes enhance wind convergence and likely enhance their development.&lt;ref&gt;{{cite web
|url=http://www.erh.noaa.gov/btv/events/15Jan2009/overview.shtml
|title=15 January 2009: Lake Champlain Sea Smoke, Steam Devils, and Waterspout: Chapters IV and V
|author=&lt;nowiki&gt;National Weather Service, Forecast Office, Burlington, Vermont&lt;/nowiki&gt;
|publisher=Eastern Region Headquarters
|date=3 February 2009
|accessdate=21 June 2009
}}&lt;/ref&gt;
{{clear}}

==Coal devils==
[[Image:Superb coal devil in Mongolia - 1.JPG|thumb|Coal devil occurs in Mongolia. Credit: [[c:user:Texasbob|Texasbob]].{{tlx|free media}}]]
Coal devils are common at the coal town of Tsagaan Khad in Ömnögovi Province, South Gobi Province, Mongolia. They occur when dust devils pick up large amounts of stockpiled coal. Their dark color makes them resemble some tornados.

In the image on the right, a coal devil occurs at the coal storage town of Tsagaan Khad, Mongolia (15km north from Mongolia-China border). On the way from Oyu Tolgoi mine to the Gashuunsukhait-Ganqimaodao border crossing.
{{clear}}

==Fire whirls==
[[Image:Fire whirl (FWS) crop.jpg|right|thumb|300px|A fire whirl has flames in the vortex. Credit: U.S. Fish and Wildlife Service.{{tlx|free media}}]]
A fire whirl or swirl, sometimes called fire devils or fire tornadoes, can be seen during intense fires in combustible building structures or, more commonly, in forest or bush fires. A fire whirl is a vortex-shaped formation of burning gases being released from the combustible material. The genesis of the vortex is probably similar to that of a dust devil. As distinct from the dust devil, it is improbable that the height reached by the fire gas vortex is greater than the visible height of the vertical flames because of turbulence in the surrounding gases that inhibit creation of a stable boundary layer between the rotating/rising gases relative to the surrounding gases.&lt;ref&gt;{{ cite web
|title=WILDFIRE MODELING, IR OBSERVATIONS AND ANALYSIS
|url=https://web.archive.org/web/20070327183634/http://jfsp.nifc.gov/conferenceproc/GISRS-12-AORadkeetal.pdf |date=2007-03-27 }}&lt;/ref&gt;

Fire whirls are sometimes colloquially called fire tornadoes, but are not usually classifiable as tornadoes as the vortex in most cases does not extend from the surface to cloud base. Also, even in such cases, those fire whirls very rarely are classic tornadoes, as their vorticity derives from surface winds and heat-induced lifting, rather than from a tornadic mesocyclone aloft, although a handful of suspected cases of the latter are known.&lt;ref name=&quot;McRae&quot;&gt;{{cite journal
|last = McRae
|first = Richard H. D.
|author2 = J. J. Sharples
|author3 = S. R. Wilkes
|author4 = A. Walker
|title = An Australian pyro-tornadogenesis event
|journal = Nat. Hazards
|volume = 65
|issue = 3
|pages = 1801–1811
|date = 2013
|doi = 10.1007/s11069-012-0443-7
}}&lt;/ref&gt;
{{clear}}

==Ash devils==
[[Image:Lages Wildfire in White Pine County, Nevada.JPG|thumb|right|250px|An ash devil results from the fire in the Schell Creek and Antelope Mountain ranges. Credit: [[c:user:Jrmichae|Jrmichae]].{{tlx|free media}}]]
Ash devils form similar to dust devils and are often seen on unstable days in burn scar areas of recent fires.

Hot cinders underneath freshly deposited ash in recently burned areas may sometimes generate numerous dust devils. The lighter weight and the darker color of the ash may create dust devils that are visible hundreds of feet into the air.
{{clear}}

==Steam devils==
[[Image:Lyons and Pease steam devils and steam fog.jpg|thumb|right|350px|Steam devils on Lake Michigan 31 January 1971, from the paper which first named and reported the phenomenon. Credit: W.A. Lyons and S.R. Pease.{{tlx|fairuse}}]]
Steam devils are phenomena often observed in the steam rising from power plants.&lt;ref name=&quot;Handy Weather Answer Book&quot;&gt;{{cite book
|last=Lyons
|first=Walter A.
|title=The Handy Weather Answer Book
|location=Detroit, MI
|publisher=Visible Ink Press
|year=1997
|isbn=0-7876-1034-8
}}&lt;/ref&gt;

Include steam devils in the International Field Year for the Great Lakes which was imminently to occur in 1972-3.&lt;ref&gt;Barrick, p.213&lt;br /&gt;Holle (2007), p.9&lt;br /&gt;Lyons &amp; Pease, pp.235, 237&lt;/ref&gt;

Steam devils are a rare and short-lived phenomenon, typically surviving no more than three or four minutes, and the smaller ones over hot springs dissipating in a matter of seconds.&lt;ref&gt;Barrick, p.213&lt;br /&gt;Bluestein, p.151&lt;br /&gt;Holle (2007), p.9&lt;br /&gt;Lyons &amp; Pease, pp.236-237&lt;br /&gt;Zurn-Birkhimer ''et al.'', p.2431&lt;/ref&gt;&lt;ref name=Bluestein&gt;Bluestein differs from other sources in almost every metric describing steam devils, so much so that he might almost be describing a different phenomenon. Bluestein gives the diameter as {{nowrap|3 feet (1 m)}}; Lyons and Pease have 50 to {{nowrap|200 m}}. Bluestein has the height as up to 20 feet, Lyons and Pease have 1,500 feet.  Bluestein states the minimum necessary termperature difference between air and water to be 68°F; Lyons and Pease give a counter-example of 39°F.  Bluestein states there is usually a clear sky; MacDougal and Lyons and Pease both provide photographs with cumulus cloud above.  Barrick gives small dimensions comparable to Bluestein, but only in relation to steam devils over geyser basins.&lt;/ref&gt;

Steam devils can become detached from their base and be blown downstream by the wind. On small bodies of water such as hot springs this can mean that the steam devil ends up over land away from the water altogether. Such steam devils continue to rotate even after they have become detached from the source of heat, but will soon dissipate.&lt;ref&gt;Holle (2007), p.9&lt;/ref&gt;

Very small steam devils may have a poorly defined column and no identifiable clear inner core. Such vortices are more properly called steam whirls by analogy with the dust whirls of land.&lt;ref&gt;Holle (1977), p.931&lt;/ref&gt;
{{clear}}

==Extratropical cyclones==
[[Image:Low pressure system over Iceland.jpg|thumb|250px|An extratropical cyclone is near Iceland on September 4, 2003. Credit: NASA’s Aqua/MODIS satellite.{{tlx|free media}}]]
A beautifully-formed low-pressure system swirls off the southwestern coast of Iceland, illustrating the maxim that &quot;nature abhors a vacuum.&quot; The vacuum in this case would be a region of low atmospheric pressure. In order to fill this void, air from a nearby high-pressure system moves in, in this case bringing clouds along for the ride. And because this low-pressure system occurred in the Northern Hemisphere, the winds spun in toward the center of the low-pressure system in a counter-clockwise direction; a phenomenon known as the Coriolis force (in the Southern Hemisphere, the Coriolis force would be manifested in a clockwise direction of movement).

The clouds in the image resembled pulled cotton and lace as they spun in a lazy hurricane-like pattern. This huge system swirled over the Denmark Strait in between Greenland and Iceland.

The process in which an extratropical cyclone undergoes a rapid drop in atmospheric pressure (24 millibars or more) in a 24-hour period is referred to as explosive cyclogenesis, and is usually present during the formation of a nor'easter.&lt;ref&gt;{{ cite web
|title= Synoptic-dynamic climatology of the &quot;Bomb&quot;
|url=http://www.meteo.mcgill.ca/atoc541/index_files/sandersgyakum1980.pdf
|author1=Sanders, F. |author2=J. R. Gyakum
|publisher=Massachusetts Institute of Technology, Cambridge
|accessdate=2012-01-21
|date=1980-06-12 }}&lt;/ref&gt;
{{clear}}

==Heat domes==
[[Image:Heat Wave.jpg|right|thumb|300px|A heat dome (&quot;H&quot; for &quot;HIGH&quot; pressure) occurs over the United States. Credit: U. S. National Weather Service/National Ocean Service.{{tlx|free media}}]]
[[Image:500-mb pressure chart 2021-06-28 700EST Heat dome Pacific NW.png|left|thumb|300px|The heat dome of the 2021 Western North America heat wave, over west Canada and the Northwest United States is shown. Credit: National Oceanic and Atmospheric Administration.{{tlx|free media}}]]
A '''heat dome''' is caused when atmosphere traps hot ocean air, as if bounded by a lid or cap.&lt;ref name=&quot;NOAH&quot;&gt;{{ cite web
|date=June 30, 2021
|title=What is a heat dome?
|url=https://oceanservice.noaa.gov/facts/heat-dome.html
|website=National Oceanic and Atmospheric Administration }}&lt;/ref&gt; They can be linked to climate change.&lt;ref name=&quot;accuweather.com&quot;&gt;{{ cite web
|last=Rosenthal |first=Zachary
|date=July 1, 2021
|title=Extreme heat
|website=AccuWeather
|url=https://www.accuweather.com/en/weather-news/what-is-a-heat-dome/971124 }}&lt;/ref&gt; The upper air weather patterns are slow to move, referred to as an Omega block.&lt;ref name=&quot;Science Alert&quot;&gt;{{ cite web |last=Freedman |first=Andrew
|date=July 25, 2019
|title=A Giant 'Heat Dome' Over Europe Is Smashing Temperature Records, And It's on The Move
|url=https://www.sciencealert.com/in-europe-a-historic-heat-wave-is-shattering-records-with-ease }}&lt;/ref&gt;

The air is compressed, and as its net heat is now in a smaller volume, it gets hotter. As the warm air attempts to rise, the high pressure above it forces it down, to get hotter, and its pressure grows higher.&lt;ref name=&quot;accuweather.com&quot;/&gt;

The high pressure acts as if a dome is causing everything below it to get hotter and hotter.&lt;ref name=&quot;World Economic Forum&quot;&gt;{{ cite web
|last=Fleming |first=Sean
|date= June 29, 2021
|title=What is the North American heat dome and how dangerous is it?
|url=https://www.weforum.org/agenda/2021/06/north-american-heat-dome-dangerous/ }}&lt;/ref&gt;
{{clear}}

==Jet streams==
[[Image:Straalstroom.jpg|thumb|250px|right|Clouds are shown along a jet stream over Canada. Credit: NASA.{{tlx|free media}}]]
'''Def.''' any of the high-speed, high-altitude air currents that circle the Earth in a westerly direction is called a '''jet stream'''.

'''Jet streams''' are fast flowing, narrow air currents found in the atmospheres of some planets, including [[Earth]]. The main jet streams are located near the tropopause, the transition between the troposphere (where temperature decreases with altitude) and the stratosphere (where temperature increases with altitude).&lt;ref name=USDOE&gt;{{ cite book
| author=United States Department of Energy
| date=26 June 2002
| url=http://www.newton.dep.anl.gov/askasci/wea00/wea00135.htm
| title=Ask a Scientist
| accessdate=5 May 2008 }}&lt;/ref&gt; The major jet streams on Earth are westerly winds (flowing west to east). Their paths typically have a meandering shape; jet streams may start, stop, split into two or more parts, combine into one stream, or flow in various directions including the opposite direction of most of the jet. The strongest jet streams are the '''polar jets''', at around {{convert|7|-|12|km|ft|abbr=on}} above sea level, and the higher and somewhat weaker '''subtropical jets''' at around {{convert|10|-|16|km|ft|abbr=on}}. The Northern Hemisphere and the Southern Hemisphere each have both a polar jet and a subtropical jet. The northern hemisphere polar jet flows over the middle to northern latitudes of North America, Europe, and Asia and their intervening oceans, while the southern hemisphere polar jet mostly circles Antarctica all year round.
{{clear}}

==Mesocyclones==
[[Image:Greensburg3 small.gif|right|thumb|250px|Storm relative motion of a tornado-producing mesocyclone over Greensburg, Kansas on May 4, 2007. The storm was producing an EF5 tornado at the time of the image. Credit: [[c:user:Spoladore|Pedro Spoladore]].{{tlx|free media}}]]
[[Image:Wall cloud with lightning - NOAA.jpg|thumb|left|Mesocyclones are sometimes visually identifiable by a rotating wall cloud like the one in this thunderstorm over Texas. Credit: Brad Smull, NOAA Photo Library, NOAA Central Library; OAR/ERL/National Severe Storms Laboratory (NSSL).{{tlx|free media}}]]
[[Image:Radar-algorithme eng.gif|thumb|Mesocyclone detection algorithm output on tornadic cells in Northern Michigan on July 3rd, 1999. Credit: Greg Stumpf, Pat Burke, Christina Hannon and Valerie McCoy of NSSL.{{tlx|free media}}]]
A mesocyclone is a vortex of air within a convective storm.&lt;ref name=&quot;MesocyloneDef&quot;&gt;{{cite web
| publisher = American Meteorological Society
| author = Glossary of Meteorology
| title = Mesocyclone
| date = June 2000
| url = http://amsglossary.allenpress.com/glossary/search?id=mesocyclone1
| accessdate = 2006-12-07
| archiveurl = https://web.archive.org/web/20060709233434/http://amsglossary.allenpress.com/glossary/search?id=mesocyclone1
| archivedate = 2006-07-09
| df =
}}&lt;/ref&gt; Mesocyclones are localized, approximately {{convert|2|km|mi|abbr=on}} to {{convert|10|km|mi|abbr=on}} in diameter within strong thunderstorms.&lt;ref name=&quot;MesocyloneDef&quot;/&gt;

Mesocyclones form when strong changes of wind speed and/or direction with height (&quot;wind shear&quot;) sets parts of the lower part of the atmosphere spinning in invisible tube-like rolls. The convective updraft of a thunderstorm then draw up this spinning air, tilting the rolls' orientation upward (from parallel to the ground to perpendicular) and causing the entire updraft to rotate as a vertical column.&lt;ref&gt;University of Illinois. [http://ww2010.atmos.uiuc.edu/(Gh)/guides/mtr/svr/comp/wind/home.rxml Vertical Wind Shear] Retrieved on 2006-10-21.&lt;/ref&gt;

The best way to detect and verify the presence of a mesocyclone is by Doppler weather radar. Nearby high values of opposite sign within velocity data are how they are detected.&lt;ref&gt;{{cite web
|url=http://amsglossary.allenpress.com/glossary/search?id=mesocyclone-signature1
|title=Mesocyclone signature
|author=Glossary of Meteorology
|date=June 2000
|accessdate=2010-02-01
|publisher=American Meteorological Society
|archiveurl=https://web.archive.org/web/20110514115419/http://amsglossary.allenpress.com/glossary/search?id=mesocyclone-signature1
|archivedate=2011-05-14
|df=
}}&lt;/ref&gt;
{{clear}}

==Polar lows==
[[Image:Sea of Japan polar low 2009-12-20 0213Z.jpg|thumb|right|250px|A polar low is over the Sea of Japan in December 2009. Credit: MODIS image captured by NASA’s Terra satellite.{{tlx|free media}}]]
[[Image:Sea of Japan polar low 2017-02-10.gif|thumb|left|250px|Evolution of the eye-like feature on a polar low. Credit: AHI images captured by the Japan Meteorology Agency’s Himawari-8 satellite.{{tlx|free media}}]]
[[Image:Polar low.jpg|thumb|right|250px|A polar low is over the Barents Sea in February 1987. Credit: U.S. National Oceanic and Atmospheric Administration.{{tlx|free media}}]]
Polar lows have been referred to by many other terms, such as '''polar mesoscale vortex''', '''Arctic hurricane''', '''Arctic low''', and '''cold air depression'''. Today the term is usually reserved for the more vigorous systems that have near-surface winds of at least 17&amp;nbsp;m/s (38&amp;nbsp;mph).&lt;ref&gt;{{Citation |last=Rasmussen |first=E. A.  |last2=Turner |first2=J. |year=2003 |title=Polar Lows: Mesoscale Weather Systems in the Polar Regions |publisher=Cambridge University Press |location=Cambridge |page=612 |isbn=0-521-62430-4 }}.&lt;/ref&gt;

During winter, when cold-core lows with temperatures in the mid-levels of the troposphere reach {{convert|-45|C|F}} move over open waters, deep convection forms which allows polar low development to become possible.&lt;ref&gt;{{cite book
|url=https://books.google.com/books?id=-tBa1DWYoDIC&amp;pg=PA227&amp;dq=cold+core+low+book#v=onepage&amp;q=cold%20core%20low%20book&amp;f=false
|title=Polar lows: mesoscale weather systems in the polar regions
|page=224
|author=Erik A. Rasmussen
|author2=John Turner
|year=2003
|publisher=Cambridge University Press
|accessdate=2011-01-27
|ISBN=978-0-521-62430-5
}}&lt;/ref&gt;

Polar lows can occur at any time during the year.  However, summer lows tend to be weaker than winter lows.&lt;ref name=&quot;HB&quot;&gt;Halldór Björnsson. [http://andvari.vedur.is/~halldor/HB/Met210old/GlobCirc.html Global circulation.] https://web.archive.org/web/20110807132251/http://andvari.vedur.is/~halldor/HB/Met210old/GlobCirc.html 2011-08-07 Veðurstofa Íslands. Retrieved on 2008-06-15.&lt;/ref&gt;
{{clear}}

==Polar vortices==
{{multiple image
| header    = The Arctic polar vortex
| align     = right

| image1    = November2013 polar vortex geopotentialheight mean Large.jpg
| width1      = 149
| alt1      = Map of a compact blob over the Arctic
| caption1  = A strong polar vortex configuration in November 2013. Credit: National Oceanic and Atmospheric Administration - Pacific Marine Environmental Laboratory.{{tlx|free media}}

| image2    = Jan52014 polar vortex geopotentialheight mean Large.jpg
| width2      = 149
| alt2      = Map of a blobs spreading from the Arcitc
| caption2  = A more typical weak polar vortex on January 5, 2014. Credit: National Oceanic and Atmospheric Administration - Pacific Marine Environmental Laboratory.{{tlx|free media}}
}}
[[Image:Polarvortexjan211985.jpg|thumb|right|250px|Low pressure area over Quebec, Maine, and New Brunswick, part of the northern polar vortex weakening, on the record-setting cold morning of January 21, 1985. Credit: National Meteorological Center, Camp Springs, MD.{{tlx|free media}}]]
[[Image:Tolar_vortex_over_the_United_Kingdom_on_December_17,_2010.png|thumb|right|300px|Polar vortex over the United Kingdom on December 17, 2010. Credit: Wetterzentrale.{{tlx|fairuse}}]]
[[Image:Polarvortexwinter.jpg|thumb|upright=1.75|250px|Polar vortex and weather impacts due to stratospheric warming. Credit: National Science Foundation.{{tlx|free media}}]]
The interface between the cold dry air mass of the pole and the warm moist air mass farther south defines the location of the polar front. The polar front is centered, roughly at 60° latitude. A polar vortex strengthens in the winter and weakens in the summer because of its dependence on the temperature difference between the equator and the poles.&lt;ref name=&quot;HB&quot;/&gt;

When the northern vortex weakens, it separates into two or more vortices, the strongest of which are near Baffin Island, Canada, and the other over northeast Siberia.&lt;ref name=&quot;glossvortex&quot;&gt;{{cite web
|website=Glossary of Meteorology
|date=June 2000
|url=http://glossary.ametsoc.org/wiki/Polar_vortex
|title=Polar vortex
|publisher=American Meteorological Society
|access-date=15 June 2008
}}&lt;/ref&gt;

When the polar vortex is weak, high-pressure zones of the mid-latitudes may push poleward, moving the polar vortex, jet stream, and polar front equatorward. The jet stream is seen to &quot;buckle&quot; and deviate south. This rapidly brings cold dry air into contact with the warm, moist air of the mid-latitudes, resulting in a rapid and dramatic change of weather known as a &quot;cold snap&quot;.&lt;ref&gt;{{cite press release
|title=Stratospheric Polar Vortex Influences Winter Cold, Researchers Say
|publisher=American Association for the Advancement of Science
|date=December 3, 2001
|url=http://www.eurekalert.org/pub_releases/2001-12/uoia-spv120301.php
|access-date=May 23, 2015
}}&lt;/ref&gt;

The polar vortex was first described as early as 1853.&lt;ref&gt;[https://books.google.com/books?id=Df4vAAAAYAAJ&amp;pg=PA430&amp;dq=%22polar+vortex%22 &quot;Air Maps&quot;], ''Littell's Living Age'' No. 495, 12 November 1853, p. 430.&lt;/ref&gt; The phenomenon's sudden stratospheric warming (SSW) develops during the winter in the Northern Hemisphere and was discovered in 1952 with radiosonde observations at altitudes higher than 20&amp;nbsp;km.&lt;ref&gt;{{cite press release
|title=GEOS-5 Analyses and Forecasts of the Major Stratospheric Sudden Warming of January 2013
|publisher=Goddard Space Flight Center
|date=
|url=http://gmao.gsfc.nasa.gov/researchhighlights/SSW/
|accessdate=January 8, 2014
}}&lt;/ref&gt;

The phenomenon was mentioned frequently in the news and weather media in the cold North American winter of 2013–2014, popularizing the term as an explanation of very cold temperatures.&lt;ref&gt;http://blog.quarkexpeditions.com/polar-vortex-the-science-myth-media-hype-behind-north-american-weather-phenomenon|date=November 2016&lt;/ref&gt;

A deep freeze that gripped much of the United States and Canada in late January 2019 has been blamed on a polar vortex. The US National Weather Service warned that frostbite is possible within just 10 minutes of being outside in such extreme temperatures, and hundreds of schools, colleges and universities in the affected areas were closed. Around 21 people died in US due to severe frostbite.&lt;ref&gt;{{cite web
|title=Casualty
|url=https://www.bbc.com/news/world-us-canada-47088684
|date=1 Feb 2019
|access-date=12 Feb 2019
|language=en
}}&lt;/ref&gt;&lt;ref&gt;{{cite web
|title=Polar vortex: What is it and how does it happen?
|url=https://www.bbc.com/news/av/world-47065461/polar-vortex-what-is-it-and-how-does-it-happen
|date=30 Jan 2019
|website=BBC video
|accessdate=31 Jan 2019
}}&lt;/ref&gt; States within the midwest region of the United States had windchills just above -50°F (-45°C), which is colder than the frozen tundra and Antarctica.&lt;ref&gt;{{Cite web
|url=https://www.theverge.com/2019/1/30/18203517/polar-vortex-weather-midwest-cold-antarctica-siberia-alaska
|title=The Midwest is colder than Antarctica, Alaska, and Siberia right now
|last=Chen
|first=Angela
|date=January 30, 2019
|website=The Verge
|access-date=
}}&lt;/ref&gt;

The Polar vortex has also thought to have had effects in Europe. For example, the 2013–14 United Kingdom winter floods were blamed on the Polar vortex bringing severe cold in the United States and Canada.&lt;ref&gt;http://climatestate.com/2014/02/09/uk-flooding-and-the-science-of-climate-change/&lt;/ref&gt; Similarly, the severe, brutal cold in the United Kingdom in the winters of 2009/10 and 2010/11 were also blamed on the Polar vortex.&lt;ref&gt;https://www.independent.co.uk/news/uk/home-news/polar-vortex-what-is-coldest-winter-uk-weather-cold-snap-why-arctic-met-office-a7402611.html&lt;/ref&gt;

Polar cyclones are low-pressure zones embedded within the polar air masses, and exist year-round. The stratospheric polar vortex develops at latitudes above the subtropical jet stream.&lt;ref&gt;{{cite journal
|last1=Hartmann
|first1=D
|last2=Schoeberl
|first2=M
|year=1991
|title=Mixing of polar vortex air into middle latitudes as revealed by tracer-tracer scatterplots
|doi=10.1029/96JD03715
|bibcode = 1997JGR...10213119W
|volume=102
|issue=D11
|journal=Journal of Geophysical Research
|pages=13119
}}&lt;/ref&gt; Horizontally, most polar vortices have a radius of less than {{convert|1000|km|mi}}.&lt;ref name=&quot;pause&quot;&gt;{{cite journal | title=Potential Vorticity Diagnosis of a Tropopause Polar Cyclone
  | author1=Cavallo, Steven M. | author2= Hakim, Gregory J. |date=April 2009|journal=Monthly Weather Review
  | volume=137|pages= 1358–71|issue=4|doi=10.1175/2008MWR2670.1|bibcode = 2009MWRv..137.1358C }}&lt;/ref&gt; Since polar vortices exist from the stratosphere downward into the mid-troposphere,&lt;ref name=&quot;glossvortex&quot;/&gt; a variety of heights/pressure levels are used to mark its position. The 50&amp;nbsp;mb pressure surface is most often used to identify its stratospheric location.&lt;ref&gt;{{cite journal
|url=https://www.academia.edu/223963
|date=April 2010
|journal=Quarterly Journal of the Royal Meteorological Society
|title=The association between stratospheric weak polar vortex events and cold air outbreaks in the Northern Hemisphere
|page=887
|first1=Erik W.
|last1=Kolstad
|first2=Tarjei
|last2=Breiteig
|first3=Adam A.
|last3=Scaife
|volume=136
|issue=649
|bibcode=2010EGUGA..12.5739K
|doi=10.1002/qj.620
|arxiv=0906.0027
}}&lt;/ref&gt; At the level of the tropopause, the extent of closed contours of potential temperature can be used to determine its strength. Others have used levels down to the 500 hPa pressure level (about {{convert|5460|m|ft}} above sea level during the winter) to identify the polar vortex.&lt;ref&gt;{{cite journal
|url=http://www.ccsenet.org/journal/index.php/jgg/article/viewFile/28960/18761
|journal=Journal of Geology and Geography
|date=2013-11-22
|author=Abdolreza Kashki
|author2=Javad Khoshhal
|title=Investigation of the Role of Polar Vortex in Iranian First and Last Snowfalls
|issn= 1916-9779
|volume=5
|number=4
}}&lt;/ref&gt;
{{clear}}

==Subtropical cyclones==
[[Image:Leslie 2018-09-29 1410Z.jpg|thumb|right|250px|Subtropical Storm Leslie is in September 2018. Credit: National Aeronautics and Space Administration (NASA).{{tlx|free media}}]]
[[Image:Gustav 2002-09-09 1805Z.jpg|thumb|left|250px|Subtropical Storm Gustav in 2002, the first system to be given a name as a subtropical cyclone. Credit: Jesse Allen, based on data from the MODIS Rapid Response Team at NASA-GSFC.{{tlx|free media}}]]
A '''subtropical cyclone''' is a weather system that has some characteristics of a tropical and an extratropical cyclone.&lt;ref name=&quot;NAtlSTClimo&quot;&gt;{{cite journal
  | title = Atlantic Subtropical Storms. Part II: Climatology
  |author1=Mark P. Guishard |author2=Jenni L. Evans |author3=Robert E. Hart | journal = Journal of Climate
  |volume=22 |issue=13 | date = July 2009
  | pages = 3574–3594
  | doi = 10.1175/2008JCLI2346.1
  |bibcode=2009JCli...22.3574G}}&lt;/ref&gt;

Subtropical cyclones have broad wind patterns with maximum sustained winds located farther from the center than typical tropical cyclones, and have no weather fronts linked into their center.&lt;ref name=&quot;NAtlSTCases&quot;&gt;{{cite journal
  | title = Atlantic Subtropical Storms. Part I: Diagnostic Criteria and Composite Analysis
  |author1=Jenni L. Evans |author2=Mark P. Guishard | journal = Monthly Weather Review
  |volume=137 |issue=7 | date = July 2009
  | pages = 2065–2080
  | doi = 10.1175/2009MWR2468.1
  |bibcode = 2009MWRv..137....1E }}&lt;/ref&gt;
{{clear}}
Subtropical cyclones are also observed to form in the South Atlantic; South Atlantic subtropical cyclones are observed in all months.&lt;ref name=&quot;SAtlSTs&quot;&gt;{{cite journal
  | title = A climatology of subtropical cyclones in the South Atlantic
  |author1=Jenni L. Evans |author2=Aviva J. Braun | journal = Journal of Climate
  |volume=25 |issue=21 | date = November 2012
  | pages = 7328–7340
  | doi = 10.1175/JCLI-D-11-00212.1
  | bibcode = 2012JCli...25.7328E}}&lt;/ref&gt;

Throughout the 1950s and 1960s, the term semi-tropical and quasi-tropical were used for what would become known as subtropical cyclones.&lt;ref&gt;David B. Spiegler (1973). Many times, subtropical cyclones have a small warm core. [http://docs.lib.noaa.gov/rescue/mwr/101/mwr-101-04-0380.pdf Reply.] Monthly Weather Review, April 1973, p. 380. Retrieved on 2008-04-20.&lt;/ref&gt; The term subtropical cyclone merely referred to any cyclone located in the subtropical belt near and just north of the horse latitudes. Intense debate ensued in the late 1960s, after a number of hybrid cyclones formed in the Atlantic Basin. In 1972, the National Hurricane Center (NHC) finally designated these storms as subtropical cyclones in real-time,&lt;ref name=&quot;NHC1972&quot;&gt;R. H. Simpson and Paul J. Hebert (1973). [http://www.aoml.noaa.gov/general/lib/lib1/nhclib/mwreviews/1972.pdf Atlantic Hurricane Season of 1972.] Monthly Weather Review, April 1973, pp. 323–332. Retrieved on 2008-06-14.&lt;/ref&gt; and updated the hurricane database to include subtropical cyclones from 1968 through 1971.
{{clear}}

==Subtropical ridges==
[[Image:Subtropicalridge2000091412.jpg|thumb|right|250px|The subtropical ridge shows up as a large area of black (dryness) on this water vapor satellite image from September 2000. Credit: National Climatic Data Center, NOAA, GIBBS satellite.{{tlx|free media}}]]
Heating of the earth near the equator forces upward motion and convection along the monsoon trough or intertropical convergence zone, then divergence over the near-equatorial trough leads to air rising aloft and moving away from the equator: as air moves towards the mid-latitudes, it cools and sinks leading to subsidence near the 30° parallel of both hemispheres, resulting in circulation known as the Hadley cell which forms the subtropical ridge.&lt;ref name=Thompson&gt;{{ cite web
|author=Owen E. Thompson
|title=Hadley cell
|date=1996
|publisher=Channel Video Productions
|url=https://web.archive.org/web/20090305122318/http://www.atmos.umd.edu/~owen/CHPI/IMAGES/circs02.html
|accessdate=2007-02-11 }}&lt;/ref&gt;

Many of the world's deserts are caused by these climatological high-pressure areas.&lt;ref&gt;{{ cite web
|author=ThinkQuest team 26634
|date=1999
|title=The Formation of Deserts
|publisher=Oracle ThinkQuest Education Foundation
|url=https://web.archive.org/web/20121017193948/http://library.thinkquest.org/26634/desert/formation.htm
|accessdate=2009-02-16 }}&lt;/ref&gt;

Because these anticyclones strengthen with height, they are known as warm core ridges.
{{clear}}

==Tornadoes==
[[Image:A tornado near Anadarko, Oklahoma, on May 3, 1999.jpg|thumb|right|250px|A tornado is near Anadarko, Oklahoma. Credit: Daphne Zaras.{{tlx|free media}}]]
A '''tornado''' is a rapidly rotating column of air that is in contact with both the surface of the Earth and a cumulonimbus cloud or, in rare cases, the base of a cumulus cloud. The windstorm is often referred to as a '''twister''', '''whirlwind''' or '''cyclone''',&lt;ref&gt;{{ cite web
|url=http://www.merriam-webster.com/dictionary/cyclone
|title=merriam-webster.com
|publisher=merriam-webster.com
|date=
|accessdate=2012-09-03 }}&lt;/ref&gt; although the word cyclone is used in meteorology to name a weather system with a low-pressure area in the center around which, from an observer looking down toward the surface of the earth, winds blow counterclockwise in the Northern Hemisphere and clockwise in the Southern.&lt;ref&gt;{{ cite book
|title=Essentials of Oceanography
|last=Garrison
|first=Tom
|publisher=Cengage Learning
|year=2012
|isbn=978-0-8400-6155-3
|location=
|pages= }}&lt;/ref&gt; Tornadoes come in many shapes and sizes, and they are often visible in the form of a condensation funnel originating from the base of a cumulonimbus cloud, with a cloud of rotating debris and dust beneath it. Most tornadoes have wind speeds less than {{convert|110|mph|km/h|sigfig=2}}, are about {{convert|250|ft|-1}} across, and travel a few miles (several kilometers) before dissipating. The most extreme tornadoes can attain wind speeds of more than {{convert|300|mph|km/h|sigfig=2}}, are more than {{convert|2|mi|km}} in diameter, and stay on the ground for dozens of miles (more than 100&amp;nbsp;km).&lt;ref name=&quot;fastest wind&quot;&gt;{{ cite web
|url=http://cswr.org/dow/DOW.htm
|title=Doppler on Wheels
|accessdate=2009-12-13
|author=Wurman, Joshua
|publisher=Center for Severe Weather Research
|date=2008-08-29
|archiveurl=https://web.archive.org/web/20070205124033/http://www.cswr.org/dow/dow.htm
|archivedate=2007-02-05 }}&lt;/ref&gt;&lt;ref name=&quot;widest tornado&quot;&gt;{{ cite web
|url=http://www.crh.noaa.gov/oax/archive/hallam/hallam.php
|title=Hallam Nebraska Tornado
|accessdate=2009-11-15
|work=National Weather Service
| publisher=National Oceanic and Atmospheric Administration
|date=2005-10-02 }}&lt;/ref&gt;&lt;ref name=&quot;SPC FAQ&quot;&gt;{{cite web
|url=http://www.spc.ncep.noaa.gov/faq/tornado/
|title=The Online Tornado FAQ
|accessdate=2006-09-08
|author=Roger Edwards
|date=2006-04-04
|work=Storm Prediction Center
|publisher=National Oceanic and Atmospheric Administration
|archiveurl=https://web.archive.org/web/20060929185156/http://www.spc.ncep.noaa.gov/faq/tornado/
|archivedate=2006-09-29 }}&lt;/ref&gt;

Various types of tornadoes include the multiple vortex tornado, landspout, and waterspout. Waterspouts are characterized by a spiraling funnel-shaped wind current, connecting to a large cumulus or cumulonimbus cloud. They are generally classified as non-supercellular tornadoes that develop over bodies of water, but there is disagreement over whether to classify them as true tornadoes. These spiraling columns of air frequently develop in tropical areas close to the equator and are less common at high latitudes.&lt;ref&gt;{{cite web
|url=http://www.erh.noaa.gov/btv/events/15Jan2009/overview.shtml
|title=15 January 2009: Lake Champlain Sea Smoke, Steam Devils, and Waterspout: Chapters IV and V
|author=National Weather Service
|publisher=National Oceanic and Atmospheric Administration
|date=2009-02-03
|accessdate=2009-06-21 }}&lt;/ref&gt;

Tornadoes occur most frequently in North America, particularly in central and southeastern regions of the United States colloquially known as tornado alley,&lt;ref&gt;{{cite web
|url=http://www.sciencenews.org/articles/20020511/bob9.asp
|title=Tornado Alley, USA: Science News Online, May 11, 2002
|author=
|date=25 August 2006
|archiveurl=https://web.archive.org/web/20060825011156/http://www.sciencenews.org/articles/20020511/bob9.asp
|archivedate=25 August 2006 }}&lt;/ref&gt; as well as in Southern Africa, northwestern and southeast Europe, western and southeastern Australia, New Zealand, Bangladesh and adjacent eastern India, and southeastern South America.&lt;ref name=&quot;EB tornado climatology&quot;&gt;{{ cite web
|url=http://www.britannica.com/eb/article-218357/tornado
|title=Tornado: Global occurrence
|accessdate=2009-12-13
|publisher=Encyclopædia Britannica Online
|year=2009 }}&lt;/ref&gt;

There are several scales for rating the strength of tornadoes. The Fujita scale rates tornadoes by damage caused and has been replaced in some countries by the updated Enhanced Fujita Scale. An F0 or EF0 tornado, the weakest category, damages trees, but not substantial structures. An F5 or EF5 tornado, the strongest category, rips buildings off their foundations and can deform large skyscrapers. The similar TORRO scale ranges from a T0 for extremely weak tornadoes to T11 for the most powerful known tornadoes.&lt;ref&gt;{{ cite web
|url=http://www.torro.org.uk/TORRO/ECSS_Slide_Show/2004%20SPAIN%20ECSS%20Post-FINAL%20slide%20show.html
|title=Wind Scales: Beaufort, T&amp;nbsp; – Scale, and Fujita's Scale
|author=Meaden, Terrance
|publisher=Tornado and Storm Research Organisation
|year=2004
|accessdate=2009-09-11
|archiveurl=https://web.archive.org/web/20100430211910/http://www.torro.org.uk/TORRO/ECSS_Slide_Show/2004%20SPAIN%20ECSS%20Post-FINAL%20slide%20show.html
|archivedate=2010-04-30 }}&lt;/ref&gt; Doppler radar data, photogrammetry, and ground swirl patterns (trochoidal marks) may also be analyzed to determine intensity and assign a rating.&lt;ref name=&quot;EF SPC&quot;&gt;{{cite web
|title=Enhanced F Scale for Tornado Damage
|work=Storm Prediction Center
|date=2007-02-01
|publisher=National Oceanic and Atmospheric Administration
|url=http://www.spc.noaa.gov/efscale/ef-scale.html
|accessdate=2009-06-21 }}&lt;/ref&gt;&lt;ref&gt;{{Cite journal
| doi=10.1175/BAMS-D-11-00006.1
|title = Tornado Intensity Estimation: Past, Present, and Future
| journal=Bulletin of the American Meteorological Society
| volume=94
| issue=5
| pages=641–653
|year = 2013
|last1 = Edwards
|first1 = Roger
| last2=Ladue
| first2=James G.
| last3=Ferree
| first3=John T.
| last4=Scharfenberg
| first4=Kevin
| last5=Maier
| first5=Chris
| last6=Coulbourne
| first6=William L.
| bibcode=2013BAMS...94..641E }}&lt;/ref&gt;

Tornado formation often occurs in one of two ways.&lt;ref name=severe&gt;{{cite web
|url=https://www.nssl.noaa.gov/education/svrwx101/tornadoes/
|title=Severe Weather 101: Tornado Basics
|publisher=NOAA National Severe Storms Laboratory
|access-date=October 2, 2018 }}&lt;/ref&gt;&lt;ref name=tornadoFAQ&gt;{{cite web
|url=https://www.spc.noaa.gov/faq/tornado/
|title=The Online Tornado FAQ
|last=Edwards
|first=Roger
|agency=NOAA Storm Prediction Center
|date=April 19, 2018
|access-date=October 2, 2018 }}&lt;/ref&gt;

In the first method, two conditions must be satisfied.&lt;ref name=&quot;nssl.noaa.gov&quot;&gt;{{ cite web
|author=&lt;nowiki&gt;National Oceanic and Atmospheric Administration&lt;/nowiki&gt;
|title=tornadoes...''Nature's Most Violent Storms''
|url=http://www.nssl.noaa.gov/edu/safety/tornadoguide.html
|date=September 1992
|work=A Preparedness Guide
|accessdate=2008-08-03
|archive-url=https://web.archive.org/web/20080624204058/http://www.nssl.noaa.gov/edu/safety/tornadoguide.html
|archive-date=2008-06-24 }}&lt;/ref&gt; One, a horizontal spinning effect must form on the Earth's surface. This usually originates in sudden changes in wind direction or speed, known as wind shear. Two, a thundercloud, or occasionally a cumulus cloud, must be present. During a thunderstorm, updrafts are occasionally powerful enough to lift the horizontal spinning row of air upwards, turning it into a vertical air column. This vertical air column then becomes the basic structure for the tornado. Tornadoes that form in this way are often weak and generally last less than 10 minutes.&lt;ref name=&quot;nssl.noaa.gov&quot;/&gt;

The second method occurs during a supercell thunderstorm, in updrafts within the storm. When winds intensify, the force released can cause the updrafts to rotate. This rotating updraft is known as a mesocyclone.&lt;ref&gt;{{ cite web
|title=Tornado Formation
|url=http://library.thinkquest.org/03oct/00758/text-only/disaster/tornado/formation.html
|author=Thinkquest
|publisher=Oracle Corporation
|date=October 2003
|accessdate=2009-08-03
|archiveurl=https://web.archive.org/web/20080421062752/http://library.thinkquest.org/03oct/00758/text-only/disaster/tornado/formation.html
|archivedate=2008-04-21 }}&lt;/ref&gt; For a tornado to form in this manner, a rear-flank downdraft enters the center of the mesocyclone from the back. Cold air, being denser than warm air is able to penetrate through the updraft. The combination of the updraft and downdraft completes the development of a tornado. Tornadoes that form in this method are often violent and can last over an hour.&lt;ref name=&quot;nssl.noaa.gov&quot; /&gt;
{{clear}}

==Tropical cyclones==
[[Image:Hurricane Isabel from ISS.jpg|upright=1.35|thumb|right|250px|Hurricane Isabel (2003) is seen from orbit during Expedition 7 of the International Space Station. The eye, eyewall, and surrounding rainbands, characteristics of tropical cyclones in the narrow sense, are clearly visible in this view from space. Credit: Mike Trenchard, Earth Sciences &amp; Image Analysis Laboratory, NASA Johnson Space Center.{{tlx|free media}}]]
A '''tropical cyclone''' is a rapidly rotating storm system characterized by a low-pressure center, a closed low-level atmospheric circulation, strong winds, and a spiral arrangement of thunderstorms that produce heavy rain. Depending on its location and strength, a tropical cyclone is referred to by different names, including hurricane,&lt;ref&gt;{{cite web
|url=http://www.oxforddictionaries.com/us/definition/english/hurricane#hurricane
|title=hurricane
|publisher=Oxford dictionary
|accessdate=October 1, 2014 }}&lt;/ref&gt;&lt;ref&gt;{{cite web
|url=http://www.merriam-webster.com/dictionary/hurricane
|title=Hurricane – Definition and More from the Free Merriam-Webster Dictionary
|accessdate=October 1, 2014 }}&lt;/ref&gt;&lt;ref&gt;{{cite web
|url=http://www.collinsdictionary.com/dictionary/english/hurricane
|title=Definition of &quot;hurricane&quot; – Collins English Dictionary
|accessdate=October 1, 2014
}}&lt;/ref&gt; typhoon, tropical storm, cyclonic storm, tropical depression, and simply cyclone.&lt;ref name=&quot;HCT&quot;&gt;{{cite web
| title = What is the difference between a hurricane, a cyclone, and a typhoon?
| work = OCEAN FACTS
| publisher = National Ocean Service
| url = http://oceanservice.noaa.gov/facts/cyclone.html
| accessdate = December 24, 2016
}}&lt;/ref&gt; A hurricane is a tropical cyclone that occurs in the Atlantic Ocean and northeastern Pacific Ocean, and a typhoon occurs in the northwestern Pacific Ocean; in the south Pacific or Indian Ocean, comparable storms are referred to simply as &quot;tropical cyclones&quot; or &quot;severe cyclonic storms&quot;.&lt;ref name=&quot;HCT&quot; /&gt;
{{clear}}

==Upper level cyclones==
[[Image:TUTTcellWestPac2007071012WV.jpg|thumb|right|250px|Satellite image of an upper tropospheric cyclonic vortex in the western North Pacific. Credit: National Climatic Data Center.{{tlx|free media}}]]
Case studies of upper tropospheric cyclones in the Atlantic and Pacific have been performed by using airplane reports (winds, temperatures and heights), radiosonde data, geostationary satellite cloud imagery, and cloud-tracked winds throughout the troposphere.&lt;ref name=&quot;dtic1&quot;&gt;{{cite web
|url=http://handle.dtic.mil/100.2/ADA247588
|title=Dust and Sand Forecasting in Iraq and Adjoining Countries
|author=MSGT Walter D. Wilkerson
|date=November 1991
|publisher=Air Weather Service
|accessdate=2009-12-23
}}&lt;/ref&gt; It was determined they were the origin of an upper tropospheric cold-core lows, or cut-off lows.&lt;ref name=&quot;JTWC&quot;&gt;{{cite web
|author=Joint Typhoon Warning Center
|url=http://www.nrlmry.navy.mil/~chu/chap2/se205.htm
|publisher=United States Navy
|year=2010
|accessdate=2009-04-24
|title=2.5 Upper Tropospheric Cyclonic Vortices }}&lt;/ref&gt;

The tropical upper tropospheric cyclone has a cold core, meaning it is stronger aloft than at the Earth's surface, or stronger in areas of the troposphere with lower pressures. This is explained by the thermal wind relationship.&lt;ref&gt;{{cite web
|url=http://amsglossary.allenpress.com/glossary/browse?s=c&amp;p=65
|title=Cold low
|date=June 2000
|accessdate=2010-05-02
|publisher=American Meteorological Society
|author=Glossary of Meteorology
|archive-url=https://web.archive.org/web/20110514103120/http://amsglossary.allenpress.com/glossary/browse?s=c&amp;p=65
|archive-date=2011-05-14 }}&lt;/ref&gt;
{{clear}}

==Warm-core cyclones==
[[Image:Global tropical cyclone tracks-edit2.jpg|thumb|right|350px|Global tropical cyclone tracks between 1985 and 2005, indicating the areas where tropical cyclones usually develop. Credit: [[c:user:Nilfanion|Nilfanion]].{{tlx|free media}}]]
The mechanisms through which tropical cyclogenesis occurs are distinctly different from those through which temperate cyclogenesis occurs. Tropical cyclogenesis involves the development of a warm-core cyclone, due to significant convection in a favorable atmospheric environment.&lt;ref name=&quot;A7&quot;&gt;{{cite web
|url=http://www.aoml.noaa.gov/hrd/tcfaq/A7.html
|title=What is an extra-tropical cyclone?
|last=Goldenberg
|first=Stan
|date=August 13, 2004
|work=Frequently Asked Questions: Hurricanes, Typhoons and Tropical Cyclones
|publisher=Atlantic Oceanographic and Meteorological Laboratory, Hurricane Research Division
|accessdate=August 30, 2008 }}&lt;/ref&gt;
{{clear}}

==Theoretical aerometeors==
[[Image:За селом 2.jpg|thumb|right|220px|A misty autumn morning in the outskirts of Rakhiv, Carpathian Biosphere Reserve, Ukraine, is shown. Credit: [[c:user:Swift11|Михайло Пецкович]].{{tlx|free media}}]]
[[Image:Chambord au lever du jour.jpg|thumb|left|220px|The Château de Chambord in central France is in mist. Credit: [[c:user:ELSA DESSAIGNE|ELSA DESSAIGNE]].{{tlx|free media}}]]
[[Image:Oberfallenberg 8.jpg|thumb|right|220px|Mist near the Austria–Switzerland border in December 2006. Credit: böhringer friedrich.{{tlx|free media}}]]
'''Def.''' &quot;movement of [atmospheric] air usually caused by convection or [subtle] differences in air pressure&quot;&lt;ref name=WindWikt&gt;{{ cite web
|author=[[wikt:User:Emperorbma|Emperorbma]]
|title=wind
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=8 December 2003
|url=https://en.wiktionary.org/wiki/wind
|accessdate=9 February 2019 }}&lt;/ref&gt; is called '''wind'''.

'''Def.''' a discrete unit of air, wind, or mist traveling or falling through or partially through an atmosphere is called an '''aerometeor'''.

'''Def.''' a &quot;wind whose direction and speed are determined by a balance of the horizontal pressure gradient force and the force due to the earth's rotation to the left in the northern hemisphere and to the right in the southern hemisphere&quot;&lt;ref name=GeostrophicWindWikt&gt;{{ cite web
|title=geostrophic wind
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=July 11, 2011
|url=http://en.wiktionary.org/wiki/geostrophic_wind
|accessdate=2013-02-17 }}&lt;/ref&gt; is called a '''geostrophic wind'''.

'''Def.''' a &quot;warm dry wind blowing down the side of a mountain&quot;&lt;ref name=FoehnWikt&gt;{{ cite web
|title=foehn
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=January 19, 2013
|url=http://en.wiktionary.org/wiki/foehn
|accessdate=2013-02-17 }}&lt;/ref&gt; is called a '''foehn''', or '''foehn wind''', or '''chinook'''.

The chinook generally blows from the southwest, but its direction may be modified by topography. When it sets in after a spell of intense cold, the temperature may rise by 20–40°F in 15 minutes due to replacement of a cold air mass with a much warmer air mass in minutes.&quot;&lt;ref name=ChinookWikt&gt;{{ cite web
|title=chinook
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=October 17, 2012
|url=http://en.wiktionary.org/wiki/chinook
|accessdate=2013-02-17 }}&lt;/ref&gt;

&quot;Wind shear is a change in wind direction, wind speed, or both, along a given direction in space (e.g., along a horizontal or vertical distance).&quot;&lt;ref name=Mireles&gt;{{ cite book
|author=Mark R. Mireles
|author2=Kirth L. Pederson
|author3=Charles H. Elford
|title=Meteorologial Techniques
|publisher=Air Force Weather Agency/DNT
|location=Offutt Air Force Base, Nebraska, USA
|date=February 21, 2007
|editor=
|pages=
|url=http://oai.dtic.mil/oai/oai?verb=getRecord&amp;metadataPrefix=html&amp;identifier=ADA466107
|arxiv=
|bibcode=
|doi=
|pmid=
|isbn=
|accessdate=2013-02-17 }}&lt;/ref&gt;

'''Def.''' a &quot;strong, abrupt rush of wind&quot;&lt;ref name=GustWikt&gt;{{ cite web
|title=gust
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=January 14, 2013
|url=http://en.wiktionary.org/wiki/gust
|accessdate=2013-02-17 }}&lt;/ref&gt; is called a '''gust'''.

==Radars==
{{main|Radiation astronomy/Radars|Radar astronomy}}
[[Image:June 2012 Radar image of Derecho.jpg|thumb|right|250px|This is a composite radar image of the June 2012 United States Derecho event. Credit: G. Carbin, NWS/Storm Prediction Center.{{tlx|free media}}]]
On the right is a composite of hourly radar images. These wind gusts averaged ~75 mph over about 450 miles. This is referred to as the Derecho event.
{{clear}}

==Mars==
{{main|Gases/Gaseous objects/Mars}}
[[Image:Marte56 01.jpg|thumb|right|250px|These are true color images of Mars taken in 1999. Credit: Antonio Cidadao.{{tlx|fairuse}}]]
[[Image:Marte56 10.jpg|thumb|left|250px|These are Hubble Space Telescope images of Mars prior to the Mars Pathfinder spacecraft and Lander. Credit: Philip James, NASA.{{tlx|fairuse}}]]
{{multiple image|center|caption_align=center|header_align=center|align=center|header= |width= |direction=horizontal
|image1=dust.devil.mars.arp.750pix.jpg
|width1=250
|caption1=Dust devil on Mars Mars Global Surveyor (MGS). Credit: Malin Space Science Systems, MGS, JPL, NASA.{{tlx|free media}}
|image2=Martian Dust Devil Trails.jpg
|width2=175
|caption2=Dust devils cause twisting dark trails on the Martian surface. Credit: NASA/JPL/University of Arizona.{{tlx|free media}}
|image3=The Serpent Dust Devil on Mars PIA15116.jpg
|width3=200
|caption3=Serpent Dust Devil of Mars (Mars Reconnaissance Orbiter (MRO)). Credit: NASA/JPL-Caltech/University of Arizona.{{tlx|free media}}
|image4=Mars-DustDevil-20170215.jpg
|width4=195
|caption4=A dust devil on hilly terrain in the Amazonis quadrangle. Credit: NASA/JPL-Caltech/University of Arizona.{{tlx|free media}}
|image5=PIA20045-Mars-DustDevils-20151105.jpg
|width5=210
|caption5=Dust devils in Valles Marineris (Mars Reconnaissance Orbiter (MRO)). Credit: NASA/JPL-Caltech/University of Arizona.{{tlx|free media}}
|footer= }}
[[Image:Marsdustdevil2.gif|thumb|center|580px|Dust devil on Mars is photographed by the Mars rover ''Spirit''. The counter in the bottom-left corner indicates time in seconds after the first photo was taken in the sequence. At the final frames, a trail is visible on the Martian surface. Three other dust devils also appear in the background. Credit: NASA.{{tlx|free media}}]]
[[Image:The Serpent Dust Devil of Mars.ogv|thumb|center|580px|The Serpent Dust Devil of Mars - video (01:16). Credit: JPL.{{tlx|free media}}]]
&quot;The [true] color images of Mars [at right] were taken in 1999, across almost 60 million miles (!) by a talented amateur astronomer in Oeiras, Portugal&amp;nbsp;– Antonio Cidadao.&quot;&lt;ref name=Hoagland&gt;{{ cite book
|author=Richard C. Hoagland
|title=Revealing Mars' True Colors ... of NASA
|publisher=TheEnterpriseMission Website
|location=
|date=2002
|url=http://www.bibliotecapleyades.net/marte/esp_marte_56.htm
|accessdate=2014-02-25 }}&lt;/ref&gt;

&quot;They were acquired with a modest 10-inch &quot;Schmidt-Cassegrain&quot; reflecting telescope, and a commercially available CCD (charge coupled device) camera. Mr. Cidadao’s total investment in his &quot;Mars imaging system&quot;—commercial telescope and electronic camera, plus computer to process the images, and the appropriate software—was approximately three thousand American dollars.&quot;&lt;ref name=Hoagland/&gt;

&quot;In 1997, before the arrival of the Mars Pathfinder spacecraft (the first NASA Lander sent to Mars since Viking), the Hubble Telescope was tasked to acquire a series of &quot;weather forecast Mars images&quot; prior to the landing [at left].&quot;&lt;ref name=Hoagland/&gt;

&quot;This long-distance reconnaissance detected a small dust storm less than a month before the Pathfinder arrival, which (with its potentially high winds) could have posed a serious threat to the Pathfinder entry and landing.&quot;&lt;ref name=Hoagland/&gt;

&quot;If dust diffuses to the landing site, the sky could turn out to be pink like that seen by Viking... otherwise [based on the Hubble images - above], Pathfinder will likely show blue sky with bright clouds.&quot;&lt;ref name=James&gt;{{ cite book
|author=Philip James
|title=Revealing Mars' True Colors ... of NASA
|publisher=TheEnterpriseMission Website
|location=
|date=2002
|url=http://www.bibliotecapleyades.net/marte/esp_marte_56.htm
|accessdate=2014-02-25 }}&lt;/ref&gt;

Dust devils also occur on Mars (see dust devil tracks) and were first photographed by the Viking orbiters in the 1970s. In 1997, the Mars Pathfinder lander detected a dust devil passing over it.&lt;ref&gt;{{cite web
| last = Metzger
| first = S. M.
| url = http://mars8.jpl.nasa.gov/MPF/science/lpsc98/1915.pdf
| title = Dust Devil Vortices at the Ares Vallis MPF Landing Site
| format = PDF
| work = Mars Exploration Program
| publisher = JPL
| accessdate = August 9, 2010}} {{dead link
|date=September 2010 }}&lt;/ref&gt;&lt;ref&gt;{{cite web
| date = March 21, 2000
| url = http://www.ruhr-uni-bochum.de/climusa/dust.htm
| archiveurl = https://web.archive.org/web/20061030133445/http://www.ruhr-uni-bochum.de/climusa/dust.htm
| archivedate = 2006-10-30
| title = Martian Dust Devils Caught
| work = Climate Research USA
| publisher = Ruhr-Universität Bochum
| accessdate = August 9, 2010 }}&lt;/ref&gt; In the image shown here, photographed by the Mars Global Surveyor, the long dark streak is formed by a moving swirling column of Martian atmosphere. The dust devil itself (the black spot) is climbing the crater wall. The streaks on the right are sand dunes on the crater floor.

Martian dust devils can be up to fifty times as wide and ten times as high as terrestrial dust devils, and large ones may pose a threat to terrestrial technology sent to Mars.&lt;ref&gt;{{cite web
| last = Smith
| first = Peter
|author2=Renno, Nilton
| date = 6 June 2001
| url = http://unisci.com/stories/20012/0606012.htm
| title = Studying Earth Dust Devils For Possible Mars Mission
| publisher = UniSci News
| accessdate = December 1, 2006 }}&lt;/ref&gt; On 7 November 2016, five such dust devils ranging in heights of 0.5 to 1.9 km were imaged in a single observation by Mars Orbiter Mission in martian southern hemisphere.&lt;ref&gt;{{Cite web
|url=https://www.hou.usra.edu/meetings/lpsc2019/pdf/1760.pdf
|title=Martian Dust Devils Observed by Mars Colour Camera Onboard Mars Orbiter Mission
|last=Singh
|first=Ramdayal
|last2=Arya
|first2=A.S.
|date=29 January 2019
|website=
|archive-url=https://web.archive.org/web/20190219212755/https://www.hou.usra.edu/meetings/lpsc2019/pdf/1760.pdf
|archive-date=19 February 2019
|access-date=19 February 2019 }}&lt;/ref&gt;

Mission members monitoring the Spirit rover on Mars reported on March 12, 2005, that a lucky encounter with a dust devil had cleaned the solar panels of that robot. Power levels dramatically increased and daily science work was anticipated to be expanded.&lt;ref&gt;{{cite web
| last = David
| first = Leonard
| date = 12 March 2005
| url = http://www.space.com/missionlaunches/spirit_dust_050312.html
| title = Spirit Gets A Dust Devil Once-Over
| publisher = Space.com
| accessdate = December 1, 2006 }}&lt;/ref&gt; A similar phenomenon (solar panels mysteriously cleaned of accumulated dust) had previously been observed with the Opportunity rover]], and dust devils had also been suspected as the cause.&lt;ref&gt;{{cite web
| url = http://athena.corll.edu/kids/did_you_know.html
| title = Did You Know?
| work = Mars Exploration Rovers
| publisher = Cornell University
| accessdate = December 1, 2006 }}&lt;/ref&gt;
{{clear}}

==Jupiter==
{{main|Jupiter}}
[[Image:PIA02863 - Jupiter surface motion animation.gif|thumb|300px|center|Zones, belts and vortices on Jupiter are shown. Credit: NASA/JPL/University of Arizona.{{tlx|free media}}]]
The wide equatorial zone is visible in the center surrounded by two dark equatorial belts (SEB and NEB).

&quot;The large grayish-blue [irregular] &quot;hot spots&quot; at the northern edge of the white Equatorial Zone change over the course of time as they march eastward across the planet.&quot;&lt;ref name=Lavoie00&gt;{{ cite book
|author=Sue Lavoie
|title=PIA02863: Planetwide Color Movie
|publisher=NASA/JPL/University of Arizona
|location=Tucson, Arizona USA
|date=28 December 2000
|url=http://photojournal.jpl.nasa.gov/catalog/PIA02863
|accessdate=30 May 2013 }}&lt;/ref&gt;

&quot;The Great Red Spot shows its counterclockwise rotation, and the uneven distribution of its high haze is obvious. To the east (right) of the Red Spot, oval storms, like ball bearings, roll over and pass each other. Horizontal bands adjacent to each other move at different rates. Strings of small storms rotate around northern-hemisphere ovals.&quot;&lt;ref name=Lavoie00/&gt;

&quot;Small, very bright features appear quickly and randomly in turbulent regions, candidates for lightning storms.&quot;&lt;ref name=Lavoie00/&gt;

&quot;The smallest visible features at the equator are about 600 kilometers (about 370 miles) across.&quot;&lt;ref name=Lavoie00/&gt;

&quot;The clip consists of 14 unevenly spaced timesteps, each a true color cylindrical projection of the complete circumference of Jupiter, from 60 degrees south to 60 degrees north. The maps are made by first assembling mosaics of six images taken by Cassini's narrow-angle camera in the same spectral filter over the course of one Jupiter rotation and, consequently, covering the whole planet. Three such global maps -- in red, green and blue filters -- are combined to make one color map showing Jupiter during one Jovian rotation. Fourteen such maps, spanning 24 Jovian rotations at uneven time intervals comprise the movie.&quot;&lt;ref name=Lavoie00/&gt;

The passage of time is accelerated by a factor of 600,000.
{{clear}}

==Saturn==
{{main|Gases/Gaseous objects/Saturn}}
[[Image:Saturn HST 2004-03-22.jpg|thumb|right|250px|The view of Saturn from Hubble, taken on March 22, 2004, is so sharp that many individual Saturnian ringlets can be seen. Credit: NASA, ESA and Erich Karkoschka (University of Arizona).{{tlx|free media}}]]
&quot;The view [at right] from Hubble [of Saturn], taken on March 22, 2004, is so sharp that many individual Saturnian ringlets can be seen.&quot;&lt;ref name=Karkoschka&gt;{{ cite book
|author=Erich Karkoschka
|title=Saturn Seen from Far and Near
|publisher=Hubble Site
|location=Baltimore, Maryland USA
|date=May 26, 2004
|url=http://hubblesite.org/newscenter/archive/releases/2004/18/image/e/
|accessdate=2014-02-26 }}&lt;/ref&gt;

&quot;Hubble's exquisite optics, coupled with the high resolution of its Advanced Camera for Surveys, allow it to take pictures of Saturn which are nearly as sharp as Cassini's, even though Hubble is nearly a billion miles farther from Saturn than Cassini.&quot;&lt;ref name=Karkoschka/&gt;

&quot;Camera exposures in four filters (blue, blue-green, green, and red) were combined to form the Hubble image, to render colors similar to what the eye would see through a telescope focused on Saturn. The subtle pastel colors of ammonia-methane clouds trace a variety of atmospheric dynamics. Saturn displays its familiar banded structure, and haze and clouds of various altitudes. Like Jupiter, all bands are parallel to Saturn's equator. Even the magnificent rings, at nearly their maximum tilt toward Earth, show subtle hues, which indicate the trace chemical differences in their icy composition.&quot;&lt;ref name=Karkoschka/&gt;
{{clear}}

==Uranus==
{{main|Gases/Gaseous objects/Uranus}}
[[Image:Uranus2.jpg|thumb|right|250px|This is an image of the planet Uranus taken by the spacecraft [[w:Voyager 2|Voyager 2]] in 1986. Credit: NASA/JPL/Voyager mission.{{tlx|free media}}]]
[[Image:Uranuscolour.png|thumb|right|250px|Uranus's southern hemisphere in approximate natural colour (left) and in shorter wavelengths (right), shows its faint cloud bands and atmospheric &quot;hood&quot; as seen by ''Voyager 2''. Credit: NASA.{{tlx|free media}}]]
[[Image:Uranus Dark spot.jpg|thumb|right|250px|The first dark spot on Uranus ever observed is in an image obtained by ACS on HST in 2006. Credit: NASA, ESA, L. Sromovsky and P. Fry (University of Wisconsin), H. Hammel (Space Science Institute), and K. Rages (SETI Institute).{{tlx|free media}}]]
[[Image:Uranus clouds.jpg|thumb|upright|250px|Uranus in 2005. Rings, southern collar and a bright cloud in the northern hemisphere are visible (HST ACS image).{{tlx|free media}}]]
In larger amateur telescopes with an objective diameter of between 15 and 23&amp;nbsp;cm, the planet appears as a pale cyan disk with distinct [[w:limb darkening|limb darkening]].

&quot;Methane possesses prominent [[w:absorption band|absorption band]]s in the [[w:visible spectrum|visible]] and [[w:near-infrared|near-infrared]] (IR) making Uranus [[w:aquamarine (color)|aquamarine]] or [[w:cyan|cyan]] in color.&quot;&lt;ref name=Lunine&gt;{{cite journal
|title=The Atmospheres of Uranus and Neptune
|author=Jonathan I. Lunine
|journal = Annual Review of Astronomy and Astrophysics
|volume=31
|pages=217–63
|year=1993
|doi=10.1146/annurev.aa.31.090193.001245
|bibcode=1993ARA&amp;A..31..217L }}&lt;/ref&gt;

In 1986 ''Voyager 2'' found that the visible southern hemisphere of Uranus can be subdivided into two regions: a bright polar cap and dark equatorial bands (see figure on the right).&lt;ref name=&quot;Smith Soderblom et al. 1986&quot;&gt;Smith, B. A.; Soderblom, L. A.; Beebe, A.; Bliss, D.; Boyce, J. M.; Brahic, A.; Briggs, G. A.; Brown, R. H. et al (4 July 1986). &quot;Voyager 2 in the Uranian System: Imaging Science Results&quot;. Science 233 (4759): 43–64. Bibcode 1986Sci...233...43S. doi:10.1126/science.233.4759.43. {{PMID|17812889}}&lt;/ref&gt; Their boundary is located at about -45 degrees of [[w:latitude|latitude]]. A narrow band straddling the latitudinal range from -45 to -50 degrees is the brightest large feature on the visible surface of the planet.&lt;ref name=&quot;Smith Soderblom et al. 1986&quot; /&gt;&lt;ref name=&quot;Hammel de Pater et al. Uranus in 2003, 2005&quot;&gt;Hammel, H. B.; de Pater, I.; Gibbard, S. G.; Lockwood, G. W.; Rages, K. (June 2005). &quot;Uranus in 2003: Zonal winds, banded structure, and discrete features&quot; (PDF). Icarus 175 (2): 534–545. Bibcode 2005Icar..175..534H. doi:10.1016/j.icarus.2004.11.012&lt;/ref&gt; It is called a southern &quot;collar&quot;. The cap and collar are thought to be a dense region of methane clouds located within the pressure range of 1.3 to 2&amp;nbsp;bar (see above).&lt;ref name=&quot;Rages Hammel et al. 2004&quot;&gt;Rages, K. A.; Hammel, H. B.; Friedson, A. J. (11 September 2004). &quot;Evidence for temporal change at Uranus' south pole&quot;. Icarus 172 (2): 548–554. Bibcode 2004Icar..172..548R. doi:10.1016/j.icarus.2004.07.009&lt;/ref&gt; Besides the large-scale banded structure, Voyager 2 observed ten small bright clouds, most lying several degrees to the north from the collar.&lt;ref name=&quot;Smith Soderblom et al. 1986&quot; /&gt; In all other respects Uranus looked like a dynamically dead planet in 1986. Unfortunately Voyager 2 arrived during the height of the planet's southern summer and could not observe the northern hemisphere. At the beginning of the 21st century, when the northern polar region came into view, the Hubble Space Telescope (HST) and [[w:Keck telescopes|Keck]] telescope initially observed neither a collar nor a polar cap in the northern hemisphere.&lt;ref name=&quot;Hammel de Pater et al. Uranus in 2003, 2005&quot; /&gt; So Uranus appeared to be asymmetric: bright near the south pole and uniformly dark in the region north of the southern collar.&lt;ref name=&quot;Hammel de Pater et al. Uranus in 2003, 2005&quot; /&gt; In 2007, when Uranus passed its equinox, the southern collar almost disappeared, while a faint northern collar emerged near 45 degrees of [[w:latitude|latitude]].&lt;ref name=&quot;Sromovsky Fry et al. 2009&quot;&gt;Sromovsky, L. A.; Fry, P. M.; Hammel, H. B.; Ahue, W. M.; de Pater, I.; Rages, K. A.; Showalter, M. R.; van Dam, M. A. (September 2009). &quot;Uranus at equinox: Cloud morphology and dynamics&quot;. Icarus 203 (1): 265–286. Bibcode 2009Icar..203..265S. doi:10.1016/j.icarus.2009.04.015.&lt;/ref&gt;

On August 23, 2006, researchers at the Space Science Institute (Boulder, CO) and the University of Wisconsin observed a dark spot on Uranus's surface, giving astronomers more insight into the planet's atmospheric activity.&lt;ref name=DarkSpot&gt;{{ cite book
|url=http://www.physorg.com/pdf78676690.pdf
|title=Hubble Discovers a Dark Cloud in the Atmosphere of Uranus
|author=L. Sromovsky
|author2=Fry P.
|author3=Hammel, H.
|author4=Rages, K
|publisher=physorg.com
|accessdate=August 22, 2007 }}&lt;/ref&gt; Why this sudden upsurge in activity should be occurring is not fully known, but it appears that Uranus's extreme axial tilt results in extreme seasonal variations in its weather.&lt;ref name=weather&gt;{{ cite book
|url=http://www.sciencedaily.com/releases/2006/10/061001211630.htm
|title=Hubble Discovers Dark Cloud In The Atmosphere Of Uranus
|publisher=Science Daily
|accessdate=April 16, 2007 }}&lt;/ref&gt;&lt;ref name=Hammel2007/&gt; Determining the nature of this seasonal variation is difficult because good data on Uranus's atmosphere have existed for less than 84 years, or one full Uranian year. A number of discoveries have been made. [[w:Photometry (astronomy)|Photometry]] over the course of half a Uranian year (beginning in the 1950s) has shown regular variation in the brightness in two [[w:spectral band|spectral band]]s, with maxima occurring at the solstices and minima occurring at the equinoxes.&lt;ref name=&quot;Lockwood &amp; Jerzykiewicz 2006&quot;&gt;Lockwood, G. W.; Jerzykiewicz, Mikołaj A. (February 2006). &quot;Photometric variability of Uranus and Neptune, 1950–2004&quot;. Icarus 180 (2): 442–452. Bibcode 2006Icar..180..442L. doi:10.1016/j.icarus.2005.09.009.&lt;/ref&gt; A similar periodic variation, with maxima at the solstices, has been noted in [[w:microwave|microwave]] measurements of the deep troposphere begun in the 1960s.&lt;ref name=&quot;Klein &amp; Hofstadter 2006&quot;&gt;Klein, M. J.; Hofstadter, M. D. (September 2006). &quot;Long-term variations in the microwave brightness temperature of the Uranus atmosphere&quot;. Icarus 184 (1): 170–180. Bibcode 2006Icar..184..170K. doi:10.1016/j.icarus.2006.04.012.&lt;/ref&gt; [[w:Stratosphere|Stratospheric]] temperature measurements beginning in the 1970s also showed maximum values near the 1986 solstice.&lt;ref name=Young2001&gt;{{ cite journal
|author=Leslie A. Young
|author2=Amanda S. Bosh
|author3=Marc Buie
|author4=et al.
|title= Uranus after Solstice: Results from the 1998&amp;nbsp;November&amp;nbsp;6 Occultation
|journal=Icarus
|volume=153
|pages=236–247
|year=2001
|doi=10.1006/icar.2001.6698
| url=http://www.boulder.swri.edu/~layoung/eprint/ur149/Young2001Uranus.pdf
|bibcode=2001Icar..153..236Y
|issue=2 }}&lt;/ref&gt; The majority of this variability is believed to occur owing to changes in the viewing geometry.&lt;ref name=&quot;Karkoschka ('Uranus') 2001&quot;&gt;Karkoschka, Erich (May 2001). &quot;Uranus' Apparent Seasonal Variability in 25 HST Filters&quot;. Icarus 151 (1): 84–92. Bibcode 2001Icar..151...84K. doi:10.1006/icar.2001.6599.&lt;/ref&gt;

There are some reasons to believe that physical seasonal changes are happening in Uranus. While the planet is known to have a bright south polar region, the north pole is fairly dim, which is incompatible with the model of the seasonal change outlined above.&lt;ref name=Hammel2007&gt;{{ cite journal
|author=H.B. Hammel, G.W. Lockwood
|title=Long-term atmospheric variability on Uranus and Neptune
|journal=Icarus
|year=2007
|volume=186
|pages=291–301
|doi=10.1016/j.icarus.2006.08.027
| bibcode=2007Icar..186..291H }}&lt;/ref&gt; During its previous northern solstice in 1944, Uranus displayed elevated levels of brightness, which suggests that the north pole was not always so dim.&lt;ref name=&quot;Lockwood &amp; Jerzykiewicz 2006&quot; /&gt; This information implies that the visible pole brightens some time before the solstice and darkens after the equinox.&lt;ref name=Hammel2007/&gt; Detailed analysis of the visible and microwave data revealed that the periodical changes of brightness are not completely symmetrical around the solstices, which also indicates a change in the [[w:meridional|meridional]] [[w:albedo|albedo]] patterns.&lt;ref name=Hammel2007/&gt; Finally in the 1990s, as Uranus moved away from its solstice, Hubble and ground based telescopes revealed that the south polar cap darkened noticeably (except the southern collar, which remained bright),&lt;ref name=&quot;Rages Hammel et al. 2004&quot; /&gt; while the northern hemisphere demonstrated increasing activity,&lt;ref name=planetary&gt;{{ cite book
|title=No Longer Boring: 'Fireworks' and Other Surprises at Uranus Spotted Through Adaptive Optics, In: ''The Planetary Society''
|author=Emily Lakdawalla
|url=http://web.archive.org/web/20060525015410/http://www.planetary.org/news/2004/1111_No_Longer_Boring_Fireworks_and_Other.html
|date=2004
|accessdate=June 13, 2007 }}&lt;/ref&gt; such as cloud formations and stronger winds, bolstering expectations that it should brighten soon.&lt;ref name=&quot;Hammel de Pater et al. Uranus in 2004, 2005&quot; &gt;Hammel, H. B.; de Pater, I.; Gibbard, S. G.; Lockwood, G. W.; Rages, K. (May 2005). &quot;New cloud activity on Uranus in 2004: First detection of a southern feature at 2.2 µm&quot; (PDF). Icarus 175 (1): 284–288. Bibcode 2005Icar..175..284H. doi:10.1016/j.icarus.2004.11.016.&lt;/ref&gt; This indeed happened in 2007 when the planet passed an equinox: a faint northern polar collar arose, while the southern collar became nearly invisible, although the zonal wind profile remained slightly asymmetric, with northern winds being somewhat slower than southern.&lt;ref name=&quot;Sromovsky Fry et al. 2009&quot; /&gt;
{{clear}}

==Neptune==
{{main|Gases/Gaseous objects/Neptune}}
[[Image:The four sides of Neptune (captured by the Hubble Space Telescope).tif|thumb|right|250px|The snapshots of Neptune were taken at roughly 4-hour intervals, offering a full view of the blue-green planet. Credit: NASA/ESA/Hubble Heritage Team (STScI/AURA).{{tlx|free media}}]]
On July 12, 2011, Neptune &quot;has arrived at the same location in space where it was discovered nearly 165 years ago. To commemorate the event, NASA's Hubble Space Telescope has taken these &quot;anniversary pictures&quot; of the blue-green giant planet.&quot;&lt;ref name=Weaver&gt;{{ cite book
|author=Donna Weaver
|author2=Ray Villard
|author3=Keith Noll
|title=Neptune Completes Its First Circuit Around The Sun Since Its Discovery
|publisher=Hubblesite Newscenter
|location=Baltimore, Maryland USA
|date=July 12, 2011
|url=http://hubblesite.org/newscenter/archive/releases/2011/19/image/a/
|accessdate=2014-02-23 }}&lt;/ref&gt;

&quot;Neptune is the most distant major planet in our solar system. German astronomer Johann Galle discovered the planet on September 23, 1846. At the time, the discovery doubled the size of the known solar system. The planet is 2.8 billion miles (4.5 billion kilometers) from the Sun, 30 times farther than Earth. Under the Sun's weak pull at that distance, Neptune plods along in its huge orbit, slowly completing one revolution approximately every 165 years.&quot;&lt;ref name=Weaver/&gt;

&quot;These four Hubble images of Neptune were taken with the Wide Field Camera 3 on June 25-26, during the planet's 16-hour rotation. The snapshots were taken at roughly four-hour intervals, offering a full view of the planet. The images reveal high-altitude clouds in the northern and southern hemispheres. The clouds are composed of methane ice crystals.&quot;&lt;ref name=Weaver/&gt;

&quot;The giant planet experiences seasons just as Earth does, because it is tilted 29 degrees, similar to Earth's 23-degree-tilt. Instead of lasting a few months, each of Neptune's seasons continues for about 40 years.&quot;&lt;ref name=Weaver/&gt;

&quot;The snapshots show that Neptune has more clouds than a few years ago, when most of the clouds were in the southern hemisphere. These Hubble views reveal that the cloud activity is shifting to the northern hemisphere. It is early summer in the southern hemisphere and winter in the northern hemisphere.&quot;&lt;ref name=Weaver/&gt;

&quot;In the Hubble images, absorption of red light by methane in Neptune's atmosphere gives the planet its distinctive aqua color. The clouds are tinted pink because they are reflecting near-infrared light.&quot;&lt;ref name=Weaver/&gt;

&quot;A faint, dark band near the bottom of the southern hemisphere is probably caused by a decrease in the hazes in the atmosphere that scatter blue light. The band was imaged by NASA's Voyager 2 spacecraft in 1989, and may be tied to circumpolar circulation created by high-velocity winds in that region.&quot;&lt;ref name=Weaver/&gt;

&quot;The temperature difference between Neptune's strong internal heat source and its frigid cloud tops, about minus 260 degrees Fahrenheit, might trigger instabilities in the atmosphere that drive large-scale weather changes.&quot;&lt;ref name=Weaver/&gt;
{{clear}}

==Meteorology==
[[Image:Earth Global Circulation - en.svg|thumb|upright=1.35|right|300px|General circulation of the Earth's atmosphere: The westerlies and trade winds are part of the Earth's atmospheric circulation. Credit: [[c:user:Kaidor|Kaidor]].{{tlx|free media}}]]
'''Def.''' &quot;the study of the atmosphere and its phenomena, especially with weather and weather forecasting&quot;&lt;ref name=MeteorologyWikt&gt;{{ cite web
|author=[[wikt:User:CORNELIUSSEON|CORNELIUSSEON]]
|title=meteorology
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=15 June 2006
|url=http://en.wiktionary.org/wiki/meteorology
|accessdate=2013-02-15 }}&lt;/ref&gt; or the &quot;atmospheric phenomena in a specific region or period&quot;&lt;ref name=MeteorologyWikt1&gt;{{ cite web
|author=[[wikt:User:DCDuring|DCDuring]]
|title=meteorology
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=26 March 2009
|url=http://en.wiktionary.org/wiki/meteorology
|accessdate=2013-02-15 }}&lt;/ref&gt; is called '''meteorology'''.

There are four main scales, or sizes of systems, dealt with in meteorology: the macroscale, the synoptic scale, the mesoscale, and the microscale.&lt;ref&gt;{{ cite web
|author=&lt;nowiki&gt;Mesoscale Dynamics and Modeling Laboratory&lt;/nowiki&gt;
|title=Part I: Introduction to Mesoscale Dynamics
|accessdate=2006-12-04
|date=2006-09-08
|url = https://web.archive.org/web/20060908102254/http://mesolab.meas.ncsu.edu/~linyl/mea713/Ch1_Note.doc }}&lt;/ref&gt;

The macroscale deals with systems with global size, such as the Madden–Julian oscillation.&lt;ref name=Arctic/&gt;

Synoptic scale systems cover a portion of a continent, such as extratropical cyclones, with dimensions of {{convert|1,000|-|2,500|km|mi|abbr=on}} across.&lt;ref name=Arctic&gt;{{ cite web
|author=&lt;nowiki&gt;Arctic Climatology and Meteorology&lt;/nowiki&gt;
|date=2006
|title=Synoptic Scale
|accessdate=2006-10-25
|url=https://web.archive.org/web/20060827170650/http://www.nsidc.org/arcticmet/glossary/synoptic_scale.html }}&lt;/ref&gt;

The mesoscale is the next smaller scale, and often is divided into two ranges: meso-alpha phenomena range from {{convert|200|-|2,000|km|mi|abbr=on}} across (the realm of the tropical cyclone), while meso-beta phenomena range from {{convert|20|–|200|km|mi|abbr=on}} across (the scale of the mesocyclone).&lt;ref name=Corporation/&gt;

The microscale is the smallest of the meteorological scales, with a size under {{convert|2|km|mi|abbr=off}} (the scale of tornadoes and waterspouts).&lt;ref name=Corporation&gt;University Corporation for Atmospheric Research. [http://meted.ucar.edu/mesoprim/mesodefn/print.htm Definition of Mesoscale.] Retrieved on 2006-10-25.&lt;/ref&gt;

==See also==
{{div col|colwidth=20em}}
* [[Gases/Gaseous objects/Mercury]]
* [[Gases/Gaseous objects/Venus]]
* [[Gases/Gaseous objects/Earth]]
* [[Gases/Gaseous objects/Mars]]
* [[Gases/Gaseous objects/Saturn]]
* [[Gases/Gaseous objects/Uranus]]
* [[Gases/Gaseous objects/Neptune]]
{{Div col end}}

==References==
{{reflist|2}}

==External links==

&lt;!-- footer templates --&gt;
{{Radiation astronomy resources}}{{Sisterlinks|Aerometeors}}

&lt;!-- footer categories --&gt;
[[Category:Atmospheric sciences/Lectures]]
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==Finding images==
Only images available on Wikimedia Commons can be embedded on Wikiversity pages. You are welcome to contribute images to Commons.

For some starting image ideas, see the [[Motivation and emotion/Gallery|motivation and emotion gallery]].

There are several ways of finding a wider range images to illustrate [[motivation and emotion/Book|motivation and emotion book chapters]]:
# Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
# Search Commons via [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search]&lt;ref&gt;[https://images.google.com/ Google Image search] provides excellent semantic search which is filterable by license (via Tools - Usage rights - Creative Commons). You can get a Wikimedia Commons search by appending &quot;site:commons.wikipedia.org&quot; to the image search term.&lt;/ref&gt;
## Search Commons whilst editing a Wikiversity page, via the Insert - Media option (while using the visual editor)
# [[commons:Commons:Upload|Upload]] free-to-use images to Wikimedia Commons: [[how to find free-to-use images]]
# Create and upload your own images to Wikimedia Commons

==Uploading==
When uploading an image to Wikimedia Commons, make sure:
* you created the image or
* you are uploading an image which has an appropriate free license (e.g., public domain or Creative Commons Attribution)

To upload:
* Go to [[c:|Wikimedia Commons]]
* Click on the blue Upload button
* Answer the series of questions about the image and upload

==Adjust the look and feel==
When embedding an image on a page, the size and position should be adjusted, figures should be numbered and cited, and a descriptive caption should be provided. In addition, make sure the topic that the image illustrates closely connnects with the text.

[[File:Basic needs.png|200px|right|thumb|'''Figure 1'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;* Figure 1 was created by a [[Motivation and emotion|Motivation and emotion]] student in 2020 wh uploadeded it to [[c:|Wikimedia Commons]]. Click on the image to find out more.&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 2'''. Implicit motives affect our thinking, feeling, and behaviour without conscious awareness. This diagram depicts major sources of implicit motives.&lt;ref&gt;* Figure 2 was created by a [[Motivation and emotion|Motivation and emotion]] student in 2020 wh uploadeded it to [[c:|Wikimedia Commons]]. Click on the image to find out more.&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 3'''. Maslow's hierarchy of needs, represented as a pyramid with basic physiological needs at the bottom.]]

* Size
** Figures 2 and 3 have been made larger than the default size, to make them easier to read
* Position
** Usually right-alignment works best, with text flowing down the left
** Centre-alignment works best for important and large images, such as a theoretical model
** Left-alignment is the least common and recommended positioning
* Captions
** Examples of captioned figures are provided in Figures 1, 2, and 3
** The captions are descriptive
*Citations
** Each figure should be referred to at least once in the main body text (e.g., see Figure 1)
* Examples
** Examples correctly presented images: [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|book chapter template]]

==Footnotes==
{{reflist}}

==See also==
* [[Motivation and emotion/Wikiversity/Figures|Figures]] (Motivation and emotion)
* [[../Tables|Tables]]

[[Category:Motivation and emotion/Assessment/Chapter]]</text>
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Tables can be an effective, efficient way of organising and presenting information.

==Examples==
An example of a captioned table is provided in Table 1. Each table should be referred to at least in the main body text. Another example is provided in the [[Template:Motivation_and_emotion/Book_chapter_structure#Tables|book chapter template]]. More examples can be found on this [[Motivation and emotion/Wikiversity/Tables|tables]] page. 

Table 1&lt;br&gt;''Ways to Cultivate Awe in Daily Life''
&lt;div align=&quot;center&quot;&gt;
{| class=&quot;wikitable&quot;
!Strategy
!Try it yourself
|-
|Connect with nature
|Take an [http://ggia.berkeley.edu/practice/awe_walk awe walk] (GGSC)
|-
|Consume awe-inspiring media
|Watch a [https://www.ted.com/talks Ted Talk] or listen to a [[w:Podcast|podcast]]&lt;br&gt;Watch an [http://ggia.berkeley.edu/practice/awe_video#data-tab-how awe-video] (GGSC), or choose from this [https://www.youtube.com/watch?v=9ZfN87gSjvI&amp;index=3&amp;list=PL_T9MO520krq5QsT1sIHdmBUNodksi8v2 YouTube playlist]&lt;br&gt;Watch a [https://www.youtube.com/watch?v=kbJcQYVtZMo flashmob] (YouTube, 5:40 mins)
|-
|Engage with the arts
|Read an [http://ggia.berkeley.edu/practice/awe_story awe story] (GGSC)&lt;br&gt;Visit a museum or gallery&lt;br&gt;Experience live music 
|-
|[[w:Savoring|Savour]] experiences of awe
|Look at photos, talk to other people, or [https://ggia.berkeley.edu/practice/awe_narrative# write about awe] (GGSC)
|}
&lt;/div&gt;

==Examples of chapters which make effective use of tables==
* [[Motivation and emotion/Book/2019/Organisational change motivation#Emerging themes|Organisational change motivation]] (2019)
* [[Motivation and emotion/Book/2019/Stimming motivation|Stimming motivation]] (2019)

==See also==
* [[Motivation and emotion/Wikiversity/Figures|Figures]]

[[Category:Motivation and emotion/Assessment/Chapter]]</text>
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{{Portal-head2|2=Welcome}}
&lt;hr&gt;
&lt;includeonly&gt;{{Shortcut|WV:NOTICE|WV:AN}}&lt;/includeonly&gt;
This page is for posting notices of interest to custodians and [[Wikiversity:Support staff|support staff]]. 

You can '''[[Special:NewSection/Wikiversity:Notices for custodians|create a new notice]].'''&lt;!-- Please sign with &lt;code&gt;&lt;nowiki&gt;--~~~~&lt;/nowiki&gt;&lt;/code&gt;. --&gt;

* Use [[Wikiversity:Chat]] to coordinate activities in real time.
* Go to [[Wikiversity:Request custodian action]] to request admin actions
* Look at [[Special:Unwatchedpages]] and add some pages of interest to your watchlist.
&lt;includeonly&gt;* Review the [[/Archive|archives]] for past discussions.&lt;/includeonly&gt;
{{RoundBoxBottom}}
{{Administering Wikiversity}}&lt;includeonly&gt;
[[Category:Wikiversity administration]]
[[Category:Wikiversity custodianship]]
&lt;/includeonly&gt;</text>
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    <title>User:Faendalimas</title>
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      <text bytes="4211" sha1="g5r3ybpitji3bhlkaxelcwerkq8zs8s" xml:space="preserve">{| class=&quot;wikitable floatright&quot;
|-
|{{Template:User ORCID|0000-0003-1279-2722}}
|-
|{{User custodian}}
|-
|{{User checkuser wikispecies}}
|-
|{{#babel:en}}
|-
|ZooBank Id:&lt;br/&gt;&lt;small&gt;57847BF7-3A48-48E6-87FA-586073B4D66E&lt;/small&gt;
|-
|Home Wiki: [https://species.wikimedia.org/wiki/User:Faendalimas Wikispecies]
|}


== Real Name: [https://species.wikimedia.org/wiki/Scott_Thomson Scott Thomson] ==
*Skype: Faendalimas
*IRC Nick: faendalimas (see me on {{Channel|wikispecies}})

I have spent the last 20 odd years working as a taxonomist and evolutionary biologist at the University of Canberra where I specialised in the systematics and nomenclature of Australasian Turtles. I have described 5 species of turtle, 3 living and 2 fossil, as well as one genus.
I have recently published a complete synonymy of the Australasian Chelidae.
Most of my editing is concerned with Reptiles and Amphibians, specifically Turtles. So I guess I shall explain a little about this. My career in Zoology started when I was employed as shop keeper looking after tropical and marine fish and invertebrates. I did this whilst I began my B.Sc. in Zoology. Afterwards I worked at Notre Dame Zoo and then Western Plains Zoo in Australia.
During my involvement with the Zoo Industry I was involved in the Australian Species Management Plan (ASMP) Reptile and Amphibian TAG. Here I wrote the Management Plan for the Alligator Snapping Turtle and assisted with the management of the Galapagos Tortoises. This led to my becoming the joint International Species Coordinator for the Galapagos Tortoise along with a fellow from a USA Zoo.
I then was taken on to finish my degree at the University of Canberra. Here I worked in the Applied Ecology Institute as a taxonomist from 1994 till 2008. I was fortunate whilst here to describe 7 species of turtles (5 living - ''Chelodina burrungandjii'', ''Chelodina canni'', ''Elseya albagula'', ''Elseya rhodini'', ''Elseya flaviventralis'' and 2 fossil - ''Elseya nadibajagu'' and ''Rheodytes devisi''). I also described one new genus (''Myuchelys'') and a subgenus (''Elseya (Hanwarachelys)'') and developed keys to species of the Chelidae and a full synonomy. This has led to my advisory role with the IUCN and my membership of the Turtle Taxonomy Working Group of the Tortoise and Freshwater Turtle Specialist Group. I take a heavy interest in nomenclatural issues and have published a number of papers in the Bulletin of Zoological Nomenclature, the journal of the ICZN.
From 2009 to 2014 I was living in northern USA and as of February, 2014 am now in Brazil, I am attached to the Museu de Zoologia in Sao Paulo where I am working on the living and fossil Chelid and Pelomedusid turtles of South America, Africa and Australasia. I am also now working hard on nomenclatural issues in Herpetology. I now spend about 3 months of each year in Florida where I am Curator of the Chelonian Research Institute, managing the largest and most comprehensive collection of turtles in the world.

== Personal Interests ==
Beta testing of software, web site development, script writing, computer programming, mod making, computer games, martial arts and of course herpetology.

==Interests on Wikiversity==
I am a current member of the editorial Board for the Wikijournal of Science. I am also interested in administrative policy development and helping this wiki any way I can.

===Lectures Created===
*[[What_is_in_a_Name:_Nomenclature_and_Taxonomy_in_Turtles|What is in a Name: Nomenclature and Taxonomy in Turtles]]: - in depth look at the relationship of taxonomy and nomenclature and how they are utilized by the science of Conservation Biology.
*[[User:Faendalimas/Drafts/Turtle|Turtle]]: Draft zoological contribution on turtles suitable for teaching material.

== Link to My Websites ==
* [http://reptilenames.wordpress.com Reptile Names (Blog)]
* [http://www.carettochelys.com Carettochelys.com]
[http://free2chate.mobie.in/ .]
[http://free2chate.mywibes.com/ .]
[https://eu.explara.com/e/andriettexrrj .]
[http://free2chat.iwopop.com/ .]
[http://aprilettetzns.teampages.com/teams/1899995-Aprilettetzns-basketball-team-website/announcements/2131748-5- .]
[https://androzel.typeform.com/to/tKTWCM .]</text>
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  <page>
    <title>Wikiversity:Interface administrators</title>
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        <username>Codename Noreste</username>
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      <text bytes="5079" sha1="oflqg7wqui5qfjl08298pcy7s1m61ss" xml:space="preserve">{{policy|WV:IA}}
{{2FA-required|enforced=yes}}
[[File:Wikiversity Interface administrator.svg|right|130px|link=]]
A Wikiversity '''interface administrator''' is a trusted user who has the technical ability to edit the [[w:Wikipedia:User scripts|JavaScript]] and [[w:Help:Cascading Style Sheets|CSS]] of all users.

== What can interface administrators do? ==
Interface administrators may use their rights as described below:&lt;ref group=n&gt;Interface administrators may '''never''' [[:w:WP:WHEEL|wheel war]] or [[:w:WP:Edit warring|edit war]] using their rights, even if they are editing according to policy. Doing so is grounds for immediate removal of access. Instead, open a discussion.&lt;/ref&gt;

{| class=&quot;wikitable&quot;
|+Interface administrator rights
! scope=&quot;col&quot; | Right
! scope=&quot;col&quot; | Description
! scope=&quot;col&quot; | Can be used for
|-
|scope=&quot;row&quot;| &lt;code&gt;editusercss&lt;/code&gt;
| Edit other users' CSS files
| rowspan=&quot;2&quot; |
*To perform uncontroversial maintenance
*To edit user scripts that are used by others, if the owner is inactive and unresponsive&lt;ref group=n&gt;Interface administrators are ''not'' permitted to edit a user's pages against the wishes of the user in question&lt;/ref&gt;
|-
|scope=&quot;row&quot;| &lt;code&gt;edituserjs&lt;/code&gt;
| Edit other users' JavaScript files
|-
|scope=&quot;row&quot;| &lt;code&gt;editsitecss&lt;/code&gt;
| Edit sitewide CSS
| rowspan=&quot;2&quot; |
*To perform uncontroversial maintenance
*To edit sitewide gadgets, following consensus&lt;ref group=n&gt;Or, in lower-stakes cases, no objections&lt;/ref&gt; regarding the edits
|-
|scope=&quot;row&quot;| &lt;code&gt;editsitejs&lt;/code&gt;
| Edit sitewide JavaScript
|-
|scope=&quot;row&quot;| &lt;code&gt;edituserjson&lt;/code&gt;
| Edit other users' JSON files
| rowspan=&quot;3&quot; | 
*These rights are already granted to custodians, and may be used in line with custodian guidelines
|-
|scope=&quot;row&quot;| &lt;code&gt;editsitejson&lt;/code&gt;
| Edit sitewide JSON
|-
|scope=&quot;row&quot;| &lt;code&gt;editinterface&lt;/code&gt;
| Edit the user interface&lt;ref group=n&gt;Specifically, this grants the ability to edit normal pages in the &lt;code&gt;MediaWiki:&lt;/code&gt; namespace.&lt;/ref&gt;
|-
|scope=&quot;row&quot;| &lt;code&gt;oathauth-enable&lt;/code&gt;
| Enable two-factor authentication
| 
*All interface administrators are required to enable two-factor authentication&lt;ref group=n&gt;Custodians also have the &lt;code&gt;oathauth-enable&lt;/code&gt; right, and should activate two-factor authentication ''before'' requesting interface administrator rights.&lt;/ref&gt;
|}
{{reflist|group=n}}

== Access ==
===How does one become an interface administrator?===
[[Wikiversity:Bureaucratship|Bureaucrats]] have the technical ability to grant interface administrator access, but should do so only in the manner described here.

* For now, Wikiversity does not have a need for permanent or long term interface administrators. Accordingly, interface administratorship '''may only be granted temporarily''' by bureaucrats (between 14 and 120 days based on discussion, at the discretion of that bureaucrat and perceived need)
* Bureaucrats '''may not grant themselves interface adminship''' - it must be granted by a different bureaucrat
** Exception: If no other bureaucrats are available within a reasonable amount of time, and other uninvolved [[Wikiversity:Support staff|support staff]] agree that the request is reasonable, a bureaucrat may grant themselves the rights
* Interface adminship should not be granted to non-support staff (non-custodian, non-curator) without prior discussion
* Users seeking interface adminship must enable two-factor authentication.

===Losing access===
Bureaucrats have the technical ability to remove interface administrator access. They should do so in the manner described here, but in an emergency are empowered to act with discretion.

* A bureaucrat may, without prior discussion, revoke interface adminship if it is being used to edit against the community's wishes, or otherwise being used improperly. The bureaucrat must then open a discussion.
* A bureaucrat may, after prior discussion, revoke interface adminship if there is consensus among support staff that it should be revoked.
* A bureaucrat may, at the request of any interface administrator, revoke their interface adminship
* Interface administrators may revoke their own interface adminship

===Discussions===
* Requests for interface adminship, and discussions regarding revoking such rights, should be made publicly in well-watched areas, such as at [[Wikiversity:Notices for custodians]] or [[Wikiversity:Request custodian action]]
* If a bureaucrat removes interface administrator rights without discussion in an emergency, they must promptly initiate a discussion for the community to review their actions
** The bureaucrat must leave a note with the user in question, alerting them to the removal of access and the discussion
** The bureaucrat must explain ''why'' they have removed access
** While a discussion is open, another bureaucrat may not restore the rights

== See also ==
* [[:meta:Interface administrators]]
* [[:meta:Help:Two-factor authentication]]
* [[Special:ListGroupRights]]

{{Official policies}}
[[Category:Wikiversity administration]]
[[Category:Wikiversity user roles]]</text>
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  <page>
    <title>Social Victorians/People/Feversham</title>
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      <text bytes="35379" sha1="oppz802jkhxbma5hd26gbgepr29cy8j" xml:space="preserve">== Overview ==
[[File:Vincent E Vanity Fair 1899-04-20.jpg|thumb|alt=Old colored drawing of a man in a 19th-century black suit with grey and black striped trousers standing very erect, his hands behind his back and a full beard and moustache, looking to his left|&quot;Eastern finance&quot; (Sir Edgar Vincent) ''Vanity Fair'', 20 April 1899]]
[[Social Victorians/People/Helmsley | Viscount Helmsley]] was the courtesy title for the eldest son and heir apparent of the Earl of Feversham (during the second half of the 19th century).

The people who attended the [[Social Victorians/1897 Fancy Dress Ball |Duchess of Devonshire's fancy-dress ball]] from this family are the Earl and Countess of Feversham, their 2 youngest daughters and their husbands. Probably one daughter was misidentified in the ''Lady's Pictorial'', so the 3 youngest daughters were present. Helen Vincent, Cynthia Graham and Ulrica Duncomble were sisters of William Duncombe, Viscount Helmsley, who died in 1881, so the Viscount Helmsley at the ball was his son, Charles Duncombe, the sisters' nephew. Charles's mother Muriel was also present.

== Acquaintances, Friends and Enemies ==

== Timeline ==
'''1851 August 7''', William Duncombe (at that time 2nd Baron Feversham of Duncombe Park) and Mabel Graham married.&lt;ref name=&quot;:0&quot;&gt;&quot;Mabel Violet Graham.&quot; {{Cite web|url=https://thepeerage.com/p2288.htm#i22879|title=Person Page|website=thepeerage.com|access-date=2020-11-23}}&lt;/ref&gt;

'''1881 July 14, Thursday afternoon, beginning about 2 p.m.''', William, Earl of Feversham, Mabel, Countess of Feversham and Lady Hermione Duncombe were invited to a [[Social Victorians/1881-07-14 Garden Party|Garden Party at Marlborough House]] hosted by [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]].

'''1881 July 22, Friday''', William, Earl of Feversham, Mabel, Countess of Feversham and Lady Hermione Duncombe were invited to — and likely attended — [[Social Victorians/1881-07-22 Marlborough House Party|the party at Marlborough House]] hosted by [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]].

'''1882 July 13, Thursday''', William, Earl of Feversham, Mabel, Countess of Feversham and Lady Hermione Duncombe were invited to a [[Social Victorians/1882-07-13 Marlborough House Garden Party|Garden Party at Marlborough House for Queen Victoria]] hosted by the [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]].

'''1884 July 03''', William, Earl of Feversham and Mabel, Countess of Feversham attended [[Social Victorians/1884-07-03 Munster Reception|Count Münster's Reception at the German Embassy]], Carlton House Terrace.

'''1886 July 21, Wednesday''', the Earl and Countess of Feversham and the Ladies Duncombe were invited to — and likely attended — [[Social Victorians/1886-07-21 Marlborough House Ball|the Ball at Marlborough House]] hosted by [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]].

'''1888 March 8''', Sir Richard James Graham's father died, so he succeeded as the 4th Baronet Graham of Netherby.&lt;ref&gt;&quot;Sir Frederick Ulric Graham, 3rd Bt.&quot; {{Cite web|url=https://thepeerage.com/p5396.htm#i53954|title=Person Page|website=thepeerage.com|access-date=2020-11-23}}&lt;/ref&gt;

'''1889 June 27''', Lady Cynthia Duncombe and Sir Richard James Graham, 4th Baronet of Netherby married.

'''1890 September 24''', Lady Helen Venetia Duncombe and Edgar Vincent married.&lt;ref name=&quot;:1&quot;&gt;&quot;Lady Helen Venetia Duncombe.&quot; {{Cite web|url=https://thepeerage.com/p23311.htm#i233108|title=Person Page|website=thepeerage.com|access-date=2020-11-23}}&lt;/ref&gt;

'''1891 July 9, Thursday''', William, Earl of Feversham seems to have been invited to a [[Social Victorians/1891-07-09 Garden Party|Garden Party at Marlborough House]] hosted by [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]], to which about 3,000 people were invited.

'''1892 May 18, Wednesday''', Mabel, Countess of Feversham attended [[Social Victorians/Timeline/1892#18 May 1892, Wednesday18 May 1892, Wednesday|the Queen's Drawing-room at Buckingham Palace]] and presented Lady Ulrica Duncombe to her Royal Highness Princess Christian of Schleswig-Holstein, who held the drawing-room on behalf of Queen Victoria.

'''1894 July 19, Thursday''', William, Earl of Feversham and Lady Ulrica Duncombe attended [[Social Victorians/Timeline/1894#19 July 1894, Thursday|a ball hosted by the Duke and Duchess of Devonshire at Devonshire House that followed a dinner for the Prince and Princess of Wales]], some of their children, the Russian Ambassador, the Portuguese Minister [is this de Soveral?] and a few British dignitaries and aristocratic friends and family.

'''1897 June 28, Monday''', William, Earl of Feversham and Mabel, Countess of Feversham were invited to [[Social Victorians/Diamond Jubilee Garden Party|Queen Victoria's immense Diamond Jubilee garden party at Buckingham Palace]].

'''1897 July 2, Friday''', Lady Helen and Sir Edgar Vincent attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House as did Lord and Lady Feversham, the Earl and Countess Feversham and an as-yet-unidentified Lady Alicia Duncombe. Sir R. and Lady C. Graham were also present.

'''1897 July 31, Saturday''', William, Earl of Feversham and Mabel, Countess of Feversham gave Mabel Wombwell a &quot;silver-gilt inkstand and candlesticks&quot;&lt;ref&gt;&quot;Marriage of Mr. H. R. Hohler and Miss Wombwell.&quot; ''Morning Post'' 2 August 1897, Monday: 6 [of 8], Col. 3a–c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970802/067/0006 (accessed June 2019).&lt;/ref&gt; for [[Social Victorians/Timeline/1897#31 July 1897, Saturday|her wedding to Henry R. Hohler]].

'''1899 April 20''', a caricature portrait (above right) by Leslie Ward (&quot;Spy&quot;) of &quot;Eastern Finance&quot; (Sir Edgar Vincent) appeared in this issue of ''Vanity Fair'', as Number 746 in its &quot;Men of the Day&quot; series.&lt;ref&gt;{{Cite journal|date=2024-01-14|title=List of Vanity Fair (British magazine) caricatures (1895–1899)|url=https://en.wikipedia.org/w/index.php?title=List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899)&amp;oldid=1195518024|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899).&lt;/ref&gt; (Note the differences between the figure and the shadow in this caricature.)

'''1926 February 20''', Edgar Vincent was created 1st Viscount D'Abernon, of Esher and Stoke D'Abernon, County Surrey.&lt;ref name=&quot;:2&quot;&gt;&quot;Edgar Vincent, 1st and last Viscount D'Abernon.&quot; {{Cite web|url=https://thepeerage.com/p23310.htm#i233094|title=Person Page|website=thepeerage.com|access-date=2020-11-23}}&lt;/ref&gt;

'''1936 March 2''', Edgar Vincent succeeded as the 16th Baronet Vincent, of D'Abernon, County Surrey.&lt;ref name=&quot;:2&quot; /&gt;

[[File:Helen-Venetia-ne-Duncombe-Viscountess-DAbernon-as-a-Genoese-Lady-after-Vandyck.jpg|thumb|alt=Black-and-white photograph of a standing woman richly dressed in an historical costume with a tiara and a black feather plume on top of her head|Helen Vincent as a Genoese Lady, after Vandyck. ©National Portrait Gallery, London.]]
== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==

=== William, Earl of Feversham and Mabel, Countess of Feversham ===
William Ernest Duncombe, 1st Earl of Feversham and Mabel Violet Graham Duncombe, Countess Feversham were present at the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], as were their daughters Lady Helen Vincent and Lady Cynthia Graham and their husbands. Nothing is known about the costumes of the Earl and Countess of Feversham.

=== Lady Helen Vincent ===
[[File:Den Haag - Mauritshuis - Anthony van Dyck (1599-1641) - Portrait of Anna Wake (1605-before 1669), wife of Peter Stevens 1618.jpg|thumb|left|alt=Old portrait of a woman richly dressed in black and white, with jewelry, in a gold frame|Portrait of Anna Wake, wife of Peter Stevens, by Antony Van Dyke (1618)]]
Lady Helen Vincent sat at Table 12 for the first seating for supper and was dressed as Contessa Valentina Gateago in the 17th-century procession.&lt;ref name=&quot;:3&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt;&lt;ref name=&quot;:4&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt; Lady Helen's high status among the group of people attending the ball is revealed by her presence in the first supper seating. 

Henry Van der Weyde's portrait (above right) of &quot;Helen Venetia (née Duncombe), Viscountess D'Abernon as a Genoese Lady, after Vandyck&quot; in costume is photogravure #83 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:5&quot;&gt;&quot;Devonshire House Fancy Dress Ball (1897): photogravures by Walker &amp; Boutall after various photographers.&quot; 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.&lt;/ref&gt; The printing on the portrait says, &quot;Lady Helen Vincent as a Genoese Lady, after Vandyck.&quot;&lt;ref&gt;&quot;Helen Venetia (née Duncombe), Viscountess D'Abernon as a Genoese Lady, after Vandyck.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158441/Helen-Venetia-ne-Duncombe-Viscountess-DAbernon-as-a-Genoese-Lady-after-Vandyck.&lt;/ref&gt;

Van Dyke's 1628 portrait of Anna Wake (left) does not look like the original of Lady Helen Vincent's dress, but it shows the painter's treatment of a similar subject.

==== Commentary on Lady Vincent's Costume ====

* No newspapers described or commented on Lady Helen's dress.
* Lady Vincent's dress is a hodgepodge of elements, many Victorian but with an approximately 17th-century collar and ruffled peplum. The waist is the most notable Victorian element. The ruffles (or little puffs) at the bottom of the bodice and the pearl belt emphasize and flatter her waist, as do the broad shoulders and collar. Similar ruffles (or little puffs or ruches) also appear at the neckline.
* Lady Helen's sleeves are Victorian in how short and high they are. Although the slashed puff is a 17th-century element, its silhouette echoes the shape of sleeves popular in the 1890s. The treatment of the sleeve below the single puff is odd, difficult to know what on earth the designer was thinking, how it was constructed and what keeps it above the elbow.
* Lady Helen has pulled her skirts to the front on both sides for the photograph, distorting the front panel of the skirt slightly. The skirt appears to have stripes made by stitching strips of the same satin fabric cut from the crosswise grain, which gives this very plain skirt more texture. The center piece of the skirt is reminiscent of an underskirt. This black-and-white photograph is too dark to permit clear analysis of the features of the skirt.
* The border at the bottom of the skirt and train is stiffened — probably with horsehair — preventing the fabric from hanging straight down, resulting in an A-line. In the 1890s,&lt;blockquote&gt;Skirts were lined with cambric or taffeta and trained gowns were weighted and disciplined by facings of horsehair which might be as deep as eighteen inches at center back.&lt;ref&gt;Payne, Blanche. ''History of Costume: From the Ancient Egyptians to the Twentieth Century''. Harper &amp; Row, 1965.&lt;/ref&gt; (532)&lt;/blockquote&gt;
* This costume lacks the sophistication that would have been present in a dress designed by [[Social Victorians/People/Dressmakers and Costumiers#Mrs. Mason|Mrs. Mason]], for example, [[Social Victorians/People/Dressmakers and Costumiers#Mr. Charles Alias|Mr. Charles Alias]] or the [[Social Victorians/People/Dressmakers and Costumiers#The House of Worth|House of Worth]]. Aesthetically, the [[Social Victorians/Terminology#Frou-frou|frou-frou]] on the top is not balanced by the simplicity of the design on the skirt and train, although, because of the stripes the costume might have looked more interesting in motion than it does in this photograph.
*The photograph appears to have been retouched on the right side of Lady Helen's waist, under her right arm, a common practice.
*Lady Helen's headdress looks like a crown because of the points made by the pearls. A single black plume rises straight up from the center of the headdress.
*Lady Helen's jewelry is primarily strands of pearls with two brooch ornaments, one pendant from one of the necklaces and the other at the center of the neckline of her bodice. Besides the several strands of pearls at her neck and on her headdress are pearls on her sleeves and at her waist.
*Lady Vincent's jewels do not display the kind of wealth that someone like the Duchess of Devonshire or Mrs. Arthur Paget, for example, had.
* The wired collar should be standing up behind her head to frame her face, but the wires cannot hold up the center back because of the cut of the lace, which should have been attached differently.

[[File:Edgar-Vincent-Viscount-dAbernon-as-a-Dutch-Stadtholder-after-Frans-Hals.jpg|thumb|alt=Black-and-white photograph of a standing man richly dressed in an historical costume with a large ruff around his neck, a large hat, and a sword|Edgar Vincent as a Dutch Stadtholder, after Frans Hals. ©National Portrait Gallery, London.]]
=== Sir Edgar Vincent ===
[[File:Frans Hals 042.jpg|thumb|left|alt=Old portrait of a proud gentleman with a big white ruff, big hat, and sword|Frans Hals, ''Willem van Heythuyzen'']]
According to the newspapers, Sir Edgar Vincent was dressed as II Conte Oravio&lt;ref name=&quot;:4&quot; /&gt; or Orayio&lt;ref name=&quot;:3&quot; /&gt; in the 17th-century procession. He is not listed as having been in the first supper seating although Lady Helen Vincent is.

Henry Van der Weyde's portrait (right) of &quot;Edgar Vincent, Viscount d'Abernon as a Dutch Stadtholder after Frans Hals&quot; in costume is photogravure #84 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:5&quot; /&gt; The printing on the portrait says, &quot;Sir Edgar Vincent as a Dutch Stadtholder after Frans Hals.&quot;&lt;ref&gt;&quot;Edgar Vincent, Viscount d'Abernon as a Dutch Stadtholder after Frans Hals.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158442/Edgar-Vincent-Viscount-dAbernon-as-a-Dutch-Stadtholder-after-Frans-Hals.&lt;/ref&gt;

Van der Weyde's photograph of Sir Edgar Vincent is similar enough to Frans Hals's 1625-1630? portrait of Willem van Heythuyzen (left) that Hals's seems to be the original. Sir Edgar Vincent is striking a very similar pose, and even the photographer's drapery and set seem to refer to the Hals painting.

==== Commentary on Sir Edgar Vincent's Costume ====
The photograph of Sir Edgar is a close copy of the portrait of Willem van Heythuyzen by Frans Hals, but the clothing worn by the Victorian has been modified, as always, for the people at this ball, to accommodate standards of beauty contemporary to their own time. The painting is very dark, affecting our sense especially of the black-on-black details.

* In spite of the similarity between the two portraits, the doublet worn by Sir Edgar reflects Victorian rather than Elizabethan fashion.
* Sir Edgar's collar is not stiffened. The folds are more limp, suggesting a [[Social Victorians/Terminology#Cavalier|Cavalier]] collar, unlike the stiffened folds on the Hals portrait. But more important is that the collar in the Hals portrait has a lot of fabric, which alone can account for the fullness. Sir Edgar's collar may be starched, but it lies flatter because the costumier used so much less fabric.
* The ornament below the collar on Sir Edgar is large and probably made of lace, as is van Heythuyzen's. We cannot tell what it is or what it symbolizes.
* The fabric used for Sir Edgar's doublet and knee breeches appears to be textured, possibly a brocade or a velvet brocade. While the cloak is black like the doublet and breeches, the fabric is a more subtle, less textured brocade. Yet another fabric was used for the lining of the cloak. The textures in the fabrics are what makes this costume so sophisticated: the color is all the same.
* Sir Edgar's sleeves were made to look like they were tied to the doublet, as Elizabethan sleeves would be, but were probably sewn to it.
* The bodice of Sir Edgar's doublet is not stiffened and pointed, which changes the line of the garment, making it looser and more Victorian than Elizabethan.
* The level (rather than pointed) bodice changes the waistline and the [[Social Victorians/Terminology#Peplum|peplum]] as well.
* The garments in both portraits have decorated belts or braid at the waist. Aglets are suspended from ribbon at the waistline on both portraits.
* Sir Edgar's knee breeches and sleeves are full, so they might be padded.
* Sir Edgar's white cuffs fold back from the wrists and have tiny starched pleats and lace edging (like the cuffs in Van den Weyde's portrait), but they are not as stiffly starched. The tiny tucks or pleats in van Heythuyzen's cuffs give them stiffness and texture; Sir Edgar's cuffs are looser and less controlled.
* The buttons on the sides of the breeches look decorative rather than functional.
* The ornament at the bottom of the knee breeches actually appears to be similar in size in both portraits, but Sir Edgar's is a simple bow that is less decorative than what looks like lacy, beaded trim on van Heythuyzen.
* The shoes are dominated by the bows, which may be velvet, in the Hals portrait. Sir Edgar's bows are placed below the tongue and are smaller.
* Sir Edgar's shoes have flat heels, and the tongue rises above the bow. Van Heythuyzen's shoes appear to have wooden pattens beneath the soles.
* The metal tips attached to ribbons at the waists of the men in both portraits are [[Social Victorians/Terminology#Aglet, Aiglet|aglets or aiglets]]. Historically, breeches could be tied to the doublet with ribbons or cords whose ends were tipped with aglets. Sir Edgar's ribboned aglets are definitely decorative, but it is not clear whether Van Heythuyzen's are decorative or functional.
* Sir Edgar and Van Heythuyzen are carrying ornate cavalier rapiers. Early cavalier rapiers were long like these are, later becoming smallswords. In the portraits, the rapiers are in scabbards. Hanging from the waist of Sir Edgar's doublet is a rapier belt to hold the rapier in its scabbard. Van Heythuyzen's scabbard is quite ornate, but Sir Edgar's is simple. Both rapiers have very ornate hand guards, which is what makes them look like cavalier weapons.
* The two swords — especially the hand guards — are so like each other, did Sir Edgar find the same sword? or have this one made? Is the sword in a collection somewhere?

==== The Historical William van Heythuyzen ====
While the ''Times'' and the ''Morning Post'' say that Sir Edgar Vincent was in the 17th-century Italian procession, the description in the [[Social Victorians/1897 Fancy Dress Ball/Photographs#The Album of Photographs|commemorative album]] associates his costume with a painting rather than a person. The man in the painting is Willem van Heythuyzen, Dutch cloth merchant and , dressed in early [[Social Victorians/Terminology#Cavalier|Cavalier style]].&lt;ref name=&quot;:10&quot;&gt;{{Cite journal|date=2023-08-27|title=Willem van Heythuysen|url=https://en.wikipedia.org/w/index.php?title=Willem_van_Heythuysen&amp;oldid=1172477813|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Willem_van_Heythuysen.&lt;/ref&gt; Van Heythuyzen was the founder of Hofje van Willem Heythuijsen. (A hofje is a group of almshouses surrounding an open courtyard in which poor, elderly people, especially women, can live.&lt;ref&gt;{{Cite journal|date=2023-08-09|title=Hofje|url=https://en.wikipedia.org/w/index.php?title=Hofje&amp;oldid=1169559641|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Hofje.&lt;/ref&gt;) Hofje van Willem Heythuijsen — the hofje founded by Willem van Heythuyzen — is still in existence.&lt;ref name=&quot;:10&quot; /&gt;

=== Lady Cynthia Graham and Sir Richard Graham ===
Lady Cynthia Graham of Netherby and [[Social Victorians/People/Pless|Princess Henry of Pless]] were dressed as the Queen of Sheba and led the [[Social Victorians/1897 Fancy Dress Ball/Quadrilles Courts#&quot;Oriental&quot; Procession|&quot;Oriental&quot; Procession]].&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:4&quot; /&gt;{{rp|p. 7, Col. 5b}} At this time, no photograph of Lady Cynthia Graham in this costume exists. (Lady Cynthia Graham is the Earl of Feversham's youngest daugther and Sir Richard Graham's second wife.)

==== Newspaper Accounts ====
Three actual accounts of Lady Cynthia's costume exist, and two are reprinted. They are not written by fashion journalists, so what her costume looked like is difficult to imagine.
* Lady Cynthia Graham &quot;was in white satin and gauze, embroidered in gold and silver and bright rose.&quot;&lt;ref&gt;&quot;Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses.&quot; London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and https://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.&lt;/ref&gt;{{rp|p. 5, Col. 7c}}
* &quot;Lady Cynthia Graham appeared as Queen of Sheba, in a robe of white Bengal satin and gauze, with embroidery of gold appliqué, satin white and cerise. The manteau was of crepon de chine, covered with embroidered gauze and appliqué of coloured satin, and studded with jewels; a ceinture and pendant were of white satin, with cerise appliqué and embroidery, and she wore a jewelled headdress.&quot;&lt;ref&gt;“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.&lt;/ref&gt;{{rp|p. 3, Col. 3c}}
* &quot;Lovely Lady Cynthia Graham was one [Queen of Sheba], in white satin embroidered in gold and silver and bright rose.&quot;&lt;ref&gt;“Girls’ Gossip.” ''Truth'' 8 July 1897, Thursday: 41 [of 70], Col. 1b – 42, Col. 2c. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0002961/18970708/089/0041.&lt;/ref&gt;{{rp|42, Col. 1b}}
* According to the ''Carlisle Patriot'', reprinting the ''Evening Standard'' description (perhaps because Lady and Lord Graham were local), &quot;Lady Cynthia Graham of Netherby also personated the famous Eastern Queen, wearing a lovely robe of white Bengal satin and gauze, with embroidery of gold applique, satin white and cerise. The manteau was of crepon de chine, covered with embroidered gauze and applique of coloured satin, and studded with jewels; a ceinture and pendent were of white satin, with cerise applique and embroidery, and she wore a jewelled headdress.&quot;&lt;ref&gt;&quot;Fancy Dress Ball: Unparalleled Splendour.&quot; ''Carlisle Patriot'' Friday 9 July 1897: 7 [of 8], Col. 4a–b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000365/18970709/084/0007.&lt;/ref&gt;
* &quot;The other Queen of Sheba, who was Lady Cynthia Graham, was charmingly attired in white and silver and rose red.&quot;&lt;ref&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 32, Col. 2c}}

Lady Cynthia Graham's original costume appeared in the Drury Lane production of ''The White Heather''.&lt;ref&gt;&quot;The Morning’s News.&quot; London ''Daily News'' 18 September 1897, Saturday: 5 [of 8], Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970918/027/0005.&lt;/ref&gt;

[[File:Sir Edward John Poynter - The visit of the Queen of Sheba to King Solomon - Google Art ProjectFXD.jpg|thumb|alt=Large oil painting showing a woman climbing some shallow steps to a man standing at the top in a commanding pose, both dressed in flowing robes|Sir Edward Poynter, ''The Visit of the Queen of Sheba to King Solomon'']]
==== The Queen of Sheba ====
Stories about the African Queen of Sheba appear in Jewish, Christian and Islamic traditions. She visited King Solomon with gifts and tested his wisdom. The [[Social Victorians/Victorian Things#Encyclopaedia Britannica|9th edition of the ''Encyclopaedia Britannica'']] does not have an article about the Queen of Sheba, although she figures in other, historical articles, like the one on Yemen. 

Sir Edward John Poynter's 1890 ''The Visit of the Queen of Sheba to King Solomon'' (right) is in the collection of the Art Gallery of New South Wales, which accessioned it in 1892, so it would have been available for viewing until then. The Queen of Sheba's clothing here, such as there is of it, is unlikely to have been an original for the costumes worn by Lady Cynthia Graham or Daisy, Princess Pless, but her headdress has some similarities to the one worn by [[Social Victorians/People/Goelet|May Goelet]] dressed as Scheherazade.

=== Alicia Duncombe ===
Lady Alicia Duncombe came dressed as a Greek Slave and walked in the &quot;Oriental&quot; procession.&lt;ref name=&quot;:32&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt;&lt;ref name=&quot;:42&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;

Besides being mentioned twice in connection with the ball, Lady Alicia Duncombe is mentioned only once in the newspapers in the 1890s–1900s. This report from the ''Lady's Pictorial'' does not seem to be correct: Lady Helen Vincent and Lady Cynthia Graham had a sister named Ulrica, but not one named Alicia:&lt;blockquote&gt;The Earl and Countess of Feversham are at Duncombe Park, Helmsley, Yorkshire, where they will have house parties throughout the month for shooting. The Duke of Cambridge is to pay them a visit: was expected there indeed this week. Lord and Lady Feversham are the parents of that family of beautiful daughters of whom the late Duchess of Leinster was the eldest. The others are Lady Helen Vincent, Lady Cynthia Graham, and Lady Alicia Duncombe. Of their three sons one alone survives, Major the Hon. Hubert Duncombe, D.S.O. Their eldest son married and left a son, the present Viscount Helmsley. Duncombe Park has twice been burnt down. On the last occasion of a fire there Lord Feversham’s grandson, the young Duke of Leinster, was only rescued with difficulty. His Grace was on a visit to his grandparents with his two brothers, and the children were only just got away in time. The Duke of Leinster is now in his sixteenth year, but is, unfortunately, not a robust lad.&lt;ref&gt;&quot;Society Notes.&quot; ''Lady's Pictorial'' 13 September 1902, Saturday: 353 [print; 43 of 54]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005980/19020913/124/0043.&lt;/ref&gt;&lt;/blockquote&gt;Ulrica, who was a sister of Lady Helen Vincent and Lady Cynthia Graham, did not marry until 1904, and she is not mentioned anywhere as having attended the ball, so a strong possibility would be that the ''Lady's Pictorial'' got the name wrong, and Alicia Duncombe was actually Ulrica Duncombe.

== Demographics ==

=== Nationality ===
*British

=== Residences ===

* Lady Cynthia and Sir Richard Graham: Netherby Hall in the Carlisle district of Cumbria (which is why the ''Carlisle Patriot'' coverage is so thorough)&lt;ref&gt;{{Cite journal|date=2021-05-08|title=Arthuret|url=https://en.wikipedia.org/w/index.php?title=Arthuret&amp;oldid=1022099353|journal=Wikipedia|language=en}} [[wikipedia:Arthuret|https://en.wikipedia.org/wiki/Arthuret#Netherby Hall]].&lt;/ref&gt;

== Family ==
*Charles Duncombe, 1st Baron Feversham of Duncombe Park (5 December 1764 – 16 July 1841)&lt;ref name=&quot;:8&quot;&gt;&quot;Charles Duncombe, 1st Baron Feversham of Duncombe Park.&quot; {{Cite web|url=https://www.thepeerage.com/p2576.htm#i25757|title=Person Page|website=www.thepeerage.com|access-date=2020-11-23}}&lt;/ref&gt;
*Lady Charlotte Legge ( – 5 November 1848)&lt;ref&gt;&quot;Lady Charlotte Legge.&quot; {{Cite web|url=https://www.thepeerage.com/p2576.htm#i25758|title=Person Page|website=www.thepeerage.com|access-date=2020-11-23}}&lt;/ref&gt;
#Hon. Frances Duncombe (– 15 June 1881)
#Hon. Louisa Duncombe ( – 18 November 1852)
#Charles Duncombe (1795 – 1819)
#'''William Duncombe, 2nd Baron Feversham of Duncombe Park''' (14 January 1798 – 11 February 1867)
#Reverend Henry Duncombe (25 August 1800 – 1 October 1832)
#Admiral Hon. Arthur Duncombe (24 March 1806 – 6 February 1889)
#Very Rev. Augustus Duncombe (2 November 1814 – 26 January 1880)
#Hon. Octavius Duncombe (8 April 1817 – 3 December 1879)


*William Duncombe, 2nd Baron Feversham of Duncombe Park (14 January 1798 – 11 February 1867)&lt;ref name=&quot;:9&quot;&gt;&quot;William Duncombe, 2nd Baron Feversham of Duncombe Park.&quot; {{Cite web|url=https://www.thepeerage.com/p1242.htm#i12415|title=Person Page|website=www.thepeerage.com|access-date=2020-11-23}}&lt;/ref&gt;
*Lady Louisa Stewart ( – 5 March 1889)&lt;ref&gt;&quot;Lady Louisa Stewart.&quot; {{Cite web|url=https://www.thepeerage.com/p1348.htm#i13478|title=Person Page|website=www.thepeerage.com|access-date=2020-11-23}}&lt;/ref&gt;
#Hon. Gertude Duncombe ( – 24 February 1916)
#Hon. Jane Duncombe ( – 3 April 1901)
#Hon. Helen Duncombe ( – 22 November 1896)
#Hon. Albert Duncombe (11 February 1826 – 14 September 1846)
#'''William Ernest Duncombe, 1st Earl Feversham of Ryedale''' (28 January 1829 – 13 January 1915)
#Hon. Cecil Duncombe (27 May 1832 – 20 May 1902)


*William Ernest Duncombe, 1st Earl of Feversham (28 January 1829 – 13 January 1915)&lt;ref name=&quot;:6&quot;&gt;&quot;William Ernest Duncombe, 1st Earl of Feversham of Ryedale.&quot; {{Cite web|url=https://thepeerage.com/p1873.htm#i18721|title=Person Page|website=thepeerage.com|access-date=2020-11-22}}&lt;/ref&gt;
*Mabel Violet Graham Duncombe (15 February 1833 – 28 August 1915)&lt;ref name=&quot;:0&quot; /&gt;
#William Reginald Duncombe, [[Social Victorians/People/Helmsley | Viscount Helmsley]] (1 August 1852 – 24 December 1881)
#Hon. James Henry Duncombe (20 October 1853 – 10 January 1886)
#Hon. Hubert Ernest Valentine Duncombe (14 February 1862 – 21 October 1918)
#Lady Hermione Wilhelmina Duncombe (30 March 1864 – 19 March 1895)
#'''Lady Helen Venetia Duncombe''' (1866 – 16 May 1954)
#'''Lady Cynthia (Mabel Cynthia) Duncombe''' (1869 – 25 April 1926)
#'''Lady Ulrica Duncombe''' (1874? [based on presentation at Queen's drawing-room May 1892] – 27 April 1935)


*Lady Helen Venetia Duncombe ( – 16 May 1954)&lt;ref name=&quot;:1&quot; /&gt;
*Edgar Vincent, 1st and last Viscount D'Abernon (19 August 1857 – 1 November 1941)&lt;ref name=&quot;:2&quot; /&gt;


* Sir Richard James Graham, 4th Bt. (24 February 1859 – 26 August 1932)&lt;ref&gt;&quot;Sir Richard James Graham, 4th Bt..&quot; {{Cite web|url=https://thepeerage.com/p7148.htm#i71471|title=Person Page|website=thepeerage.com|access-date=2020-11-23}}&lt;/ref&gt;
* Olivia Baring (14 May 1863 – 21 March 1887)&lt;ref&gt;&quot;Olivia Baring.&quot; {{Cite web|url=https://thepeerage.com/p7148.htm#i71472|title=Person Page|website=thepeerage.com|access-date=2020-11-23}}&lt;/ref&gt;
* Lady Cynthia (Mabel Cynthia) Duncombe (1869 – 25 April 1926)&lt;ref&gt;&quot;Lady Mabel Cynthia Duncombe.&quot; {{Cite web|url=https://thepeerage.com/p1604.htm#i16038|title=Person Page|website=thepeerage.com|access-date=2020-11-23}}&lt;/ref&gt;
*# Lt.-Col. Sir Fergus Frederick Graham, 5th Bt. (10 March 1893 – 1 August 1978)
*# Richard Preston Graham-Vivian (10 August 1896 – 30 September 1979)
*# Daphne Graham (17 March 1903 – )


*Charles William Reginald Duncombe, 2nd Earl of Feversham (8 May 1879 – 15 September 1916)&lt;ref name=&quot;:7&quot;&gt;&quot; Charles William Reginald Duncombe, 2nd Earl of Feversham of Ryedale.&quot; {{Cite web|url=https://thepeerage.com/p2288.htm#i22880|title=Person Page|website=thepeerage.com|access-date=2020-11-23}}&lt;/ref&gt;
*Marjorie Blanche Eva Greville Duncombe (25 October 1884 – 25 July 1964)&lt;ref&gt;&quot;Lady Marjorie Blanche Eva Greville.&quot; {{Cite web|url=https://thepeerage.com/p2289.htm#i22881|title=Person Page|website=thepeerage.com|access-date=2020-11-23}}&lt;/ref&gt;
#Lady Mary Diana Duncombe (19 March 1905 – October 1943)
#Charles William Slingsby Duncombe, 3rd Earl of Feversham (2 November 1906 – 4 September 1963)
#Hon. David William Ernest Duncombe (8 February 1910 – September 1927)

== Also Known As ==
*Family name: Duncombe
*Earl Feversham of Ryedale
**William Ernest Duncombe, 1st Earl of Feversham (25 July 1868 – 13 January 1915)&lt;ref name=&quot;:6&quot; /&gt;
**Charles William Reginald Duncombe, 2nd Earl of Feversham (13 January 1915 – 15 September 1916)&lt;ref name=&quot;:7&quot; /&gt;
*[[Social Victorians/People/Helmsley | Viscount Helmsley]]
**William Ernest Duncombe (25 July 1868 – 1881)&lt;ref name=&quot;:6&quot; /&gt;
**Charles William Reginald Duncombe, 2nd Earl of Feversham (24 December 1881 – 13 January 1915)&lt;ref&gt;{{Cite journal|date=2020-09-12|title=Charles Duncombe, 2nd Earl of Feversham|url=https://en.wikipedia.org/w/index.php?title=Charles_Duncombe,_2nd_Earl_of_Feversham&amp;oldid=978075739|journal=Wikipedia|language=en}}&lt;/ref&gt;
*Baron of Feversham
**William Ernest Duncombe (11 February 1867 – )&lt;ref name=&quot;:6&quot; /&gt;
*Baron Feversham of Duncombe Park
**Charles Duncombe, 1st Baron Feversham of Duncombe Park ( – 16 July 1841)&lt;ref name=&quot;:8&quot; /&gt;
**William Duncombe, 2nd Baron Feversham of Duncombe Park (16 July 1841 – 11 February 1867)&lt;ref name=&quot;:9&quot; /&gt;
*Other [[Social Victorians/People/Duncombe | Duncombe]] families existed as well.
== Questions and Notes ==
#The newspapers call the Earl and Countess Feversham ''Lord and Lady Feversham''.
#The ''Times'' article lists Sir R. and Lady C. Graham&lt;ref name=&quot;:3&quot; /&gt;: if Lady C. Graham is Lady Cynthia, then Sir R. Graham is Sir Richard James Graham.
#Also present at the ball and accounted for on the [[Social Victorians/People/Duncombe | Duncombe page]] are the following: Alicia Duncombe, Lady and Mr. Florence Duncombe.
#Present at other social events and not accounted for were the following: Caroline Duncombe and the Misses Duncombe.
#William Duncombe, 1st Earl of Feversham is #443 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who attended]] the Duchess of Devonshire's 2 July 1897 fancy-dress ball; Mabel, Countess Feversham is #444; Lady Helen Vincent is #215; Sir Edgar Vincent is #226; Sir Edgar Vincent is #226; Lady Cynthia Graham of Netherby is #220; Sir Richard James Graham is #464.

== Footnotes ==
{{reflist}}</text>
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  <page>
    <title>Social Victorians/People/Helmsley</title>
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      <text bytes="8336" sha1="qkexp3w7rzvs8gcsml242aafsi51qo9" xml:space="preserve">== Overview ==
Charles Duncombe became Viscount Helmsley in 1881, when his father died and when he was 2 years old, and he did not marry until 1904. His father was not the later Earl of Feversham, and so Charles Duncombe was not the heir apparent to the earldom. His mother would still have been called Lady Helmsley or Viscountess Helmsley until he married.

== Also Known As ==
*Family name: Duncombe
*Viscount Helmsley was a courtesy title for the eldest son and heir apparent of the [[Social Victorians/People/Feversham | Earl of Feversham]] (but not throughout the entire 19th century).&lt;ref&gt;{{Cite journal|date=2020-10-14|title=Baron Feversham|url=https://en.wikipedia.org/w/index.php?title=Baron_Feversham&amp;oldid=983534946|journal=Wikipedia|language=en}}&lt;/ref&gt;
*Viscount Helmsley
**William Reginald Duncombe ( – 24 December 1881)
**Charles William Reginald Duncombe (24 December 1881 –1915)&lt;ref&gt;{{Cite journal|date=2020-09-12|title=Charles Duncombe, 2nd Earl of Feversham|url=https://en.wikipedia.org/w/index.php?title=Charles_Duncombe,_2nd_Earl_of_Feversham&amp;oldid=978075739|journal=Wikipedia|language=en}}&lt;/ref&gt;
*Viscountess Helmsley
**Muriel Frances Louisa Chetwynd-Talbot Duncombe (23 December 1876 – 19 January 1904
**Marjorie Blanche Eva Greville Duncombe (19 January 1904 – )
*Dowager Viscountess Helmesley
**Muriel Frances Louisa Chetwynd-Talbot Duncombe Owen (19 January 1904 – 2 March 1925)

== Acquaintances, Friends and Enemies ==
== Timeline ==
'''1876 December 23''', Muriel Frances Louisa Chetwynd-Talbot and William Reginald Duncombe married.&lt;ref name=&quot;:0&quot;&gt;&quot;Lady Muriel Frances Louisa Talbot.&quot; {{Cite web|url=https://thepeerage.com/p1278.htm#i12776|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}&lt;/ref&gt;

'''1885 June 6''', Muriel Frances Louisa Chetwynd-Talbot Duncombe and Hugh Darby Annesley Owen married.&lt;ref name=&quot;:0&quot; /&gt;

'''1896 February 12''', Mabel Theresa Duncombe and Sir William Gervase Beckett married.&lt;ref&gt;&quot;Hon. Mabel Theresa Duncombe.&quot; {{Cite web|url=https://thepeerage.com/p1727.htm#i17261|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}&lt;/ref&gt;

'''1897 July 2''', Lord and Lady Helmsley attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House. (Lord Charles Duncombe, Viscount Helmsley is #353 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who were present]]; Muriel Duncombe, Lady Helmsley is #354.)

'''1904 January 19''', Charles Duncombe and Marjorie Blanche Eva Greville married.&lt;ref name=&quot;:1&quot;&gt;&quot;Lady Marjorie Blanche Eva Greville.&quot; {{Cite web|url=https://thepeerage.com/p2289.htm#i22881|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}&lt;/ref&gt;
[[File:Martin van Meytens 003.jpg|alt=Old painting of a 9-year-old boy dressed very formally and richly, seated at a table with a crown nearby and holding a book.|thumb|Archduke Charles Joseph of Austria, c. 1747–1749.]]

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==

=== Lord Charles Duncombe, Viscount Helmsley ===
At the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], Lord Charles Duncombe, Viscount Helmsley was dressed as Archduke Charles in the Archduchess Marie-Karoline and Emperor Joseph II section of the Austrian Court of Maria Theresa Quadrille.&lt;ref name=&quot;:2&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt;&lt;ref name=&quot;:3&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;

* He was in &quot;Court costume.&quot;&lt;ref&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;
* &quot;V&lt;small&gt;ISCOUNT&lt;/small&gt; H&lt;small&gt;ELMSLEY&lt;/small&gt; was in a Court costume.&quot;&lt;ref&gt;“Additional Costumes Worn at the Duchess of Devonshire’s Fancy Ball.” The ''Queen, The Lady’s Newspaper''17 July 1897, Saturday: 63 [of 97 BNA; p. 138 on the print page], Col. 2a–3a [3 of 3 cols.]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0002627/18970717/283/0064.&lt;/ref&gt;{{rp|Col. 3a}}
If Charles Duncombe, Viscount Helmsley was dressed as Archduke Charles Joseph of Austria, second son of Maria Therese and Francis I, then Marie-Karoline and Emperor Joseph II (the leaders of one of the sections of the Maria Thérèse Quadrille) were his historical siblings. Charles Duncombe was 18 at the time of the ball. Archduke Charles Joseph — nearly 16 when he died — is shown at perhaps 9 years old in a portrait by Martin van Meytens (right).

=== Muriel Duncombe, Lady Helmsley ===
Muriel Duncombe, Lady Helmsley was dressed as Princess Charlotte of Lorraine, also in the Austrian Court of Maria Theresa Quadrille.&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; Muriel Duncombe was not Charles's wife but his mother.

No photographs of their costumes exist at this time.

An Anne Charlotte of Lorraine-Brionne, known as Mademoiselle de Brionne, was at the court of Marie Antoinette and would more likely have been in the Countess of Warwick's procession. A few other Princess Charlottes or Princess Anne Charlottes of Lorraine existed. They were all at least one generation older than Marie Antoinette but associated with the French rather than Austrian court. So it is not clear who she was dressed as.

== Demographics ==
*Nationality: English

== Family ==

* William Ernest Duncombe, 1st Earl Feversham of Ryedale (28 January 1829 – 13 January 1915)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p1873.htm#i18721|title=&quot;William Ernest Duncombe, 1st Earl Feversham of Ryedale.&quot; Person Page 1872|website=www.thepeerage.com|access-date=2026-06-09}}&lt;/ref&gt;
* Mabel Violet Graham (15 February 1833 – 28 August 1915)
*# Lady Ulrica Duncombe ( – 27 April 1935)
*# '''William Reginald Duncombe, Viscount Helmsley''' (1 August 1852 – 24 December 1881)
*# Hon. James Henry Duncombe (20 October 1853 – 10 January 1886)
*# Hon. Hubert Ernest Valentine Duncombe (14 February 1862 – 21 October 1918)
*# Lady Hermione Wilhelmina Duncombe (30 March 1864 – 19 March 1895)
*# Lady Helen Venetia Duncombe (1866 – 16 May 1954)
*# Lady Mabel Cynthia Duncombe (1869 – 25 April 1926)


*Muriel Frances Louisa Chetwynd-Talbot Duncombe Owen (c. 1860 – 2 March 1925)&lt;ref name=&quot;:0&quot; /&gt;
*William Reginald Duncombe, Viscount Helmsley (1 August 1852 – 24 December 1881)&lt;ref&gt;{{Cite journal|date=2019-08-11|title=William Duncombe, Viscount Helmsley|url=https://en.wikipedia.org/w/index.php?title=William_Duncombe,_Viscount_Helmsley&amp;oldid=910373349|journal=Wikipedia|language=en}}&lt;/ref&gt;
*#'''Mabel Theresa Duncombe''' (1877–1913)
*#'''Charles William Reginald Duncombe''', 2nd [[Social Victorians/People/Feversham | Earl of Feversham]] (1879–1916)
*Hugh Darby Annesley Owen ( – 12 March 1908)&lt;ref&gt;&quot;Hugh Darby Annesley Owen.&quot; {{Cite web|url=https://thepeerage.com/p1873.htm#i18722|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}&lt;/ref&gt;


*Charles William Reginald Duncombe, 2nd Earl of Feversham (8 May 1879 – 15 September 1916)&lt;ref&gt;&quot;Charles William Reginald Duncombe, 2nd Earl of Feversham of Ryedale.&quot; {{Cite web|url=https://thepeerage.com/p2288.htm#i22880|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}&lt;/ref&gt;
*Marjorie Blanche Eva Greville Duncombe (25 October 1884 – 25 July 1964)&lt;ref name=&quot;:1&quot; /&gt;
#Lady Mary Diana Duncombe (19 March 1905 – October 1943)
#Charles William Slingsby Duncombe, 3rd Earl of Feversham (2 November 1906 – 4 September 1963)
#Hon. David William Ernest Duncombe (8 February 1910 – September 1927)

== Notes and Questions ==
#Muriel Duncombe Owen seems likely to have been Viscountess Helmsley, or Lady Helmsley. Her son Charles William Reginald Duncombe was Viscount Helmsley by this time, but he did not marry until 1904, so no other Lady Helmsely seems likely. She married Hugh Darby Annesley Owen, however, in 1885, but she would still be eligible to use her title.

== Footnotes ==
{{reflist}}</text>
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    <revision>
      <id>2815586</id>
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        <username>Scogdill</username>
        <id>1331941</id>
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      <text bytes="8388" sha1="ad1wrmxz1ildpj4nboogfvcvn454aoe" xml:space="preserve">== Overview ==
Charles Duncombe became Viscount Helmsley in 1881, when his father died and when he was 2 years old, and he did not marry until 1904. His father was not the later Earl of Feversham, and so Charles Duncombe was not the heir apparent to the earldom. His mother would still have been called Lady Helmsley or Viscountess Helmsley until he married.

== Also Known As ==
*Family name: Duncombe
*Viscount Helmsley was a courtesy title for the eldest son and heir apparent of the [[Social Victorians/People/Feversham | Earl of Feversham]] (but not throughout the entire 19th century).&lt;ref&gt;{{Cite journal|date=2020-10-14|title=Baron Feversham|url=https://en.wikipedia.org/w/index.php?title=Baron_Feversham&amp;oldid=983534946|journal=Wikipedia|language=en}}&lt;/ref&gt;
*Viscount Helmsley
**William Reginald Duncombe ( – 24 December 1881)
**Charles William Reginald Duncombe (24 December 1881 –1915)&lt;ref&gt;{{Cite journal|date=2020-09-12|title=Charles Duncombe, 2nd Earl of Feversham|url=https://en.wikipedia.org/w/index.php?title=Charles_Duncombe,_2nd_Earl_of_Feversham&amp;oldid=978075739|journal=Wikipedia|language=en}}&lt;/ref&gt;
*Viscountess Helmsley
**Muriel Frances Louisa Chetwynd-Talbot Duncombe (23 December 1876 – 19 January 1904
**Marjorie Blanche Eva Greville Duncombe (19 January 1904 – )
*Dowager Viscountess Helmesley
**Muriel Frances Louisa Chetwynd-Talbot Duncombe Owen (19 January 1904 – 2 March 1925)

== Acquaintances, Friends and Enemies ==
== Timeline ==
'''1876 December 23''', Muriel Frances Louisa Chetwynd-Talbot and William Reginald Duncombe married.&lt;ref name=&quot;:0&quot;&gt;&quot;Lady Muriel Frances Louisa Talbot.&quot; {{Cite web|url=https://thepeerage.com/p1278.htm#i12776|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}&lt;/ref&gt;

'''1885 June 6''', Muriel Frances Louisa Chetwynd-Talbot Duncombe and Hugh Darby Annesley Owen married.&lt;ref name=&quot;:0&quot; /&gt;

'''1896 February 12''', Mabel Theresa Duncombe and Sir William Gervase Beckett married.&lt;ref&gt;&quot;Hon. Mabel Theresa Duncombe.&quot; {{Cite web|url=https://thepeerage.com/p1727.htm#i17261|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}&lt;/ref&gt;

'''1897 July 2''', Lord and Lady Helmsley attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House.

'''1904 January 19''', Charles Duncombe and Marjorie Blanche Eva Greville married.&lt;ref name=&quot;:1&quot;&gt;&quot;Lady Marjorie Blanche Eva Greville.&quot; {{Cite web|url=https://thepeerage.com/p2289.htm#i22881|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}&lt;/ref&gt;
[[File:Martin van Meytens 003.jpg|alt=Old painting of a 9-year-old boy dressed very formally and richly, seated at a table with a crown nearby and holding a book.|thumb|Archduke Charles Joseph of Austria, c. 1747–1749.]]

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==

=== Lord Charles Duncombe, Viscount Helmsley ===
At the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], Lord Charles Duncombe, Viscount Helmsley was dressed as Archduke Charles in the Archduchess Marie-Karoline and Emperor Joseph II section of the Austrian Court of Maria Theresa Quadrille.&lt;ref name=&quot;:2&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt;&lt;ref name=&quot;:3&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;

* He was in &quot;Court costume.&quot;&lt;ref&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;
* &quot;V&lt;small&gt;ISCOUNT&lt;/small&gt; H&lt;small&gt;ELMSLEY&lt;/small&gt; was in a Court costume.&quot;&lt;ref&gt;“Additional Costumes Worn at the Duchess of Devonshire’s Fancy Ball.” The ''Queen, The Lady’s Newspaper''17 July 1897, Saturday: 63 [of 97 BNA; p. 138 on the print page], Col. 2a–3a [3 of 3 cols.]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0002627/18970717/283/0064.&lt;/ref&gt;{{rp|Col. 3a}}
If Charles Duncombe, Viscount Helmsley was dressed as Archduke Charles Joseph of Austria, second son of Maria Therese and Francis I, then Marie-Karoline and Emperor Joseph II (the leaders of one of the sections of the Maria Thérèse Quadrille) were his historical siblings. Charles Duncombe was 18 at the time of the ball. Archduke Charles Joseph — nearly 16 when he died — is shown at perhaps 9 years old in a portrait by Martin van Meytens (right).

=== Muriel Duncombe, Lady Helmsley ===
Muriel Duncombe, Lady Helmsley was dressed as Princess Charlotte of Lorraine, also in the Austrian Court of Maria Theresa Quadrille.&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; Muriel Duncombe was not Charles's wife but his mother.

No photographs of their costumes exist at this time.

An Anne Charlotte of Lorraine-Brionne, known as Mademoiselle de Brionne, was at the court of Marie Antoinette and would more likely have been in the Countess of Warwick's procession. A few other Princess Charlottes or Princess Anne Charlottes of Lorraine existed. They were all at least one generation older than Marie Antoinette but associated with the French rather than Austrian court. So it is not clear who she was dressed as.

== Demographics ==
*Nationality: English

== Family ==

* William Ernest Duncombe, 1st Earl Feversham of Ryedale (28 January 1829 – 13 January 1915)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p1873.htm#i18721|title=&quot;William Ernest Duncombe, 1st Earl Feversham of Ryedale.&quot; Person Page 1872|website=www.thepeerage.com|access-date=2026-06-09}}&lt;/ref&gt;
* Mabel Violet Graham (15 February 1833 – 28 August 1915)
*# Lady Ulrica Duncombe ( – 27 April 1935)
*# '''William Reginald Duncombe, Viscount Helmsley''' (1 August 1852 – 24 December 1881)
*# Hon. James Henry Duncombe (20 October 1853 – 10 January 1886)
*# Hon. Hubert Ernest Valentine Duncombe (14 February 1862 – 21 October 1918)
*# Lady Hermione Wilhelmina Duncombe (30 March 1864 – 19 March 1895)
*# Lady Helen Venetia Duncombe (1866 – 16 May 1954)
*# Lady Mabel Cynthia Duncombe (1869 – 25 April 1926)


*Muriel Frances Louisa Chetwynd-Talbot Duncombe Owen (c. 1860 – 2 March 1925)&lt;ref name=&quot;:0&quot; /&gt;
*William Reginald Duncombe, Viscount Helmsley (1 August 1852 – 24 December 1881)&lt;ref&gt;{{Cite journal|date=2019-08-11|title=William Duncombe, Viscount Helmsley|url=https://en.wikipedia.org/w/index.php?title=William_Duncombe,_Viscount_Helmsley&amp;oldid=910373349|journal=Wikipedia|language=en}}&lt;/ref&gt;
*#'''Mabel Theresa Duncombe''' (1877–1913)
*#'''Charles William Reginald Duncombe''', 2nd [[Social Victorians/People/Feversham | Earl of Feversham]] (1879–1916)
*Hugh Darby Annesley Owen ( – 12 March 1908)&lt;ref&gt;&quot;Hugh Darby Annesley Owen.&quot; {{Cite web|url=https://thepeerage.com/p1873.htm#i18722|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}&lt;/ref&gt;


*Charles William Reginald Duncombe, 2nd Earl of Feversham (8 May 1879 – 15 September 1916)&lt;ref&gt;&quot;Charles William Reginald Duncombe, 2nd Earl of Feversham of Ryedale.&quot; {{Cite web|url=https://thepeerage.com/p2288.htm#i22880|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}&lt;/ref&gt;
*Marjorie Blanche Eva Greville Duncombe (25 October 1884 – 25 July 1964)&lt;ref name=&quot;:1&quot; /&gt;
#Lady Mary Diana Duncombe (19 March 1905 – October 1943)
#Charles William Slingsby Duncombe, 3rd Earl of Feversham (2 November 1906 – 4 September 1963)
#Hon. David William Ernest Duncombe (8 February 1910 – September 1927)

== Notes and Questions ==
#Muriel Duncombe Owen seems likely to have been Viscountess Helmsley, or Lady Helmsley. Her son Charles William Reginald Duncombe was Viscount Helmsley by this time, but he did not marry until 1904, so no other Lady Helmsely seems likely. She married Hugh Darby Annesley Owen, however, in 1885, but she would still be eligible to use her title.
#Lord Charles Duncombe, Viscount Helmsley is #353 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who were present]] at the Duchess of Devonshire's 1897 fancy-dress ball; Muriel Duncombe, Lady Helmsley is #354.

== Footnotes ==
{{reflist}}</text>
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        <username>Scogdill</username>
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      <text bytes="3802" sha1="ncsjlajy1ydgi9kskmghgqhptsksfu3" xml:space="preserve">== Also Known As ==
*Family name: Duncombe
*Duncombe is also the family name of the [[Social Victorians/People/Feversham | Earl of Feversham]] and [[Social Victorians/People/Helmsley|Viscount Helmsley]].

== Acquaintances, Friends and Enemies ==

== Timeline ==
'''1876 December 5''', Alfred Charles Duncombe and Lady Florence Montagu married.&lt;ref name=&quot;:1&quot;&gt;&quot;Lady Anne Florence Adelaide Montagu.&quot; {{Cite web|url=https://www.thepeerage.com/p6893.htm#i68923|title=Person Page|website=www.thepeerage.com|access-date=2020-11-25}}&lt;/ref&gt;

'''1897 July 2, Friday''', Lady Florence Duncombe and Mr. Alfred Duncombe attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House.

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==
[[File:Lady-Anne-Florence-Adelaide-Duncombe-ne-Montagu-as-a-Lady-of-the-Court-of-Marie-Stuart.jpg|thumb|alt=Black-and-white photograph of a seated woman richly dressed in an historical costume|Lady Florence Duncombe as a Lady of the Court of Marie Stuart. ©National Portrait Gallery, London.]]
Lady Florence Duncombe and Alfred Duncombe — called Mr. and Lady F. Duncombe — attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]].

Lady Florence Duncombe went, according to the ''Gentlewoman'', as an &quot;Elizabethan Court lady,&quot; wearing &quot;black silk velvet, white quilted satin studded with pearls.&quot;&lt;ref&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 34, Col. 1c}}

Elliott &amp; Fry's portrait of &quot;Lady Anne Florence Adelaide Duncombe (née Montagu) as a Lady of the Court of Marie Stuart&quot; in costume is photogravure #258 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref&gt;&quot;Devonshire House Fancy Dress Ball (1897): photogravures by Walker &amp; Boutall after various photographers.&quot; 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.&lt;/ref&gt; The printing on the portrait says, &quot;Lady Florence Duncombe as a Lady of the Court of Marie Stuart.&quot;&lt;ref&gt;&quot;Lady Florence Duncombe as a Lady of the Court of Marie Stuart.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158621/Lady-Anne-Florence-Adelaide-Duncombe-ne-Montagu-as-a-Lady-of-the-Court-of-Marie-Stuart.&lt;/ref&gt;

This portrait was not taken at the ball or in a photographer's studio using sets and props. It looks like it was taken in someone's home.

== Demographics ==
*Nationality: British
=== Residences ===
*Calwich Abbey&lt;ref name=&quot;:0&quot;&gt;&quot;Alfred Charles Duncombe.&quot; {{Cite web|url=https://www.thepeerage.com/p6893.htm#i68922|title=Person Page|website=www.thepeerage.com|access-date=2020-11-25}}&lt;/ref&gt;
== Family ==
*Alfred Charles Duncombe (5 June 1843 – 22 February 1925)&lt;ref name=&quot;:0&quot; /&gt;
*Lady Florence (Anne Florence Adelaide) Montagu ( – 16 January 1940)&lt;ref name=&quot;:1&quot; /&gt;

=== Relations ===
*Lady Anne Duncombe's father was John William Montagu, [[Social Victorians/People/Sandwich|7th Earl of Sandwich]].&lt;ref name=&quot;:1&quot; /&gt;
*Her mother was [[Social Victorians/People/Paget Family|Lady Mary Paget]].&lt;ref name=&quot;:1&quot; /&gt;

== Questions and Notes ==
#Lady Florence Duncombe is #456 and Alfred Duncombe is #454 in the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of attendees]] at the ball. Lady Alicia Duncombe is #453, so in the ''Morning Post'', at least, she is mentioned when Lady Florence and Alfred Duncombe are mentioned, suggesting that they were together in the Borthwicks' minds.

== Footnotes ==
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        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* To Do */</comment>
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      <text bytes="3374" sha1="nhxq8dslzo3xikb6tqpxnktnvyj6lmu" xml:space="preserve">&lt;!--Info--&gt;
https://www.archive.org
{{list subpages|Alandmanson|User}}

===[[African Arthropods|Project: African Arthropods]]===

;[[African Arthropods/Chelicerates|African Chelicerates]]
:Arachnids and sea spiders — No sub-pages yet.
;[[African Arthropods/Crustaceans|African Crustaceans]]
:Including branchiopods, barnacles, crabs, lobsters, crayfish, shrimp, fish lice, tongue worms, and ostracods — No sub-pages yet.
;[[African Arthropods/Hexapods|African Hexapods]]
:[[African Arthropods/Insects|African Insects]]
:* '''[[African Arthropods/Diptera|Diptera]]''' 
:**[[African Arthropods/Acalyptrate flies|Acalyptrate flies]]

:* '''[[African Arthropods/Hymenoptera|Hymenoptera]]'''
:**[[African Arthropods/Chalcidoidea|African Chalcidoidea]]
:***[[African Arthropods/Eulophidae|African Eulophidae]]
:***[[African Arthropods/Encyrtidae|African Encyrtidae]]
:***[[African Arthropods/Afrotropical Encyrtidae Key|Key to the genera of Afrotropical Encyrtidae]]
:***[[African Arthropods/Chalcid wasps with branched antennae|African chalcid wasps with branched antennae]]
:***[[African Arthropods/Wasps associated with plant galls|Wasps associated with plant galls]]
:**[[African Arthropods/Diaprioidea|African Diaprioidea]]
:**[[African Arthropods/Platygastroidea|African Platygastroidea]]
:**[[African Arthropods/Aculeata|African Aculeata]]
:***[[African Arthropods/Crabroninae|African Crabroninae]] 
:***[[African Arthropods/Eumeninae|African potter wasps]]
:***[[African Arthropods/Philanthus|South African species of Philanthus]]

:* '''[[African Arthropods/Lepidoptera|Lepidoptera]]'''
;[[African Arthropods/Myriapods|African Myriapods]]
:Centipedes, Millipedes, Pauropodans, Symphylans — No sub-pages yet.&lt;br&gt;&lt;br&gt;

;Arthropods in South Africa
:[[African Arthropods/Ferncliffe Nature Reserve|Ferncliffe Nature Reserve]]
:[[African Arthropods/Arthropods on ''Ficus burkei''|Arthropods on ''Ficus burkei'']]
:[[African Arthropods/Hymenoptera of South Africa|Hymenoptera of South Africa]]
:[[African Arthropods/Pompilidae of South Africa|Pompilidae of South Africa]]
::[[African Arthropods/Pompilidae of SA with yellow wings tipped black|Pompilidae of SA with yellow wings, wingtips black]]
::[[African Arthropods/Pompilidae of SA with dark, blackish wings|Pompilidae of South Africa with dark, blackish wings]]

&lt;br&gt;

===To Do===
Working on:
User:Alandmanson/Hymenoptera of Africa

Microgastrine cocoons in a net: &lt;br&gt;
* http://www.waspweb.org/Chalcidoidea/Eupelmidae/Eupelminae/Eupelmus/Eupelmus/Eupelmus_species_2.htm
* https://www.waspweb.org/Ichneumonoidea/Braconidae/Microgastrinae/Glyptapanteles/Glyptapanteles_acraeae.htm
* https://commons.wikimedia.org/wiki/File:Microgastrinae_cocooncocoon_iNat_42943906.jpg
* https://www.inaturalist.org/observations/38150348
* https://www.inaturalist.org/observations/144355729
* https://www.inaturalist.org/observations/39807090
* https://www.inaturalist.org/observations/145817446&lt;br&gt;


[[Crop_production_in_KwaZulu-Natal|Project: Crop_production_in_KwaZulu-Natal]]

[[Crop production in KwaZulu-Natal Annotated Bibliography]]

[[Information for smallholders in KwaZulu-Natal]]

[[Crop_production_in_KwaZulu-Natal/Climate-smart_Agriculture|Climate-smart Agriculture in KZN]]

[[Plant propagation]]&lt;br&gt;
&lt;br&gt;


[[Animal Phyla/Arthropoda]]&lt;br&gt;

[[:Category:Animals]]&lt;br&gt;

[[:Category:Zoology]]&lt;br&gt;

[[:Category:Entomology]]</text>
      <sha1>nhxq8dslzo3xikb6tqpxnktnvyj6lmu</sha1>
    </revision>
    <revision>
      <id>2815539</id>
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      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* To Do */</comment>
      <origin>2815539</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3378" sha1="m3yhernapjxc36cz14k8quze3vconjg" xml:space="preserve">&lt;!--Info--&gt;
https://www.archive.org
{{list subpages|Alandmanson|User}}

===[[African Arthropods|Project: African Arthropods]]===

;[[African Arthropods/Chelicerates|African Chelicerates]]
:Arachnids and sea spiders — No sub-pages yet.
;[[African Arthropods/Crustaceans|African Crustaceans]]
:Including branchiopods, barnacles, crabs, lobsters, crayfish, shrimp, fish lice, tongue worms, and ostracods — No sub-pages yet.
;[[African Arthropods/Hexapods|African Hexapods]]
:[[African Arthropods/Insects|African Insects]]
:* '''[[African Arthropods/Diptera|Diptera]]''' 
:**[[African Arthropods/Acalyptrate flies|Acalyptrate flies]]

:* '''[[African Arthropods/Hymenoptera|Hymenoptera]]'''
:**[[African Arthropods/Chalcidoidea|African Chalcidoidea]]
:***[[African Arthropods/Eulophidae|African Eulophidae]]
:***[[African Arthropods/Encyrtidae|African Encyrtidae]]
:***[[African Arthropods/Afrotropical Encyrtidae Key|Key to the genera of Afrotropical Encyrtidae]]
:***[[African Arthropods/Chalcid wasps with branched antennae|African chalcid wasps with branched antennae]]
:***[[African Arthropods/Wasps associated with plant galls|Wasps associated with plant galls]]
:**[[African Arthropods/Diaprioidea|African Diaprioidea]]
:**[[African Arthropods/Platygastroidea|African Platygastroidea]]
:**[[African Arthropods/Aculeata|African Aculeata]]
:***[[African Arthropods/Crabroninae|African Crabroninae]] 
:***[[African Arthropods/Eumeninae|African potter wasps]]
:***[[African Arthropods/Philanthus|South African species of Philanthus]]

:* '''[[African Arthropods/Lepidoptera|Lepidoptera]]'''
;[[African Arthropods/Myriapods|African Myriapods]]
:Centipedes, Millipedes, Pauropodans, Symphylans — No sub-pages yet.&lt;br&gt;&lt;br&gt;

;Arthropods in South Africa
:[[African Arthropods/Ferncliffe Nature Reserve|Ferncliffe Nature Reserve]]
:[[African Arthropods/Arthropods on ''Ficus burkei''|Arthropods on ''Ficus burkei'']]
:[[African Arthropods/Hymenoptera of South Africa|Hymenoptera of South Africa]]
:[[African Arthropods/Pompilidae of South Africa|Pompilidae of South Africa]]
::[[African Arthropods/Pompilidae of SA with yellow wings tipped black|Pompilidae of SA with yellow wings, wingtips black]]
::[[African Arthropods/Pompilidae of SA with dark, blackish wings|Pompilidae of South Africa with dark, blackish wings]]

&lt;br&gt;

===To Do===
Working on:
[[User:Alandmanson/Hymenoptera of Africa]]

Microgastrine cocoons in a net: &lt;br&gt;
* http://www.waspweb.org/Chalcidoidea/Eupelmidae/Eupelminae/Eupelmus/Eupelmus/Eupelmus_species_2.htm
* https://www.waspweb.org/Ichneumonoidea/Braconidae/Microgastrinae/Glyptapanteles/Glyptapanteles_acraeae.htm
* https://commons.wikimedia.org/wiki/File:Microgastrinae_cocooncocoon_iNat_42943906.jpg
* https://www.inaturalist.org/observations/38150348
* https://www.inaturalist.org/observations/144355729
* https://www.inaturalist.org/observations/39807090
* https://www.inaturalist.org/observations/145817446&lt;br&gt;


[[Crop_production_in_KwaZulu-Natal|Project: Crop_production_in_KwaZulu-Natal]]

[[Crop production in KwaZulu-Natal Annotated Bibliography]]

[[Information for smallholders in KwaZulu-Natal]]

[[Crop_production_in_KwaZulu-Natal/Climate-smart_Agriculture|Climate-smart Agriculture in KZN]]

[[Plant propagation]]&lt;br&gt;
&lt;br&gt;


[[Animal Phyla/Arthropoda]]&lt;br&gt;

[[:Category:Animals]]&lt;br&gt;

[[:Category:Zoology]]&lt;br&gt;

[[:Category:Entomology]]</text>
      <sha1>m3yhernapjxc36cz14k8quze3vconjg</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Wikiversity/Figures</title>
    <ns>0</ns>
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      <id>2815623</id>
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      <timestamp>2026-06-14T04:01:22Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Merge [[Motivation and emotion/Assessment/Chapter/Figures]]</comment>
      <origin>2815623</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4708" sha1="4nnglc4y1utstqnmxuso23awzie53bb" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example.jpg|150px|right|thumb|'''Figure X'''. An example image with an APA-style caption, right justified and 150px in size.]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]].

The simplest way to embed images is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload images directly to Wikiversity. Upload all images to Wikimedia Commons instead.

==Finding images==
Only images available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable figures:

Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
Use Wikimedia Commons search via Wikimedia tools:
Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
Visual Editor search (Insert → Media → File name)
Use external search tools with licensing filters:
[https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
[https://search.creativecommons.org/ Creative Commons search engine]
Google Image search restricted to Commons via query: site:commons.wikimedia.org
Consult curated learning resources such as the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for starting points

Reasoning: combining internal Commons browsing with external licensed search increases coverage while maintaining reuse compliance.

==Uploading to Wikimedia Commons==
Images may be uploaded only if:

You created the image, or
The image is licensed for free reuse (e.g., Creative Commons Attribution or public domain)

Upload process:

Go to [[c:|Wikimedia Commons]]
Select Upload Wizard
Provide required metadata:
Copyright owner
Licence type
Description of image and educational purpose

Once uploaded, images can be embedded in Wikiversity pages.

Reasoning: metadata ensures legal reuse and discoverability across Wikimedia projects.

==Embedding an image==
Images should be inserted using the Visual Editor:

Edit page → Insert → Media → File
Add:
Caption
Size
Position

Best practice:

Add at least one in-text citation (e.g., “see Figure 1”)
Ensure image content directly supports surrounding text
Use consistent numbering across figures within a page

==Adjusting presentation (layout and academic style)==
Figures should follow clear academic conventions:

[[File:Basic needs.png|200px|right|thumb|'''Figure 1'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 2'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 3'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

Key conventions:

Size
** Increase size for complex diagrams requiring readability
Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
Captions
** Must be descriptive and interpretable without the main text
Referencing
** Each figure must be explicitly cited in text (e.g., Figure 1)
Integration
** Figures must directly support the argument or explanation in adjacent text

Reasoning: consistent visual grammar improves cognitive integration of text and diagrammatic information.

==Examples==

[[Motivation and emotion/Gallery|Motivation and emotion gallery]]
[[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==

[[Motivation and emotion/Assessment/Chapter/Figures|Figures (assessment guidance)]]
[[Help:Media]]
[[Help:Wikitext quick reference#Images, tables, video, and sounds]]
[[Motivation and emotion/Wikiversity/Figures|Figures (Motivation and emotion)]]
[[../Tables|Tables]]

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>4nnglc4y1utstqnmxuso23awzie53bb</sha1>
    </revision>
    <revision>
      <id>2815624</id>
      <parentid>2815623</parentid>
      <timestamp>2026-06-14T04:02:13Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <origin>2815624</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4712" sha1="301a3t9wj02nhkx32q7r9azytukbylw" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example.jpg|150px|right|thumb|'''Figure X'''. An example image with an APA-style caption, right justified and 150px in size.]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]].

The simplest way to embed images is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload images directly to Wikiversity. Upload all images to Wikimedia Commons instead.

==Finding images==
Only images available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable figures:

Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
Use Wikimedia Commons search via Wikimedia tools:
Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
Visual Editor search (Insert → Media → File name)
Use external search tools with licensing filters:
[https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
[https://search.creativecommons.org/ Creative Commons search engine]
Google Image search restricted to Commons via query: site:commons.wikimedia.org
Consult curated learning resources such as the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for starting points

Reasoning: combining internal Commons browsing with external licensed search increases coverage while maintaining reuse compliance.

==Uploading to Wikimedia Commons==
Images may be uploaded only if:

You created the image, or
The image is licensed for free reuse (e.g., Creative Commons Attribution or public domain)

Upload process:

Go to [[c:|Wikimedia Commons]]
Select Upload Wizard
Provide required metadata:
Copyright owner
Licence type
Description of image and educational purpose

Once uploaded, images can be embedded in Wikiversity pages.

Reasoning: metadata ensures legal reuse and discoverability across Wikimedia projects.

==Embedding an image==
Images should be inserted using the Visual Editor:

Edit page → Insert → Media → File
Add:
Caption
Size
Position

Best practice:

Add at least one in-text citation (e.g., “see Figure 1”)
Ensure image content directly supports surrounding text
Use consistent numbering across figures within a page

==Adjusting presentation (layout and academic style)==
Figures should follow clear academic conventions:

[[File:Basic needs.png|200px|right|thumb|'''Figure 1'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 1 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 2'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 3'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

Key conventions:

Size
** Increase size for complex diagrams requiring readability
Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
Captions
** Must be descriptive and interpretable without the main text
Referencing
** Each figure must be explicitly cited in text (e.g., Figure 1)
Integration
** Figures must directly support the argument or explanation in adjacent text

Reasoning: consistent visual grammar improves cognitive integration of text and diagrammatic information.

==Examples==

[[Motivation and emotion/Gallery|Motivation and emotion gallery]]
[[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==

[[Motivation and emotion/Assessment/Chapter/Figures|Figures (assessment guidance)]]
[[Help:Media]]
[[Help:Wikitext quick reference#Images, tables, video, and sounds]]
[[Motivation and emotion/Wikiversity/Figures|Figures (Motivation and emotion)]]
[[../Tables|Tables]]

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>301a3t9wj02nhkx32q7r9azytukbylw</sha1>
    </revision>
    <revision>
      <id>2815625</id>
      <parentid>2815624</parentid>
      <timestamp>2026-06-14T04:05:26Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Finding images */ Tidy</comment>
      <origin>2815625</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4604" sha1="ob4wnx0ghs1b258vph5k3x6fjfkw9am" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example.jpg|150px|right|thumb|'''Figure X'''. An example image with an APA-style caption, right justified and 150px in size.]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]].

The simplest way to embed images is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload images directly to Wikiversity. Upload all images to Wikimedia Commons instead.

==Finding images==
Only images available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable figures:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Images may be uploaded only if:

You created the image, or
The image is licensed for free reuse (e.g., Creative Commons Attribution or public domain)

Upload process:

Go to [[c:|Wikimedia Commons]]
Select Upload Wizard
Provide required metadata:
Copyright owner
Licence type
Description of image and educational purpose

Once uploaded, images can be embedded in Wikiversity pages.

Reasoning: metadata ensures legal reuse and discoverability across Wikimedia projects.

==Embedding an image==
Images should be inserted using the Visual Editor:

Edit page → Insert → Media → File
Add:
Caption
Size
Position

Best practice:

Add at least one in-text citation (e.g., “see Figure 1”)
Ensure image content directly supports surrounding text
Use consistent numbering across figures within a page

==Adjusting presentation (layout and academic style)==
Figures should follow clear academic conventions:

[[File:Basic needs.png|200px|right|thumb|'''Figure 1'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 1 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 2'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 3'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

Key conventions:

Size
** Increase size for complex diagrams requiring readability
Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
Captions
** Must be descriptive and interpretable without the main text
Referencing
** Each figure must be explicitly cited in text (e.g., Figure 1)
Integration
** Figures must directly support the argument or explanation in adjacent text

Reasoning: consistent visual grammar improves cognitive integration of text and diagrammatic information.

==Examples==

[[Motivation and emotion/Gallery|Motivation and emotion gallery]]
[[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==

[[Motivation and emotion/Assessment/Chapter/Figures|Figures (assessment guidance)]]
[[Help:Media]]
[[Help:Wikitext quick reference#Images, tables, video, and sounds]]
[[Motivation and emotion/Wikiversity/Figures|Figures (Motivation and emotion)]]
[[../Tables|Tables]]

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>ob4wnx0ghs1b258vph5k3x6fjfkw9am</sha1>
    </revision>
    <revision>
      <id>2815626</id>
      <parentid>2815625</parentid>
      <timestamp>2026-06-14T04:06:49Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Uploading to Wikimedia Commons */ Tidy</comment>
      <origin>2815626</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4543" sha1="c4ysiu7sfswgzdgmuif76o0f1e59jzw" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example.jpg|150px|right|thumb|'''Figure X'''. An example image with an APA-style caption, right justified and 150px in size.]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]].

The simplest way to embed images is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload images directly to Wikiversity. Upload all images to Wikimedia Commons instead.

==Finding images==
Only images available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable figures:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Images may only be uploaded to Commons if:
* You created the image, or
* The image is licensed for free reuse (e.g., Creative Commons Attribution or public domain)

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata:
** Copyright owner
** Licence type
** Description of image and educational purpose

Once uploaded, images can be embedded in Wikiversity pages

==Embedding an image==
Images should be inserted using the Visual Editor:

Edit page → Insert → Media → File
Add:
Caption
Size
Position

Best practice:

Add at least one in-text citation (e.g., “see Figure 1”)
Ensure image content directly supports surrounding text
Use consistent numbering across figures within a page

==Adjusting presentation (layout and academic style)==
Figures should follow clear academic conventions:

[[File:Basic needs.png|200px|right|thumb|'''Figure 1'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 1 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 2'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 3'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

Key conventions:

Size
** Increase size for complex diagrams requiring readability
Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
Captions
** Must be descriptive and interpretable without the main text
Referencing
** Each figure must be explicitly cited in text (e.g., Figure 1)
Integration
** Figures must directly support the argument or explanation in adjacent text

Reasoning: consistent visual grammar improves cognitive integration of text and diagrammatic information.

==Examples==

[[Motivation and emotion/Gallery|Motivation and emotion gallery]]
[[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==

[[Motivation and emotion/Assessment/Chapter/Figures|Figures (assessment guidance)]]
[[Help:Media]]
[[Help:Wikitext quick reference#Images, tables, video, and sounds]]
[[Motivation and emotion/Wikiversity/Figures|Figures (Motivation and emotion)]]
[[../Tables|Tables]]

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>c4ysiu7sfswgzdgmuif76o0f1e59jzw</sha1>
    </revision>
    <revision>
      <id>2815627</id>
      <parentid>2815626</parentid>
      <timestamp>2026-06-14T04:08:20Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Embedding an image */ Tidy</comment>
      <origin>2815627</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4556" sha1="9exbccd1dsga5td9bzgqj6827icv7xe" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example.jpg|150px|right|thumb|'''Figure X'''. An example image with an APA-style caption, right justified and 150px in size.]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]].

The simplest way to embed images is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload images directly to Wikiversity. Upload all images to Wikimedia Commons instead.

==Finding images==
Only images available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable figures:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Images may only be uploaded to Commons if:
* You created the image, or
* The image is licensed for free reuse (e.g., Creative Commons Attribution or public domain)

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata:
** Copyright owner
** Licence type
** Description of image and educational purpose

Once uploaded, images can be embedded in Wikiversity pages

==Embedding an image==
Images can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure image content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., “see Figure 1”)

==Adjusting presentation (layout and academic style)==
Figures should follow clear academic conventions:

[[File:Basic needs.png|200px|right|thumb|'''Figure 1'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 1 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 2'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 3'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

Key conventions:

Size
** Increase size for complex diagrams requiring readability
Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
Captions
** Must be descriptive and interpretable without the main text
Referencing
** Each figure must be explicitly cited in text (e.g., Figure 1)
Integration
** Figures must directly support the argument or explanation in adjacent text

Reasoning: consistent visual grammar improves cognitive integration of text and diagrammatic information.

==Examples==

[[Motivation and emotion/Gallery|Motivation and emotion gallery]]
[[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==

[[Motivation and emotion/Assessment/Chapter/Figures|Figures (assessment guidance)]]
[[Help:Media]]
[[Help:Wikitext quick reference#Images, tables, video, and sounds]]
[[Motivation and emotion/Wikiversity/Figures|Figures (Motivation and emotion)]]
[[../Tables|Tables]]

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>9exbccd1dsga5td9bzgqj6827icv7xe</sha1>
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      </contributor>
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[[File:Example.jpg|150px|right|thumb|'''Figure X'''. An example image with an APA-style caption, right justified and 150px in size.]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]].

The simplest way to embed images is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload images directly to Wikiversity. Upload all images to Wikimedia Commons instead.

==Finding images==
Only images available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable figures:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Images may only be uploaded to Commons if:
* You created the image, or
* The image is licensed for free reuse (e.g., Creative Commons Attribution or public domain)

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata:
** Copyright owner
** Licence type
** Description of image and educational purpose

Once uploaded, images can be embedded in Wikiversity pages

==Embedding an image==
Images can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure image content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., “see Figure 1”)

==Adjusting presentation (layout and academic style)==
Figures should follow clear academic conventions:

[[File:Basic needs.png|200px|right|thumb|'''Figure 1'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 1 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 2'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 3'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==

[[Motivation and emotion/Gallery|Motivation and emotion gallery]]
[[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==

[[Motivation and emotion/Assessment/Chapter/Figures|Figures (assessment guidance)]]
[[Help:Media]]
[[Help:Wikitext quick reference#Images, tables, video, and sounds]]
[[Motivation and emotion/Wikiversity/Figures|Figures (Motivation and emotion)]]
[[../Tables|Tables]]

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>2uggwgtj845e92e8foe8yhj84jshadm</sha1>
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        <id>10242</id>
      </contributor>
      <minor />
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      <text bytes="4495" sha1="1aastk9i1q2fzj4r6rjrsvyyd69t913" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example.jpg|150px|right|thumb|'''Figure X'''. An example image with an APA-style caption, right justified and 150px in size.]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]].

The simplest way to embed images is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload images directly to Wikiversity. Upload all images to Wikimedia Commons instead.

==Finding images==
Only images available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable figures:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Images may only be uploaded to Commons if:
* You created the image, or
* The image is licensed for free reuse (e.g., Creative Commons Attribution or public domain)

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata:
** Copyright owner
** Licence type
** Description of image and educational purpose

Once uploaded, images can be embedded in Wikiversity pages

==Embedding an image==
Images can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure image content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., “see Figure 1”)

==Adjusting presentation (layout and academic style)==
Figures should follow clear academic conventions:

[[File:Basic needs.png|200px|right|thumb|'''Figure 1'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 1 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 2'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 3'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==

[[Motivation and emotion/Assessment/Chapter/Figures|Figures (assessment guidance)]]
[[Help:Media]]
[[Help:Wikitext quick reference#Images, tables, video, and sounds]]
[[Motivation and emotion/Wikiversity/Figures|Figures (Motivation and emotion)]]
[[../Tables|Tables]]

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>1aastk9i1q2fzj4r6rjrsvyyd69t913</sha1>
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        <id>10242</id>
      </contributor>
      <comment>/* See also */ Tidy</comment>
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      <text bytes="4375" sha1="48x30vqibbgii4fg20ppfnrjivln8a7" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example.jpg|150px|right|thumb|'''Figure X'''. An example image with an APA-style caption, right justified and 150px in size.]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]].

The simplest way to embed images is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload images directly to Wikiversity. Upload all images to Wikimedia Commons instead.

==Finding images==
Only images available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable figures:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Images may only be uploaded to Commons if:
* You created the image, or
* The image is licensed for free reuse (e.g., Creative Commons Attribution or public domain)

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata:
** Copyright owner
** Licence type
** Description of image and educational purpose

Once uploaded, images can be embedded in Wikiversity pages

==Embedding an image==
Images can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure image content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., “see Figure 1”)

==Adjusting presentation (layout and academic style)==
Figures should follow clear academic conventions:

[[File:Basic needs.png|200px|right|thumb|'''Figure 1'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 1 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 2'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 3'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Help:Media]]
* [[Help:Wikitext quick reference#Images, tables, video, and sounds]]
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]]

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>48x30vqibbgii4fg20ppfnrjivln8a7</sha1>
    </revision>
    <revision>
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      <timestamp>2026-06-14T04:13:53Z</timestamp>
      <contributor>
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        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* Adjusting presentation (layout and academic style) */</comment>
      <origin>2815631</origin>
      <model>wikitext</model>
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[[File:Example.jpg|150px|right|thumb|'''Figure X'''. An example image with an APA-style caption, right justified and 150px in size.]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]].

The simplest way to embed images is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload images directly to Wikiversity. Upload all images to Wikimedia Commons instead.

==Finding images==
Only images available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable figures:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Images may only be uploaded to Commons if:
* You created the image, or
* The image is licensed for free reuse (e.g., Creative Commons Attribution or public domain)

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata:
** Copyright owner
** Licence type
** Description of image and educational purpose

Once uploaded, images can be embedded in Wikiversity pages

==Embedding an image==
Images can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure image content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., “see Figure 1”)

==Adjusting presentation (layout and academic style)==
[[File:Basic needs.png|200px|right|thumb|'''Figure 1'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 1 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 2'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 3'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Help:Media]]
* [[Help:Wikitext quick reference#Images, tables, video, and sounds]]
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]]

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>dbukfh69x6zqb853ov8x242es6gyiwe</sha1>
    </revision>
    <revision>
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      <timestamp>2026-06-14T04:14:39Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Adjusting presentation (layout and academic style) */ Adjust heading</comment>
      <origin>2815632</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4354" sha1="qits4udt46w16k4dzuzj2pquhlgt4yj" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example.jpg|150px|right|thumb|'''Figure X'''. An example image with an APA-style caption, right justified and 150px in size.]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]].

The simplest way to embed images is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload images directly to Wikiversity. Upload all images to Wikimedia Commons instead.

==Finding images==
Only images available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable figures:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Images may only be uploaded to Commons if:
* You created the image, or
* The image is licensed for free reuse (e.g., Creative Commons Attribution or public domain)

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata:
** Copyright owner
** Licence type
** Description of image and educational purpose

Once uploaded, images can be embedded in Wikiversity pages

==Embedding an image==
Images can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure image content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., “see Figure 1”)

==Customise image presentation==
[[File:Basic needs.png|200px|right|thumb|'''Figure 1'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 1 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 2'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 3'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Help:Media]]
* [[Help:Wikitext quick reference#Images, tables, video, and sounds]]
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]]

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>qits4udt46w16k4dzuzj2pquhlgt4yj</sha1>
    </revision>
    <revision>
      <id>2815634</id>
      <parentid>2815632</parentid>
      <timestamp>2026-06-14T04:18:43Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Uploading to Wikimedia Commons */ Tidy</comment>
      <origin>2815634</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4421" sha1="tug6p02vvsnfsc74vyzpt12e0e0o7z0" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example.jpg|150px|right|thumb|'''Figure X'''. An example image with an APA-style caption, right justified and 150px in size.]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]].

The simplest way to embed images is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload images directly to Wikiversity. Upload all images to Wikimedia Commons instead.

==Finding images==
Only images available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable figures:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Images may be uploaded to Commons if:
* An equivalent image isn't already available on Commons
* You created the image or it is licensed for free reuse (e.g., Creative Commons Attribution or public domain)
* The image is educational

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata, including:
** Copyright owner
** Licence type
** Description of image

Once uploaded to Commons, images can be embedded in Wikiversity pages

==Embedding an image==
Images can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure image content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., “see Figure 1”)

==Customise image presentation==
[[File:Basic needs.png|200px|right|thumb|'''Figure 1'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 1 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 2'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 3'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Help:Media]]
* [[Help:Wikitext quick reference#Images, tables, video, and sounds]]
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]]

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
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[[File:Example.jpg|150px|right|thumb|'''Figure X'''. An example image with an APA-style caption, right justified and 150px in size.]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]].

The simplest way to embed images is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload images directly to Wikiversity. Upload all images to Wikimedia Commons instead.

==Finding images==
Only images available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable figures:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Images may be uploaded to Commons if:
* An equivalent image isn't already available on Commons
* You created the image or it is licensed for free reuse (e.g., Creative Commons Attribution or public domain)
* The image is educational

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata, including:
** Copyright owner
** Licence type
** Description of image

Once uploaded to Commons, images can be embedded in Wikiversity pages

==Embedding an image==
[[File:Basic needs.png|200px|right|thumb|'''Figure 1'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 1 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 2'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 3'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

Images can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure image content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., “see Figure 1”)

==Customise image presentation==
Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Help:Media]]
* [[Help:Wikitext quick reference#Images, tables, video, and sounds]]
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]]

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>o30579xyvh79mt350xu649nntrsonwr</sha1>
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[[File:Example.jpg|150px|right|thumb|'''Figure 1'''. An example image with an APA-style caption, right justified and 150px in size.]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]] (e.g., see Figure 1).

The simplest way to embed images is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload images directly to Wikiversity. Upload all images to Wikimedia Commons instead.

==Finding images==
Only images available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable figures:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Images may be uploaded to Commons if:
* An equivalent image isn't already available on Commons
* You created the image or it is licensed for free reuse (e.g., Creative Commons Attribution or public domain)
* The image is educational

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata, including:
** Copyright owner
** Licence type
** Description of image

Once uploaded to Commons, images can be embedded in Wikiversity pages (e.g., see Figures 2 to 4)

==Embedding an image==
[[File:Basic needs.png|200px|right|thumb|'''Figure 2'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 3'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 3 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 4'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

Images can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure image content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., “see Figure 1”)

==Customise image presentation==
Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Help:Media]]
* [[Help:Wikitext quick reference#Images, tables, video, and sounds]]
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]]

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
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[[File:Example.jpg|150px|right|thumb|'''Figure 1'''. An example image with an APA-style caption, right justified and 150px in size.]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]] (e.g., see Figure 1).

The simplest way to embed images is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload images directly to Wikiversity. Upload all images to Wikimedia Commons instead.

==Finding images==
Only images available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable figures:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Images may be uploaded to Commons if:
* An equivalent image isn't already available on Commons
* You created the image or it is licensed for free reuse (e.g., Creative Commons Attribution or public domain)
* The image is educational

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata, including:
** Copyright owner
** Licence type
** Description of image

Once uploaded to Commons, images can be embedded in Wikiversity pages (e.g., see Figures 2 to 4)

==Embedding an image==
Images can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure image content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., see Figure 1)

==Customise image presentation==
Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
[[File:Basic needs.png|200px|right|thumb|'''Figure 2'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 3'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 3 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 4'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Help:Media]]
* [[Help:Wikitext quick reference#Images, tables, video, and sounds]]
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]]

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>pkhvoj8a9zngvykqv6x1x6pohhl6mv5</sha1>
    </revision>
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      <timestamp>2026-06-14T10:36:42Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>images -&gt; media</comment>
      <origin>2815645</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4540" sha1="d6jfwirh23z850co58p8sqfjd0awxvt" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example.jpg|150px|right|thumb|'''Figure 1'''. An example image with an APA-style caption, right justified and 150px in size.]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]] (e.g., see Figure 1). Figures can show a variety media, including:
* photographs
* diagrams
* animated gifs
* audio
* video

The simplest way to embed media is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload media directly to Wikiversity; upload to Wikimedia Commons instead.

==Finding media==
Only media available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable media:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Media may be uploaded to Commons if:
* Equivalent media isn't already available on Commons
* You created the media or it is licensed for free reuse (e.g., Creative Commons Attribution or public domain)
* The media is educational

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata, including:
** Copyright owner
** Licence type
** Description of image

Once uploaded to Commons, media can be embedded in Wikiversity pages (e.g., see Figures 2 to 4)

==Embedding media==
Media can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure media content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., see Figure 1)

==Customise figure presentation==
Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
[[File:Basic needs.png|200px|right|thumb|'''Figure 2'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 3'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 3 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 4'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Help:Media]]
* [[Help:Wikitext quick reference#Images, tables, video, and sounds]]
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]]

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>d6jfwirh23z850co58p8sqfjd0awxvt</sha1>
    </revision>
    <revision>
      <id>2815646</id>
      <parentid>2815645</parentid>
      <timestamp>2026-06-14T10:53:18Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>+ Figure 2 (audio)</comment>
      <origin>2815646</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4730" sha1="r6jmvdf4tnb23h74q7p7ks8y2ytqen8" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example.jpg|150px|right|thumb|'''Figure 1'''. An example image with an APA-style caption, right justified and 150px in size.]]

[[File:Alzheimer's Disease.ogg|150px|right|thumb|'''Figure 2'''. This is a spoken word version of the [[w:Alzheimer's disease|Alzheimer's disease]] Wikipedia article, 2008]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]] (e.g., see Figure 1). Figures can show a variety media, including:
* photographs
* diagrams
* animated gifs
* audio (see Figure 2)
* video

The simplest way to embed media is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload media directly to Wikiversity; upload to Wikimedia Commons instead.

==Finding media==
Only media available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable media:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Media may be uploaded to Commons if:
* Equivalent media isn't already available on Commons
* You created the media or it is licensed for free reuse (e.g., Creative Commons Attribution or public domain)
* The media is educational

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata, including:
** Copyright owner
** Licence type
** Description of image

Once uploaded to Commons, media can be embedded in Wikiversity pages (e.g., see Figures 3 to 5)

==Embedding media==
Media can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure media content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., see Figure 1)

==Customise figure presentation==
Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
[[File:Basic needs.png|200px|right|thumb|'''Figure 3'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 4'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 3 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 5'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Help:Media]]
* [[Help:Wikitext quick reference#Images, tables, video, and sounds]]
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]]

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>r6jmvdf4tnb23h74q7p7ks8y2ytqen8</sha1>
    </revision>
    <revision>
      <id>2815647</id>
      <parentid>2815646</parentid>
      <timestamp>2026-06-14T10:55:46Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Update Figure 1 image to svg</comment>
      <origin>2815647</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4733" sha1="3f83jowluy13rm45sa3aqpsceecvu3x" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example en.svg|150px|right|thumb|'''Figure 1'''. An example image with an APA-style caption, right justified and 150px in size.]]

[[File:Alzheimer's Disease.ogg|150px|right|thumb|'''Figure 2'''. This is a spoken word version of the [[w:Alzheimer's disease|Alzheimer's disease]] Wikipedia article, 2008]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]] (e.g., see Figure 1). Figures can show a variety media, including:
* photographs
* diagrams
* animated gifs
* audio (see Figure 2)
* video

The simplest way to embed media is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload media directly to Wikiversity; upload to Wikimedia Commons instead.

==Finding media==
Only media available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable media:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Media may be uploaded to Commons if:
* Equivalent media isn't already available on Commons
* You created the media or it is licensed for free reuse (e.g., Creative Commons Attribution or public domain)
* The media is educational

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata, including:
** Copyright owner
** Licence type
** Description of image

Once uploaded to Commons, media can be embedded in Wikiversity pages (e.g., see Figures 3 to 5)

==Embedding media==
Media can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure media content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., see Figure 1)

==Customise figure presentation==
Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
[[File:Basic needs.png|200px|right|thumb|'''Figure 3'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 4'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 3 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 5'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Help:Media]]
* [[Help:Wikitext quick reference#Images, tables, video, and sounds]]
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]]

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>3f83jowluy13rm45sa3aqpsceecvu3x</sha1>
    </revision>
    <revision>
      <id>2815648</id>
      <parentid>2815647</parentid>
      <timestamp>2026-06-14T10:58:36Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* See also */ [[Template:Motivation and emotion/Book chapter structure#Figures|Figures]] (Motivation and emotion book chapter template)</comment>
      <origin>2815648</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4857" sha1="9l5ceac4raqi9557tnwz801fjq2xesd" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example en.svg|150px|right|thumb|'''Figure 1'''. An example image with an APA-style caption, right justified and 150px in size.]]

[[File:Alzheimer's Disease.ogg|150px|right|thumb|'''Figure 2'''. This is a spoken word version of the [[w:Alzheimer's disease|Alzheimer's disease]] Wikipedia article, 2008]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]] (e.g., see Figure 1). Figures can show a variety media, including:
* photographs
* diagrams
* animated gifs
* audio (see Figure 2)
* video

The simplest way to embed media is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload media directly to Wikiversity; upload to Wikimedia Commons instead.

==Finding media==
Only media available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable media:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Media may be uploaded to Commons if:
* Equivalent media isn't already available on Commons
* You created the media or it is licensed for free reuse (e.g., Creative Commons Attribution or public domain)
* The media is educational

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata, including:
** Copyright owner
** Licence type
** Description of image

Once uploaded to Commons, media can be embedded in Wikiversity pages (e.g., see Figures 3 to 5)

==Embedding media==
Media can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure media content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., see Figure 1)

==Customise figure presentation==
Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
[[File:Basic needs.png|200px|right|thumb|'''Figure 3'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 4'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 3 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 5'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Help:Media]]
* [[Help:Wikitext quick reference#Images, tables, video, and sounds]]
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]]
* [[Template:Motivation and emotion/Book chapter structure#Figures|Figures]] (Motivation and emotion book chapter template)

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>9l5ceac4raqi9557tnwz801fjq2xesd</sha1>
    </revision>
    <revision>
      <id>2815649</id>
      <parentid>2815648</parentid>
      <timestamp>2026-06-14T10:59:41Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* See also */</comment>
      <origin>2815649</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4945" sha1="jwvw8kg4lc6g36nr8ovhof6sot8nh12" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example en.svg|150px|right|thumb|'''Figure 1'''. An example image with an APA-style caption, right justified and 150px in size.]]

[[File:Alzheimer's Disease.ogg|150px|right|thumb|'''Figure 2'''. This is a spoken word version of the [[w:Alzheimer's disease|Alzheimer's disease]] Wikipedia article, 2008]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]] (e.g., see Figure 1). Figures can show a variety media, including:
* photographs
* diagrams
* animated gifs
* audio (see Figure 2)
* video

The simplest way to embed media is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload media directly to Wikiversity; upload to Wikimedia Commons instead.

==Finding media==
Only media available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable media:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Media may be uploaded to Commons if:
* Equivalent media isn't already available on Commons
* You created the media or it is licensed for free reuse (e.g., Creative Commons Attribution or public domain)
* The media is educational

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata, including:
** Copyright owner
** Licence type
** Description of image

Once uploaded to Commons, media can be embedded in Wikiversity pages (e.g., see Figures 3 to 5)

==Embedding media==
Media can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure media content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., see Figure 1)

==Customise figure presentation==
Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
[[File:Basic needs.png|200px|right|thumb|'''Figure 3'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 4'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 3 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 5'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Template:Motivation and emotion/Book chapter structure#Figures|Figures]] (Motivation and emotion book chapter template)
* [[Help:Wikitext quick reference#Images, tables, video, and sounds|Images, tables, video, and sounds]] (Wikiversity help page)
* [[Help:Media|Media]] (Wikiversity help page)
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]]

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>jwvw8kg4lc6g36nr8ovhof6sot8nh12</sha1>
    </revision>
    <revision>
      <id>2815650</id>
      <parentid>2815649</parentid>
      <timestamp>2026-06-14T11:00:10Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* See also */</comment>
      <origin>2815650</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4993" sha1="bvmk7ks1ie9wq2kd3nmxak73tim1ete" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example en.svg|150px|right|thumb|'''Figure 1'''. An example image with an APA-style caption, right justified and 150px in size.]]

[[File:Alzheimer's Disease.ogg|150px|right|thumb|'''Figure 2'''. This is a spoken word version of the [[w:Alzheimer's disease|Alzheimer's disease]] Wikipedia article, 2008]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]] (e.g., see Figure 1). Figures can show a variety media, including:
* photographs
* diagrams
* animated gifs
* audio (see Figure 2)
* video

The simplest way to embed media is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload media directly to Wikiversity; upload to Wikimedia Commons instead.

==Finding media==
Only media available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable media:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Media may be uploaded to Commons if:
* Equivalent media isn't already available on Commons
* You created the media or it is licensed for free reuse (e.g., Creative Commons Attribution or public domain)
* The media is educational

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata, including:
** Copyright owner
** Licence type
** Description of image

Once uploaded to Commons, media can be embedded in Wikiversity pages (e.g., see Figures 3 to 5)

==Embedding media==
Media can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure media content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., see Figure 1)

==Customise figure presentation==
Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
[[File:Basic needs.png|200px|right|thumb|'''Figure 3'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 4'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 3 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 5'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Template:Motivation and emotion/Book chapter structure#Figures|Figures]] (Motivation and emotion book chapter template)
* [[Help:Wikitext quick reference#Images, tables, video, and sounds|Images, tables, video, and sounds]] (Wikiversity help page)
* [[Help:Media|Media]] (Wikiversity help page)
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]] (Motivation and emotion book chapter help page)

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>bvmk7ks1ie9wq2kd3nmxak73tim1ete</sha1>
    </revision>
    <revision>
      <id>2815653</id>
      <parentid>2815650</parentid>
      <timestamp>2026-06-14T11:09:59Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>+ animated gif example</comment>
      <origin>2815653</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5156" sha1="ailu28pw4d9d0qn3pgz6xlhnv8hj8ic" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example en.svg|150px|right|thumb|'''Figure 1'''. An example image with an APA-style caption, right justified and 150px in size.]]

[[File:Hypothalamus.gif|150px|right|thumb|'''Figure 2'''. The hypothalamus regulates fundamental motivations, including eating, drinking, and sex.

[[File:Alzheimer's Disease.ogg|150px|right|thumb|'''Figure 3'''. This is a spoken word version of the [[w:Alzheimer's disease|Alzheimer's disease]] Wikipedia article, 2008]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]] (e.g., see Figure 1). Figures can show a variety media, including:
* photographs
* diagrams
* animated gifs (see Figure 2)
* audio (see Figure 3)
* video

The simplest way to embed media is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload media directly to Wikiversity; upload to Wikimedia Commons instead.

==Finding media==
Only media available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable media:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Media may be uploaded to Commons if:
* Equivalent media isn't already available on Commons
* You created the media or it is licensed for free reuse (e.g., Creative Commons Attribution or public domain)
* The media is educational

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata, including:
** Copyright owner
** Licence type
** Description of image

Once uploaded to Commons, media can be embedded in Wikiversity pages (e.g., see Figures 4 to 6)

==Embedding media==
Media can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure media content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., see Figure 1)

==Customise figure presentation==
Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
[[File:Basic needs.png|200px|right|thumb|'''Figure 4'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 5'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 3 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 6'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Template:Motivation and emotion/Book chapter structure#Figures|Figures]] (Motivation and emotion book chapter template)
* [[Help:Wikitext quick reference#Images, tables, video, and sounds|Images, tables, video, and sounds]] (Wikiversity help page)
* [[Help:Media|Media]] (Wikiversity help page)
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]] (Motivation and emotion book chapter help page)

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>ailu28pw4d9d0qn3pgz6xlhnv8hj8ic</sha1>
    </revision>
    <revision>
      <id>2815654</id>
      <parentid>2815653</parentid>
      <timestamp>2026-06-14T11:10:18Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <origin>2815654</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5158" sha1="gis93jp78byl7cfss1vrc2fghr9t52q" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example en.svg|150px|right|thumb|'''Figure 1'''. An example image with an APA-style caption, right justified and 150px in size.]]

[[File:Hypothalamus.gif|150px|right|thumb|'''Figure 2'''. The hypothalamus regulates fundamental motivations, including eating, drinking, and sex.]]

[[File:Alzheimer's Disease.ogg|150px|right|thumb|'''Figure 3'''. This is a spoken word version of the [[w:Alzheimer's disease|Alzheimer's disease]] Wikipedia article, 2008]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]] (e.g., see Figure 1). Figures can show a variety media, including:
* photographs
* diagrams
* animated gifs (see Figure 2)
* audio (see Figure 3)
* video

The simplest way to embed media is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload media directly to Wikiversity; upload to Wikimedia Commons instead.

==Finding media==
Only media available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable media:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Media may be uploaded to Commons if:
* Equivalent media isn't already available on Commons
* You created the media or it is licensed for free reuse (e.g., Creative Commons Attribution or public domain)
* The media is educational

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata, including:
** Copyright owner
** Licence type
** Description of image

Once uploaded to Commons, media can be embedded in Wikiversity pages (e.g., see Figures 4 to 6)

==Embedding media==
Media can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure media content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., see Figure 1)

==Customise figure presentation==
Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
[[File:Basic needs.png|200px|right|thumb|'''Figure 4'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 2 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 5'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 3 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Maslow's Hierarchy of Needs.png|400px|right|thumb|'''Figure 6'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Template:Motivation and emotion/Book chapter structure#Figures|Figures]] (Motivation and emotion book chapter template)
* [[Help:Wikitext quick reference#Images, tables, video, and sounds|Images, tables, video, and sounds]] (Wikiversity help page)
* [[Help:Media|Media]] (Wikiversity help page)
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]] (Motivation and emotion book chapter help page)

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>gis93jp78byl7cfss1vrc2fghr9t52q</sha1>
    </revision>
    <revision>
      <id>2815655</id>
      <parentid>2815654</parentid>
      <timestamp>2026-06-14T11:19:08Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Move Figure 6 to Figure 2 and add an example photograph</comment>
      <origin>2815655</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5360" sha1="8fft3vi6ot9n98q8e4q8b4zg0n9v5a4" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example en.svg|150px|right|thumb|'''Figure 1'''. An example image with an APA-style caption, right justified and 150px in size.]]

[[File:Cycling Time Trial effort.jpg|150px|right|thumb|'''Figure 2'''. A key indicator of motivation is the intensity and persistance of effort.]]

[[File:Maslow's Hierarchy of Needs.png|360px|right|thumb|'''Figure 2'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

[[File:Hypothalamus.gif|150px|right|thumb|'''Figure 3'''. The hypothalamus regulates fundamental motivations, including eating, drinking, and sex.]]

[[File:Alzheimer's Disease.ogg|150px|right|thumb|'''Figure 4'''. This is a spoken word version of the [[w:Alzheimer's disease|Alzheimer's disease]] Wikipedia article, 2008]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]] (e.g., see Figure 1). Figures can show a variety media, including:
* photographs (e.g., see Figure 2)
* diagrams (e.g., see Figure 3)
* animated gifs (e.g., see Figure 4)
* audio (e.g., see Figure 5)
* video

The simplest way to embed media is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload media directly to Wikiversity; upload to Wikimedia Commons instead.

==Finding media==
Only media available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable media:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Media may be uploaded to Commons if:
* Equivalent media isn't already available on Commons
* You created the media or it is licensed for free reuse (e.g., Creative Commons Attribution or public domain)
* The media is educational

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata, including:
** Copyright owner
** Licence type
** Description of image

Once uploaded to Commons, media can be embedded in Wikiversity pages (e.g., see Figures 6 to 7)

==Embedding media==
Media can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure media content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., see Figure 1)

==Customise figure presentation==
Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
[[File:Basic needs.png|200px|right|thumb|'''Figure 6'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 5 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 7'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 6 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Template:Motivation and emotion/Book chapter structure#Figures|Figures]] (Motivation and emotion book chapter template)
* [[Help:Wikitext quick reference#Images, tables, video, and sounds|Images, tables, video, and sounds]] (Wikiversity help page)
* [[Help:Media|Media]] (Wikiversity help page)
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]] (Motivation and emotion book chapter help page)

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>8fft3vi6ot9n98q8e4q8b4zg0n9v5a4</sha1>
    </revision>
    <revision>
      <id>2815656</id>
      <parentid>2815655</parentid>
      <timestamp>2026-06-14T11:19:43Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <origin>2815656</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5360" sha1="05p4b3ud64ueafm2hdpcb0pd7yj7rxp" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example en.svg|150px|right|thumb|'''Figure 1'''. An example image with an APA-style caption, right justified and 150px in size.]]

[[File:Cycling Time Trial effort.jpg|150px|right|thumb|'''Figure 2'''. A key indicator of motivation is the intensity and persistance of effort.]]

[[File:Maslow's Hierarchy of Needs.png|360px|right|thumb|'''Figure 3'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

[[File:Hypothalamus.gif|150px|right|thumb|'''Figure 4'''. The hypothalamus regulates fundamental motivations, including eating, drinking, and sex.]]

[[File:Alzheimer's Disease.ogg|150px|right|thumb|'''Figure 5'''. This is a spoken word version of the [[w:Alzheimer's disease|Alzheimer's disease]] Wikipedia article, 2008]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]] (e.g., see Figure 1). Figures can show a variety media, including:
* photographs (e.g., see Figure 2)
* diagrams (e.g., see Figure 3)
* animated gifs (e.g., see Figure 4)
* audio (e.g., see Figure 5)
* video

The simplest way to embed media is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload media directly to Wikiversity; upload to Wikimedia Commons instead.

==Finding media==
Only media available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable media:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Media may be uploaded to Commons if:
* Equivalent media isn't already available on Commons
* You created the media or it is licensed for free reuse (e.g., Creative Commons Attribution or public domain)
* The media is educational

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata, including:
** Copyright owner
** Licence type
** Description of image

Once uploaded to Commons, media can be embedded in Wikiversity pages (e.g., see Figures 6 to 7)

==Embedding media==
Media can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure media content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., see Figure 1)

==Customise figure presentation==
Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
[[File:Basic needs.png|200px|right|thumb|'''Figure 6'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 5 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 7'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 6 created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Template:Motivation and emotion/Book chapter structure#Figures|Figures]] (Motivation and emotion book chapter template)
* [[Help:Wikitext quick reference#Images, tables, video, and sounds|Images, tables, video, and sounds]] (Wikiversity help page)
* [[Help:Media|Media]] (Wikiversity help page)
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]] (Motivation and emotion book chapter help page)

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>05p4b3ud64ueafm2hdpcb0pd7yj7rxp</sha1>
    </revision>
    <revision>
      <id>2815657</id>
      <parentid>2815656</parentid>
      <timestamp>2026-06-14T11:24:01Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>+ example video</comment>
      <origin>2815657</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5556" sha1="1x2qsx9hugfsu2b6bfhzl1mx82p3klj" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example en.svg|150px|right|thumb|'''Figure 1'''. An example image with an APA-style caption, right justified and 150px in size.]]

[[File:Cycling Time Trial effort.jpg|150px|right|thumb|'''Figure 2'''. A key indicator of motivation is the intensity and persistance of effort.]]

[[File:Maslow's Hierarchy of Needs.png|360px|right|thumb|'''Figure 3'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base.]]

[[File:Hypothalamus.gif|150px|right|thumb|'''Figure 4'''. The hypothalamus regulates fundamental motivations, including eating, drinking, and sex.]]

[[File:Alzheimer's Disease.ogg|150px|right|thumb|'''Figure 5'''. This is a spoken word version of the [[w:Alzheimer's disease|Alzheimer's disease]] Wikipedia article, 2008]]

[[File:EN simpleshow foundation Fear of Flying explainer video.webm|150px|right|thumb|'''Figure 6'''. This video explains the psychology behind the fear of flying.]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]] (e.g., see Figure 1). Figures can show a variety media, including:
* photographs (e.g., see Figure 2)
* diagrams (e.g., see Figure 3)
* animated gifs (e.g., see Figure 4)
* audio (e.g., see Figure 5)
* video (e.g., see Figure 6)

The simplest way to embed media is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload media directly to Wikiversity; upload to Wikimedia Commons instead.

==Finding media==
Only media available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable media:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Media may be uploaded to Commons if:
* Equivalent media isn't already available on Commons
* You created the media or it is licensed for free reuse (e.g., Creative Commons Attribution or public domain)
* The media is educational

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata, including:
** Copyright owner
** Licence type
** Description of image

Once uploaded to Commons, media can be embedded in Wikiversity pages (e.g., see Figures 7 to 8)

==Embedding media==
Media can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure media content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., see Figure 1)

==Customise figure presentation==
Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
[[File:Basic needs.png|200px|right|thumb|'''Figure 7'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 7 was created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 8'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 8 was created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Template:Motivation and emotion/Book chapter structure#Figures|Figures]] (Motivation and emotion book chapter template)
* [[Help:Wikitext quick reference#Images, tables, video, and sounds|Images, tables, video, and sounds]] (Wikiversity help page)
* [[Help:Media|Media]] (Wikiversity help page)
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]] (Motivation and emotion book chapter help page)

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>1x2qsx9hugfsu2b6bfhzl1mx82p3klj</sha1>
    </revision>
    <revision>
      <id>2815658</id>
      <parentid>2815657</parentid>
      <timestamp>2026-06-14T11:26:10Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Expand captions</comment>
      <origin>2815658</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5770" sha1="i5h9bzmmyfrt7ghgt9atvf18mhmtaeg" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example en.svg|150px|right|thumb|'''Figure 1'''. An example image with an APA-style caption, right justified and 150px in size. &lt;nowiki&gt;[Example image]&lt;/nowiki&gt;]]

[[File:Cycling Time Trial effort.jpg|150px|right|thumb|'''Figure 2'''. A key indicator of motivation is the intensity and persistance of effort. &lt;nowiki&gt;[Example photography]&lt;/nowiki&gt;]]

[[File:Maslow's Hierarchy of Needs.png|360px|right|thumb|'''Figure 3'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base. &lt;nowiki&gt;[Example diagram]&lt;/nowiki&gt;]]

[[File:Hypothalamus.gif|150px|right|thumb|'''Figure 4'''. The hypothalamus regulates fundamental motivations, including eating, drinking, and sex. &lt;nowiki&gt;[Example animated gif]&lt;/nowiki&gt;]]

[[File:Alzheimer's Disease.ogg|150px|right|thumb|'''Figure 5'''. This is a spoken word version of the [[w:Alzheimer's disease|Alzheimer's disease]] Wikipedia article, 2008. &lt;nowiki&gt;[Example audio]&lt;/nowiki&gt;]]

[[File:EN simpleshow foundation Fear of Flying explainer video.webm|150px|right|thumb|'''Figure 6'''. This video explains the psychology behind the fear of flying. &lt;nowiki&gt;[Example video]&lt;/nowiki&gt;]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]] (e.g., see Figure 1). Figures can show a variety media, including:
* photographs (e.g., see Figure 2)
* diagrams (e.g., see Figure 3)
* animated gifs (e.g., see Figure 4)
* audio (e.g., see Figure 5)
* video (e.g., see Figure 6)

The simplest way to embed media is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload media directly to Wikiversity; upload to Wikimedia Commons instead.

==Finding media==
Only media available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable media:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Media may be uploaded to Commons if:
* Equivalent media isn't already available on Commons
* You created the media or it is licensed for free reuse (e.g., Creative Commons Attribution or public domain)
* The media is educational

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata, including:
** Copyright owner
** Licence type
** Description of image

Once uploaded to Commons, media can be embedded in Wikiversity pages (e.g., see Figures 7 to 8)

==Embedding media==
Media can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure media content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., see Figure 1)

==Customise figure presentation==
Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
[[File:Basic needs.png|200px|right|thumb|'''Figure 7'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 7 was created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 8'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 8 was created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Template:Motivation and emotion/Book chapter structure#Figures|Figures]] (Motivation and emotion book chapter template)
* [[Help:Wikitext quick reference#Images, tables, video, and sounds|Images, tables, video, and sounds]] (Wikiversity help page)
* [[Help:Media|Media]] (Wikiversity help page)
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]] (Motivation and emotion book chapter help page)

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>i5h9bzmmyfrt7ghgt9atvf18mhmtaeg</sha1>
    </revision>
    <revision>
      <id>2815659</id>
      <parentid>2815658</parentid>
      <timestamp>2026-06-14T11:26:41Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <origin>2815659</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5769" sha1="qxbg3ktlxroh3xzhvelj4hugnz84vu0" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example en.svg|150px|right|thumb|'''Figure 1'''. An example image with an APA-style caption, right justified and 150px in size. &lt;nowiki&gt;[Example image]&lt;/nowiki&gt;]]

[[File:Cycling Time Trial effort.jpg|150px|right|thumb|'''Figure 2'''. A key indicator of motivation is the intensity and persistance of effort. &lt;nowiki&gt;[Example photograph]&lt;/nowiki&gt;]]

[[File:Maslow's Hierarchy of Needs.png|360px|right|thumb|'''Figure 3'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base. &lt;nowiki&gt;[Example diagram]&lt;/nowiki&gt;]]

[[File:Hypothalamus.gif|150px|right|thumb|'''Figure 4'''. The hypothalamus regulates fundamental motivations, including eating, drinking, and sex. &lt;nowiki&gt;[Example animated gif]&lt;/nowiki&gt;]]

[[File:Alzheimer's Disease.ogg|150px|right|thumb|'''Figure 5'''. This is a spoken word version of the [[w:Alzheimer's disease|Alzheimer's disease]] Wikipedia article, 2008. &lt;nowiki&gt;[Example audio]&lt;/nowiki&gt;]]

[[File:EN simpleshow foundation Fear of Flying explainer video.webm|150px|right|thumb|'''Figure 6'''. This video explains the psychology behind the fear of flying. &lt;nowiki&gt;[Example video]&lt;/nowiki&gt;]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]] (e.g., see Figure 1). Figures can show a variety media, including:
* photographs (e.g., see Figure 2)
* diagrams (e.g., see Figure 3)
* animated gifs (e.g., see Figure 4)
* audio (e.g., see Figure 5)
* video (e.g., see Figure 6)

The simplest way to embed media is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload media directly to Wikiversity; upload to Wikimedia Commons instead.

==Finding media==
Only media available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable media:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Media may be uploaded to Commons if:
* Equivalent media isn't already available on Commons
* You created the media or it is licensed for free reuse (e.g., Creative Commons Attribution or public domain)
* The media is educational

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata, including:
** Copyright owner
** Licence type
** Description of image

Once uploaded to Commons, media can be embedded in Wikiversity pages (e.g., see Figures 7 to 8)

==Embedding media==
Media can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure media content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., see Figure 1)

==Customise figure presentation==
Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
[[File:Basic needs.png|200px|right|thumb|'''Figure 7'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 7 was created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 8'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 8 was created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Template:Motivation and emotion/Book chapter structure#Figures|Figures]] (Motivation and emotion book chapter template)
* [[Help:Wikitext quick reference#Images, tables, video, and sounds|Images, tables, video, and sounds]] (Wikiversity help page)
* [[Help:Media|Media]] (Wikiversity help page)
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]] (Motivation and emotion book chapter help page)

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>qxbg3ktlxroh3xzhvelj4hugnz84vu0</sha1>
    </revision>
    <revision>
      <id>2815660</id>
      <parentid>2815659</parentid>
      <timestamp>2026-06-14T11:29:55Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Replace File:Maslow's Hierarchy of Needs.png with svg</comment>
      <origin>2815660</origin>
      <model>wikitext</model>
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      <text bytes="5769" sha1="ho5fd1jfv9iaty0cdrovx3p4q48wvy5" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example en.svg|150px|right|thumb|'''Figure 1'''. An example image with an APA-style caption, right justified and 150px in size. &lt;nowiki&gt;[Example image]&lt;/nowiki&gt;]]

[[File:Cycling Time Trial effort.jpg|150px|right|thumb|'''Figure 2'''. A key indicator of motivation is the intensity and persistance of effort. &lt;nowiki&gt;[Example photograph]&lt;/nowiki&gt;]]

[[File:Maslow's Hierarchy of Needs.svg|360px|right|thumb|'''Figure 3'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base. &lt;nowiki&gt;[Example diagram]&lt;/nowiki&gt;]]

[[File:Hypothalamus.gif|150px|right|thumb|'''Figure 4'''. The hypothalamus regulates fundamental motivations, including eating, drinking, and sex. &lt;nowiki&gt;[Example animated gif]&lt;/nowiki&gt;]]

[[File:Alzheimer's Disease.ogg|150px|right|thumb|'''Figure 5'''. This is a spoken word version of the [[w:Alzheimer's disease|Alzheimer's disease]] Wikipedia article, 2008. &lt;nowiki&gt;[Example audio]&lt;/nowiki&gt;]]

[[File:EN simpleshow foundation Fear of Flying explainer video.webm|150px|right|thumb|'''Figure 6'''. This video explains the psychology behind the fear of flying. &lt;nowiki&gt;[Example video]&lt;/nowiki&gt;]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]] (e.g., see Figure 1). Figures can show a variety media, including:
* photographs (e.g., see Figure 2)
* diagrams (e.g., see Figure 3)
* animated gifs (e.g., see Figure 4)
* audio (e.g., see Figure 5)
* video (e.g., see Figure 6)

The simplest way to embed media is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload media directly to Wikiversity; upload to Wikimedia Commons instead.

==Finding media==
Only media available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable media:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Media may be uploaded to Commons if:
* Equivalent media isn't already available on Commons
* You created the media or it is licensed for free reuse (e.g., Creative Commons Attribution or public domain)
* The media is educational

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata, including:
** Copyright owner
** Licence type
** Description of image

Once uploaded to Commons, media can be embedded in Wikiversity pages (e.g., see Figures 7 to 8)

==Embedding media==
Media can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure media content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., see Figure 1)

==Customise figure presentation==
Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==Examples==
[[File:Basic needs.png|200px|right|thumb|'''Figure 7'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 7 was created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 8'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 8 was created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

* [[Motivation and emotion/Gallery|Motivation and emotion gallery]]
* [[Template:Motivation_and_emotion/Book_chapter_structure#Figures|Book chapter figure examples]]

==See also==
* [[Template:Motivation and emotion/Book chapter structure#Figures|Figures]] (Motivation and emotion book chapter template)
* [[Help:Wikitext quick reference#Images, tables, video, and sounds|Images, tables, video, and sounds]] (Wikiversity help page)
* [[Help:Media|Media]] (Wikiversity help page)
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]] (Motivation and emotion book chapter help page)

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>ho5fd1jfv9iaty0cdrovx3p4q48wvy5</sha1>
    </revision>
    <revision>
      <id>2815661</id>
      <parentid>2815660</parentid>
      <timestamp>2026-06-14T11:33:15Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Remove redundant section for examples</comment>
      <origin>2815661</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5613" sha1="4dq7xlb4hhud2f1f86a1uwmpog7iz8u" xml:space="preserve">&lt;noinclude&gt;{{title|Working with figures}}&lt;/noinclude&gt;

[[File:Example en.svg|150px|right|thumb|'''Figure 1'''. An example image with an APA-style caption, right justified and 150px in size. &lt;nowiki&gt;[Example image]&lt;/nowiki&gt;]]

[[File:Cycling Time Trial effort.jpg|150px|right|thumb|'''Figure 2'''. A key indicator of motivation is the intensity and persistance of effort. &lt;nowiki&gt;[Example photograph]&lt;/nowiki&gt;]]

[[File:Maslow's Hierarchy of Needs.svg|360px|right|thumb|'''Figure 3'''. Maslow's hierarchy of needs represented as a pyramid, with physiological needs at the base. &lt;nowiki&gt;[Example diagram]&lt;/nowiki&gt;]]

[[File:Hypothalamus.gif|150px|right|thumb|'''Figure 4'''. The hypothalamus regulates fundamental motivations, including eating, drinking, and sex. &lt;nowiki&gt;[Example animated gif]&lt;/nowiki&gt;]]

[[File:Alzheimer's Disease.ogg|150px|right|thumb|'''Figure 5'''. This is a spoken word version of the [[w:Alzheimer's disease|Alzheimer's disease]] Wikipedia article, 2008. &lt;nowiki&gt;[Example audio]&lt;/nowiki&gt;]]

[[File:EN simpleshow foundation Fear of Flying explainer video.webm|150px|right|thumb|'''Figure 6'''. This video explains the psychology behind the fear of flying. &lt;nowiki&gt;[Example video]&lt;/nowiki&gt;]]

[[File:Basic needs.png|200px|right|thumb|'''Figure 7'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential ingredients for intrinsic motivation and psychological well-being.&lt;ref&gt;Figure 7 was created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

[[File:Implicit Motives associations.png|400px|right|thumb|'''Figure 8'''. Implicit motives affect cognition, affect, and behaviour outside conscious awareness. This diagram illustrates major sources of implicit motives.&lt;ref&gt;Figure 8 was created by a Motivation and emotion student in 2020 and uploaded to [[c:|Wikimedia Commons]].&lt;/ref&gt;]]

Figures can be a powerful way to communicate and illustrate concepts in [[Motivation and emotion/Book|motivation and emotion book chapters]] (e.g., see Figure 1). Figures can show a variety media, including:
* photographs (e.g., see Figure 2)
* diagrams (e.g., see Figure 3, Figure 7, and Figure 8)
* animated gifs (e.g., see Figure 4)
* audio (e.g., see Figure 5)
* video (e.g., see Figure 6)

The simplest way to embed media is to use material already available in [[commons:Wiki commons|Wikimedia Commons]].

Do not upload media directly to Wikiversity; upload to Wikimedia Commons instead.

==Finding media==
Only media available on Wikimedia Commons can be embedded on Wikiversity pages. Several pathways can be used to locate suitable media:
* Explore and search [[commons:Wikimedia Commons|Wikimedia Commons]] directly
* Use Wikimedia Commons search via Wikimedia tools:
** Category browsing (often structured but uneven in quality), e.g. [[commons:Category:Emotions|Category:Emotions]]
** Search results with thumbnails, e.g. [https://commons.wikimedia.org/w/index.php?search=worried worried]
** Visual Editor search (Insert → Media → File name)
* Use external search tools with licensing filters:
** [https://www.google.com/advanced_image_search Google Images Advanced Search] (set Tools → Usage rights → Creative Commons)
** [https://search.creativecommons.org/ Creative Commons search engine]
** [https://www.google.com/search?tbm=isch&amp;q=site%3Acommons.wikimedia.org Google Image search restricted to Commons]
* See the [[Motivation and emotion/Gallery|motivation and emotion gallery]] for some starting images

==Uploading to Wikimedia Commons==
Media may be uploaded to Commons if:
* Equivalent media isn't already available on Commons
* You created the media or it is licensed for free reuse (e.g., Creative Commons Attribution or public domain)
* The media is educational

Upload process:
* Go to [[c:|Wikimedia Commons]]
* Select Upload Wizard
* Provide required metadata, including:
** Copyright owner
** Licence type
** Description of image

Once uploaded to Commons, media can be embedded in Wikiversity pages (e.g., see Figures 7 to 8)

==Embedding media==
Media can be inserted using the Visual Editor:
* Edit page → Insert → Media → File
* Add:
** Caption
** Size
**Position

Best practice:
* Ensure media content directly supports surrounding text
* Use consistent numbering across figures within a page
* Add at least one in-text citation (e.g., see Figure 1)

==Customise figure presentation==
Figures should follow clear academic conventions:
* Size
** Optimise for viewing (not too large or small)
** Increase size for complex diagrams requiring readability
* Position
** Right alignment is standard for text flow
** Centre alignment is suitable for conceptual models or key figures
** Left alignment is rarely used
* Captions
** Must be descriptive and interpretable without the main text
* Referencing
** Each figure must be explicitly cited in text (e.g., see Figure 1)
* Integration
** Figures must directly support the argument or explanation in adjacent text

==See also==
* [[Template:Motivation and emotion/Book chapter structure#Figures|Figures]] (Motivation and emotion book chapter template)
* [[Help:Wikitext quick reference#Images, tables, video, and sounds|Images, tables, video, and sounds]] (Wikiversity help page)
* [[Help:Media|Media]] (Wikiversity help page)
* [[Motivation and emotion/Assessment/Chapter/Tables|Tables]] (Motivation and emotion book chapter help page)

==References==
{{reflist}}
&lt;noinclude&gt;
[[Category:Motivation and emotion/Wikiversity]]
[[Category:Motivation and emotion/Assessment/Chapter]]&lt;/noinclude&gt;</text>
      <sha1>4dq7xlb4hhud2f1f86a1uwmpog7iz8u</sha1>
    </revision>
  </page>
  <page>
    <title>Eventmath/Lesson plans/Proportions and voting power under the Electoral College</title>
    <ns>0</ns>
    <id>282260</id>
    <revision>
      <id>2815568</id>
      <parentid>2443870</parentid>
      <timestamp>2026-06-13T19:31:07Z</timestamp>
      <contributor>
        <username>Thomascampbell123</username>
        <id>3093541</id>
      </contributor>
      <minor />
      <comment>([[c:GR|GR]]) [[c:COM:FR|File renamed]]: [[File:California Presidential Election Results 2020.svg]] → [[File:2020 United States presidential election in California results map by county.svg]] [[c:Commons:WikiProject Elections and Referendums/USA election map naming conventions]]</comment>
      <origin>2815568</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21562" sha1="r2yw5wftuwfsqlmfknpz4ppkrdpgyb4" xml:space="preserve">&lt;!-- DON'T MESS WITH THIS BIT --&gt;
{{Eventmath draft header}}
&lt;div style=&quot;background-color: #DAF3EF; padding: 1em; overflow:hidden;&quot;&gt;
&lt;div style=&quot;overflow:hidden;&quot;&gt;
&lt;!--CONTRIBUTOR INSTRUCTIONS
1. This text is a comment: it'll be hidden from the published lesson plan, so it's just for you!
   If you see unfamiliar code, look for an explanation in a comment like this.
2. Heads up: you'll find a comment at the bottom with simple instructions on adding &quot;category tags,&quot; 
   which will help people find this lesson plan. Please follow them :)
--&gt;
&lt;div style=&quot;width:50%; margin-right:4em; float:left;&quot;&gt;
{{Eventmath lesson plan overview
&lt;!--EDIT BELOW THIS LINE ONLY--&gt;
&lt;!-- The summary below is displayed as a guide for visitors and contributors. You can publish without it, but it's required for removal from draft status.--&gt;

|assumed-knowledge    = &lt;!--Below, please briefly describe any mathematical skills or background knowledge students should have before completing this lesson.--&gt;

Proportions and basic spreadsheet calculations. Basic mechanics of the Electoral College in the United States (see [[Eventmath/Lesson_plans/Proportions_and_voting_power_under_the_Electoral_College#Resources|Resources]]).

|activities           = &lt;!--Below, please briefly summarize what students will do during this lesson and what they will learn.--&gt;

Students will propose different measures of the &quot;individual voter power&quot; of residents in a state, based on its population and its representation in the Electoral College. They will calculate these measures in a spreadsheet and compare their findings to the article's main assertion.

|class-time           = 60-90 minutes &lt;!--Below, please type one of the following, exactly as it appears: 
                         00-15 minutes, 15-30 minutes, 30-45 minutes, 45-60 minutes, 60-90 minutes, more than 90 minutes.
                         Your best guess is fine! It can be changed later.--&gt;

&lt;!--Next, you'll provide all citation data for the source (article/social media post/video).

Article example:
| source-website      = Huffington Post
| source-title        = Voters In Wyoming Have 3.6 Times The Voting Power That I Have. It’s Time To End The Electoral College.
| source-date         = 2016-11-10
| source-url          = https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764
| source-archive-url  = https://web.archive.org/web/20201210161426/https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764
| source-archive-date = 2021-10-28

Post example:
| source-website      = Twitter
| source-title        = Remember: Germany is supposed to be a SUCCESS STORY.
| source-date         = 2020-12-22
| source-url          = https://twitter.com/Neoavatara/status/1341581271991013376
| source-archive-url  = https://web.archive.org/web/20201223064324/https://twitter.com/Neoavatara/status/1341581271991013376
| source-archive-date = 2021-02-16

Notes:
1. To ensure lesson plans can be sorted by date, please enter all dates in the format YYYY-MM-DD,
   including leading zeros: January is 01, not 1, for example. 

2. If you cannot find a date, use n.d. as the date (no date).

3. To ensure that the article/post will remain accessible for educators and students...
   a. Archive it at https://archive.org/web/ (use the &quot;Save Page Now&quot; form to generate a permanent URL).
   b. Paste that URL here as the source-archive-url.
   c. Enter the date you archived the source as the source-archive date.

4. Linking to a website helps the site. If the lesson plan is debunking disinformation, you may 
   avoid spreading disinformation by omitting the url of the original site entirely. In that case, 
   use the URL of the archived version as the value of source-url, and omit values for archive-url and archive-date.
--&gt;

| source-website      = Huffington Post
| source-title        = Voters In Wyoming Have 3.6 Times The Voting Power That I Have. It's Time To End The Electoral College.
| source-date         = 2016-11-10
| source-url          = https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764
| source-archive-url  = https://web.archive.org/web/20201210161426/https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764
| source-archive-date = 2020-12-10
}}
&lt;/div&gt;
&lt;div style=&quot;float:left; width:20%;&quot;&gt;
[[File:Eventmath_logo_bitmap_graphic.png|250px]]
&lt;/div&gt;
&lt;/div&gt;
&lt;div style=&quot;width:70%; float:left;&quot;&gt;
&lt;!--INSTRUCTIONS FOR THE BODY OF THE LESSON PLAN
1. Note that &quot;==My heading==&quot; is a level 2 section heading, &quot;===My subheading===&quot; is level 3, etc. Level 1 is reserved for the title.
2. After you fill in a section, please delete the ''italicized instructions.''
3. Check out the &quot;Advanced&quot; and &quot;Help&quot; menus at the top of the editor if you don't know how to format lists, etc.
--&gt;
==Activities==
===Discussion of articles===
# '''Preparation:''' Read the following article before class: [https://thehill.com/opinion/judiciary/519826-let-us-abolish-the-electoral-college &quot;Let us abolish the Electoral College,&quot;] October 6, 2020, ''The Hill''. Discuss with students whether the stated opinions are supported by evidence, especially numerical evidence.
# '''Preparation:''' Read the main article from ''Huffington Post'', although it's so short that perhaps this could be done together during class. Discuss with students whether this provides more contextual information than the first article.
# '''Discussion:''' Ask students whether the value of 3.6 in the title of the ''Huffington Post'' article refers to the ratio of a Wyomingite voter's power to a Californian voter's power or whether it refers to the reciprocal.
# '''Discussion:''' Ask students to define a quantified measure of individual voter power by state. Encourage them to think of more than one way to do this, emphasizing that there may not necessarily be &quot;one correct way.&quot; However, for the sake of the rest of the lesson, make sure that students eventually decide to include at least these two measures:
## Number of residents / number of Electoral College votes.
## Number of Electoral College votes / number of residents. (This is the reciprocal of the first one.)

===Data collection and analysis===
[[File:Faenza-x-office-spreadsheet.svg|right|350px|spreadsheet]]
The main activity now is to create a spreadsheet containing the most recent census data showing each state's population, the number of Electoral College votes each state has, and the individual voter power in each state according to each of the proposed measures stated above. (For the instructor's convenience, [https://docs.google.com/spreadsheets/d/1HOca2MKYheQ6gJirmaSazNGvAlYOj_OLnr8rYaRiDvI/edit?usp=sharing here is a Google Sheets file] with each state's population -- both 2010 and 2020 census data -- and their # of Electoral College votes.)

For the data collection, you may consider having students work on this in groups so they can continue the task as homework later. Or, you may consider working together as a class to create one spreadsheet, perhaps by assigning individual tasks to groups of students (e.g. &quot;This group will find recent census data while this group sets up the spreadsheet calculations.&quot;)

For the data analysis, you may consider having some prepared discussion questions or a worksheet of problems for groups of students to solve using the spreadsheet. For example, you might ask:
* Which state's individuals have the smallest voting power by measure #1? What about measure by #2?
* Which state's individuals have the largest voting power by measure #1, and by measure #2? Could there theoretically be a state with a larger value than that? How large do you think it could be?
* Which states are traditionally considered &quot;influential&quot; in presidential elections? What are the values of their individual voting power, and what does that tell us, if anything?

===Concluding discussion===
[[File:2020 United States presidential election in California results map by county.svg|right|300px|California Presidential Election Results 2020]]
Ask students some probing questions to inspire further discussion, such as:
* Which of these measures is more helpful for quantifying what we actually want to measure?
* Which of these measures is more helpful for understanding the real world situation in question?
* Which of these measures do the articles actually use?
* And do the articles make all of this information clear?

Finally, use all of the data and any points raised in discussion to compare Wyoming and California, as the article does:
* Ask students whether they agree with the article’s headline, or whether they might have written it differently.
* Discuss with the students why they think Wyoming and California were chosen, out of all the states. How might the headline have been different if other states were chosen as comparison points?

===Instructor notes===
====Comparison of the proposed measures====
&lt;!--Table copied from https://en.wikipedia.org/wiki/Scientific_notation
    and modified because the Val template on Wikiversity needs to be updated --&gt;
{| class=&quot;wikitable&quot; style=&quot;float:right; margin:5px;&quot;
|-
!Decimal notation
!Scientific notation
|-
|&lt;math&gt;2&lt;/math&gt;
|&lt;math&gt;2\times 10^0&lt;/math&gt;
|-
|&lt;math&gt;300&lt;/math&gt;
|&lt;math&gt;3\times 10^2&lt;/math&gt;
|-
|&lt;math&gt;4321.768&lt;/math&gt;
|&lt;math&gt;4.321768\times 10^3&lt;/math&gt;
|-
|&lt;math&gt;-53000&lt;/math&gt;
||&lt;math&gt;-5.3\times 10^4&lt;/math&gt;
|-
|&lt;math&gt;6720000000&lt;/math&gt;
|&lt;math&gt;6.72\times 10^9&lt;/math&gt;
|-
|&lt;math&gt;0.2&lt;/math&gt;
|&lt;math&gt;2\times 10^{-1}&lt;/math&gt;
|-
|&lt;math&gt;987&lt;/math&gt;
|&lt;math&gt;9.87\times 10^2&lt;/math&gt;
|- 
|&lt;math&gt;0.00000000751&lt;/math&gt;
|&lt;math&gt;7.51\times 10^{-9}&lt;/math&gt;
|}
Discussing these finer points with students might help them better understand how we make conventional choices about definitions in mathematics.
* If you define individual voter power in a state as the number of residents per Electoral College vote (calculation: number of residents / number of Electoral College votes), then you will find that  Wyoming has the absolute minimum and California has the absolute maximum. However, this is arguably &quot;incorrect&quot; for two reasons:
** The ratio of individual voter power in Wyoming to individual voter power in California would be 1/3.6 and not 3.6, as the headline states. So, this measure is at least inconsistent with the article's claims.
** Moreover, this measure is inconsistent with our intuition about how individual voter power might change. For example, if the number of Electoral College votes allotted to a given state were to instantly increase (say, due to a legislative action), then we would expect the power of each individual voter in that state to increase, as well. However, the number of residents per Electoral College vote would decrease.
* If you define individual voter power in a state as the amount of an Electoral College vote that each voter represents (calculation: number of Electoral College votes / number of residents), then you will find that Wyoming has the absolute maximum and California has the absolute minimum, and the ratio of those ratios is 3.6.
** This supports the article’s claim. Since this definition is consistent with intuition, as well, it better captures the quantity we are trying to measure.
** However, this measure leads to tiny numbers. This may prompt a mini-lesson on scientific notation or other ways of expressing and working with large/small numbers. Moreover, you may have to help students format the spreadsheet to display the values correctly and helpfully.

====Limitations of the proposed measures====
Each of the measures is subject to limitations. For example, Wyoming and California are currently [https://en.wikipedia.org/wiki/United_States_Electoral_College#Summary winner-take-all states], and [https://en.wikipedia.org/wiki/United_States_presidential_elections_in_Wyoming Wyoming consistently votes for Republicans in presidential elections], whereas [https://en.wikipedia.org/wiki/United_States_presidential_elections_in_California California consistently votes for Democrats]. So, it is likely that all of Wyoming’s electors in a modern election will vote for the Republican candidate, in which case a Wyomingite who votes for a Democrat would not be represented in the Electoral College vote. The same can be said of a voter in California who votes for a Republican. Neither measure discussed in this lesson plan accounts for this issue, or others like it.

Despite such limitations, these quantitative measures do provide a starting point for discussion. An essay prompt in the [[Eventmath/Lesson_plans/Proportions_and_voting_power_under_the_Electoral_College#Assignments|''Assignments'' section]] is intended to help students explore this tension.

====Why Wyoming and California?====
Why were Wyoming and California chosen as the highlights of the article? For one, the author lives in California. For another, they make for the most striking and disparate comparison between any pair of states, at least for the 2010 census data.

==Assignments==
There are many directions an assignment could take with this lesson. Here are just some examples:
&lt;div style=&quot;padding-top: 0.5em;&quot;&gt;
[[File:Composition Notebook (5736841980).jpg|left|250px|alt=Composition Notebook]]
&lt;/div&gt;
&lt;div style=&quot;overflow-x: hidden; padding-left: 1em;&quot;&gt;
* ''Discussion/Essay prompt:'' Ask students to write a headline that would more accurately represent what the methods in the article itself imply about the comparison between California and Wyoming, and to describe why they believe that headline is a more accurate representation.
* ''Essay prompt:'' Ask students to write an op-ed of their own that either rebuts the original article or provides more evidence in support of it. (It may be important to remind students that they can disagree with an argument without necessarily disagreeing with the conclusion, for instance.)
* ''Assignment prompt:'' Ask students to create an informative, shareable explanation of how the headline of the original ''Huffington Post'' article may mislead someone who doesn’t read the rest of the article. It may help to ask them to pretend they’re making something for their friends or family to see, like a Twitter thread or TikTok video.
* ''Project prompt:'' Ask students to recreate the data collection and analysis for presidential elections in the past. It may be interesting to look at population and Electoral College data from 20, 50, or 100 years ago and compare individual voter power in different states. At what point in time was there the largest disparity between states? Is this 3.6 value (WY/CA) the largest ratio, historically? And how has the voting power of individuals in a state changed over time?
* ''Extended project prompt:'' This idea goes well beyond the basic mathematics of the earlier parts of this lesson, but it may work for your context: What if the USA used a different apportionment scheme for allocating representatives in Congress? Students could run various algorithms to allocate reps and adjust the data in the spreadsheet accordingly. (Information about how representatives are apportioned to the states can be found [https://en.wikipedia.org/wiki/United_States_congressional_apportionment#Apportionment_methods on Wikipedia] and on the [https://www.census.gov/topics/public-sector/congressional-apportionment/about.html website of the U.S. Census Bureau].)
* ''Essay prompt:'' In the article [https://digitalcommons.law.villanova.edu/cgi/viewcontent.cgi?article=1780&amp;context=vlr &quot;One Man, 3.312 Votes: A Mathematical Analysis of the Electoral College,&quot;] John F. Banzhaf III states that &quot;the theoretical voting power of an individual may not coincide with his actual ability to affect the outcome of any particular election,&quot; citing issues such as &quot;the political impotence of a Republican in a solidly Democratic state.&quot; Does this statement apply to the measure of individual voter power you chose? Describe any limitations of your chosen measure, and discuss the extent to which they affect its overall utility and explanatory power.
&lt;/div&gt;

==Resources==

===Background===
&lt;!-- Electoral College image and caption copied from https://en.wikipedia.org/wiki/United_States_Electoral_College --&gt;
[[File:ElectoralCollege2024.svg|thumb|upright=1.35|Electoral votes, out of 538, allocated to each [[U.S. state|state]] and the [[District of Columbia]] for [[United States presidential election|presidential elections]] to be held in [[2024 United States presidential election|2024]] and 2028, based on representation, which depends on population data from the [[2020 United States census|2020 census]]. Every jurisdiction is entitled to at least 3.]]
There is not much mathematical background presumed for this lesson plan, only a facility with calculations and proportions. Here is a [https://www.khanacademy.org/test-prep/praxis-math/praxis-math-lessons/praxis-math-number-and-quantity/a/gtp--praxis-math--article--ratios-and-proportions--lesson Khan Academy lesson on ratios and proportions].

Part of implementing this lesson plan may involve first teaching students the basic mechanics of the Electoral College. Here are a few resources for that:
* &quot;How the Electoral College works: A guide to the complex system the United States uses to select a president,&quot; from [https://graphics.reuters.com/USA-ELECTION/ELECTORAL-COLLEGE/qzjpqaeqapx/ Reuters]
* &quot;How the Electoral College Works,&quot; from [https://youtu.be/OUS9mM8Xbbw CGP Grey on YouTube]
* &quot;Does your vote count? The Electoral College explained - Christina Greer,&quot; from [https://youtu.be/W9H3gvnN468 TED-Ed on YouTube]

===Explorations===
Students may ask whether there are other resources that have defined this notion of &quot;individual voter power,&quot; or you may wish to assign a project that asks students to find any and compare to their findings from this lesson. 

* Example from the popular press: ''Wallet Hub'' has a list of [https://wallethub.com/edu/how-much-is-your-vote-worth/7932 &quot;States with the Most &amp; Least Powerful Voters&quot;] and a methodology section at the end that describes how they defined &quot;powerful.&quot; Here's an [https://web.archive.org/web/20220324095524/https://wallethub.com/edu/how-much-is-your-vote-worth/7932 archived version of the same page], in case the first link doesn't work.
* Example from the research literature: Banzhaf III, John F. &quot;[https://digitalcommons.law.villanova.edu/cgi/viewcontent.cgi?article=1780&amp;context=vlr One man, 3.312 votes: a mathematical analysis of the Electoral College.]&quot; ''Vill. L. Rev.'' 13 (1968): 304.
Finally, students may also be interested in public opinion around these issues. For example, a recent Pew Research report claims [https://www.pewresearch.org/fact-tank/2022/08/05/majority-of-americans-continue-to-favor-moving-away-from-electoral-college/ &quot;Majority of Americans continue to favor moving away from Electoral College&quot;]. Students may want to read this as part of the class discussion about this topic, or they may want to use this information when writing a response essay of their own.

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  <page>
    <title>C language in plain view</title>
    <ns>0</ns>
    <id>285380</id>
    <revision>
      <id>2815491</id>
      <parentid>2815346</parentid>
      <timestamp>2026-06-13T13:26:55Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2815491</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="rnux31z090p4iwr4mo8ty71gjnvulf0" xml:space="preserve">=== 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 &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; 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.20260613.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


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

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]]

&lt;/br&gt;</text>
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      <contributor>
        <username>Alexis Jazz</username>
        <id>791434</id>
      </contributor>
      <comment>Updating gadget usage statistics from [[Special:GadgetUsage]] ([[phab:T121049]])</comment>
      <origin>2815526</origin>
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      <text bytes="1280" sha1="ton4zw0tb73llasyohsdkqqawx90x5o" xml:space="preserve">{{#ifexist:Project:GUS2Wiki/top|{{/top}}|This page provides a historical record of [[Special:GadgetUsage]] through its page history. To get the data in CSV format, see wikitext. To customize this message or add categories, create [[/top]].}}
The following data is cached, and was last updated 2026-06-13T11:02:15Z. A maximum of {{PLURAL:5000|one result is|5000 results are}} available in the cache.
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  <page>
    <title>User:Alandmanson/Hymenoptera of Africa</title>
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      <id>2815508</id>
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      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Superfamily Apoidea */</comment>
      <origin>2815508</origin>
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      <text bytes="15149" sha1="8xnbzf3pd2yt6pe7vppb1tkugoh5uug" xml:space="preserve">= Superfamily Apoidea =
== African Ampulicidae ==
== African Astatidae ==
== African Bembicidae ==
===Subfamily Bembicinae===	
&lt;gallery mode=packed heights=150&gt;
Bembecinus iN 153052948.jpg|''Bembecinus'' sp.
Bembix iN 195581985.jpg|''Bembix'' sp.
Hoplisoides thalia iN 268375748 a.jpg|''Hoplisoides thalia''
&lt;/gallery&gt;		
Tribe Gorytini	
*		''Afrogorytes''
*		''Gorytes''
*		''Harpactus''
*		''Hoplisoides''
*		''Lestiphorus''
		
Tribe Spheciini	
*		''Ammatomus''
*		''Kohlia''
*		''Sphecius''
		
Tribe Handlirschiini	
*		''Handlirschia''
		
Tribe Bembicini	
*		''Bembix''
		
Tribe Stizini	
*		''Bembecinus''
*		''Stizoides''
*		''Stizus''

===Subfamily Nyssoninae===	
*		''Brachystegus''
*		''Hovanysson''
*		''Nysson''
		
===Subfamily Alyssontinae===	
*		''Alysson''
*		''Didineis''

== African Crabronidae ==
== African Pemphredonidae ==
== African Philanthidae ==
== African Psenidae ==
== African Sphecidae ==

= Superfamily Chalcidoidea =
 
Chalcidoidea is an incredibly diverse group of wasps with a wide variety of life histories; many are parasitoids, making them ecologically important; some are used for biological control of insect pests. About 22 000 species have been described, but as many as 500 000 are thought to exist worldwide.&lt;ref name=Noyes2019/&gt;

Most chalcidoids are small (body length less than 5 mm), the smallest being males of the mymarid ''Dicopomorpha echmepterygis'', which is the smallest known flying insect (0.13 mm).&lt;ref name=Noyes2019&gt;Noyes, J.S. 2019. Universal Chalcidoidea Database. [http://www.nhm.ac.uk/chalcidoids]&lt;/ref&gt; However, some species may be as long as 20 mm (excluding antennae and ovipositor), such as the pelecinellid ''Doddifoenus wallacei''.&lt;ref name=Krogmann&amp;Burks2009&gt;Krogmann, Lars; Burks, Roger A. (2009-12-31). &quot;Doddifoenus wallacei, a new giant parasitoid wasp of the subfamily Leptofoeninae (Chalcidoidea: Pteromalidae), with a description of its mesosomal skeletal anatomy and a molecular characterization&quot;. Zootaxa. 2194: 21–36. doi:10.5281/zenodo.189452 [https://www.researchgate.net/profile/Lars-Krogmann/publication/250916337 PDF]&lt;/ref&gt; Many have bright metallic colours, hence the name Chalcidoidea or chalcid - the greek word χαλκός (chalcos) means copper or bronze.&lt;ref name=Perseus2023&gt;[http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0073:entry=xalko/s Georg Autenrieth, A Homeric Dictionary]&lt;/ref&gt;

Many new chalcidoid families were proposed in 2022.&lt;ref name=Burks2022&gt;Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. [https://doi.org/10.3897/jhr.94.94263 DOI]&lt;/ref&gt; This was largely based on phylogenetic evidence (both molecular and morphological) accumulated since the families of the Superfamily were outlined in 1993&lt;ref name=Gibson1993&gt;Gibson, G.A.P. (1993) Superfamilies Mymarommatoidea and Chalcidoidea (pp. 570-655). In Goulet, H. &amp; Huber, J. (eds). Hymenoptera of the World: an identification guide to families. Research Branch, Agriculture Canada, Ottawa, Canada, 668 pp. [https://www.researchgate.net/publication/259227143 PDF]&lt;/ref&gt;.


Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. https://doi.org/10.3897/jhr.94.94263

Cruaud, A., Rasplus, J.Y., Zhang, J., Burks, R., Delvare, G., Fusu, L., Gumovsky, A., Huber, J.T., Janšta, P., Mitroiu, M.D. and Noyes, J.S., 2022. The Chalcidoidea bush of life–a massive radiation blurred by mutational saturation. bioRxiv. https://doi.org/10.1101/2022.09.11.507458 https://www.biorxiv.org/content/10.1101/2022.09.11.507458.full.pdf


= Superfamily Ichneumonoidea =

Quicke, D.L., 2015. The braconid and ichneumonid parasitoid wasps: biology, systematics, evolution and ecology. John Wiley &amp; Sons.
https://www.researchgate.net/publication/281662102_The_Braconid_and_Ichneumonid_Parasitoid_Wasps_Biology_Systematics_Evolution_and_Ecology

=== Diplazontinae ===
Quote from Fitton &amp; Rotheray (1982):
&quot;Diplazontines are distinguished easily from other ichneumonids by their general appearance (Fig. 1). The main confirmatory characters are the bifid upper tooth of the mandible (Fig. 2) and the sub-rectangular shape of the first tergite of the gaster (Figs. 1,3,4, 13, 15 and 17). In fact, the diplazontines are so distinct that they can be separated from all other Hymenoptera-Apocrita by possession of the following five characters in combination: (1) antenna of seventeen or more segments (Fig. I); (2) forewing with costal cell obliterated (that is, veins C and Sc + R + Rs contiguous from the wing base to the base of the pterostigma (Fig. 6 ) ; (3) forewing with cross-vein 2 mcu present (Fig. 6); (4) mandible with its upper tooth subdivided (bifid), making the mandible tridentate (Fig. 2); (5) first tergite of gaster sub-rectangular in shape (that is, subequal in width anteriorly and posteriorly) (Figs. 1 , 3 , 4 , 13,15 and 17).&quot;

Fitton, M. G., &amp; Rotheray, G. E. (1982). A key to the European genera of diplazontine ichneumon‐flies, with notes on the British fauna. Systematic Entomology, 7(3), 311-320.

=== Superfamily Pompiloidea ===
Family Mutillidae (Velvet ants)
Family Pompilidae (Spider hunting wasps)
Family Sapygidae (Sapygid wasps)


== Spider-hunting wasps ==
Two wasps from the genus ''Auplopus'' from Europe. Most of the ''Auplopus'' species, along with others from the Tribe Ageniellini, amputate the legs of the spiders they capture; this makes it much easier to move the spider to the nest.
&lt;gallery mode=packed heights=200&gt;
Auplopus carbonarius IMG 1624.jpg
Auplopus carbonarius fg01 20060623 Nied Garten.jpg
&lt;/gallery&gt;
African genera of Ageniellini:
* ''[[Auplopus]]'' &lt;small&gt;Spinola, 1841&lt;/small&gt;
* ''[[Cyemagenia]]'' &lt;small&gt;Arnold, 1946&lt;/small&gt;
* ''[[Dichragenia]]'' &lt;small&gt;Haupt, 1950&lt;/small&gt;
* ''[[Phanagenia]]'' &lt;small&gt;Banks, 1933&lt;/small&gt;   Madagascar
* ''[[Poecilagenia]]'' &lt;small&gt;Haupt, 1926&lt;/small&gt;

== Superfamily Scolioidea ==
=== Family Bradynobaenidae (Bradynobaenid wasps) ===
=== Family Scoliidae (Mammoth wasps) ===

== Superfamily Tiphioidea ==
Family Tiphiidae (Tiphiid wasps)
== Superfamily Thynnoidea ==
Family Thynnidae(Thynnid wasps)
== Superfamily Vespoidea ==
=== Family Rhopalosomatidae (Rhopalosomatid wasps) ===
===Family Vespidae (Paper, Potter &amp; Pollen wasps)===


== '''&lt;big&gt;Hymenoptera&lt;/big&gt;''' ==
About 20 000 described species of [[w:Hymenoptera|Hymenoptera]] (wasps, bees, ants and sawflies) are known from the [[w:Afrotropical|Afrotropical]] region. Estimates of the actual species count for the region range from 100 000 species to as high as 500 000 species.&lt;ref name=WaspWebOverview&gt;[https://www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm van Noort, Simon (2023). WaspWeb: Afrotropical Hymenoptera Initiative. www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm Retrieved 22 February 2023.]&lt;/ref&gt; Nineteen different superfamilies are illustrated on [[User:Alandmanson/Hymenoptera of southern Africa|this gallery]] in Wikimedia Commons.

The huge number of undescribed species means that many species will be extinct before we are even aware of them, as there are too few taxonomists employed to tackle the task of describing all of them in the next few decades.

==Classification==
The Order can be split into two Suborders - The [[w:Symphyta|Symphyta]] (Woodwasps, Horntails, Sawflies) and the [[w:Apocrita|Apocrita]] (Narrow-waisted wasps, ants and bees).&lt;br&gt;

When compared to the temperate regions of the northern hemisphere, the diversity of Symphyta in Africa is relatively poor, although sawflies of the Superfamily [[w:Tenthredinoidea|Tenthredinoidea]] are fairly common in forests and other woody vegetation types.

Africa has a rich diversity of Apocrita. The cladogram shown below&lt;ref name=Peters&gt;{{Cite journal |last1=Peters |first1=Ralph S. |last2=Krogmann |first2=Lars |last3=Mayer |first3=Christoph |last4=Donath |first4=Alexander |last5=Gunkel |first5=Simon |last6=Meusemann |first6=Karen |last7=Kozlov |first7=Alexey |last8=Podsiadlowski |first8=Lars |last9=Petersen |first9=Malte |title=Evolutionary History of the Hymenoptera |journal=Current Biology |volume=27 |issue=7 |pages=1013–1018 |doi=10.1016/j.cub.2017.01.027|pmid=28343967 |year=2017 |doi-access=free }}&lt;/ref&gt; indicates the possible relationships between 11 of the superfamilies that comprise Apocrita; These 11 superfamilies are all represented in Africa. This breakdown is used by iNaturalist.&lt;ref name=inat-Aculeata&gt;https://www.inaturalist.org/taxa/326777-Aculeata&lt;/ref&gt; It is, however, not accepted by all hymenopterists, and may change as more phylogenetic evidence is accumulated.&lt;ref name=waspwebClass&gt;[https://www.waspweb.org/Classification/index.htm van Noort, Simon (2023). WaspWeb: Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants). www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
{{Clade|style=font-size:85%; line-height:85%
|label1=Apocrita
            |1={{Clade
               |label1=[[w:Parasitoida|Parasitoida]]
               |1={{Clade
                  |1={{Clade
                     |1=[[w:Ceraphronoidea|Ceraphronoidea]]
                     |2=[[w:Ichneumonoidea|Ichneumonoidea]]
                     }}
                  |label2=[[w:Proctotrupomorpha|Proctotrupomorpha]]
                  |2={{Clade
                     |1=[[w:Cynipoidea|Cynipoidea]]
                     |2={{Clade
                        |1=[[w:Platygastroidea|Platygastroidea]]
                        |2={{Clade
                           |1=[[w:Chalcidoidea|Chalcidoidea]]
                           |2={{Clade
                              |1=[[w:Diaprioidea|Diaprioidea]]
                              |2=[[w:Proctotrupoidea|Proctotrupoidea]]
                  }} }} }} }} }}
               |2={{Clade
                  |1={{Clade
                     |1=[[w:Evanioidea|Evanioidea]]
                     |2=[[w:Stephanoidea|Stephanoidea]]
                     }}
                  |2={{Clade
                     |1=[[w:Trigonaloidea|Trigonaloidea]]
                     |label2=[[w:Aculeata|Aculeata]]
                     |2={{Clade
                        |1=[[w:Chrysidoidea|Chrysidoidea]]
                        |2={{Clade
                           |1=[[w:Vespoidea|Vespoidea]] (potter, honey and social wasps)
                           |2={{Clade
                              |1={{Clade
                                 |1=[[w:Pompiloidea|Pompiloidea]] (velvet ants, spider wasps and relatives)
                                 |2={{Clade
                                    |1=[[w:Thynnoidea|Thynnoidea]]
                                    |2=[[w:Tiphioidea|Tiphioidea]]
                                 }} }}
                              |2={{Clade
                                 |1=[[w:Scolioidea|Scolioidea]]
                                 |2={{Clade
                                    |1=[[w:Formicoidea|Formicoidea]] (ants)
                                    |2=[[w:Apoidea|Apoidea]] (bees and related wasps)
}} }} }} }} }} }} }} }} }}
&lt;br&gt;

==Some common African Symphyta==
&lt;gallery mode=packed heights=200&gt;
Sawfly, Argidae 2021 12 19 12 39 42.jpg|Sawfly, ''Arge'' sp. (Argidae, Superfamily Tenthredinoidea) 
Athalia 2019 11 21 0815.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Athalia 2019 08 23 8964.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Sirex noctilio F (prof.).JPG|''Sirex noctilio'', a pest in ''Pinus'' plantations (introduced to Africa)&lt;ref name=WaspWebSn&gt;[https://www.waspweb.org/Siricoidea/Siricidae/Sirex/Sirex_noctilio.htm van Noort, Simon (2023). WaspWeb: ''Sirex noctilio''. www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
&lt;/gallery&gt;

== Frequently reported African Apocrita ==
&lt;br&gt;
[[w:Aculeata|Aculeata]] (ants, bees and stinging wasps) are the most commonly observed Hymenoptera in Africa. There are many more photographs of African Aculeata on the web - See [https://www.waspweb.org/Classification/index.htm African Aculeata on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=326777 African Aculeata on iNaturalist] &lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Slender Ant (Tetraponera natalensis) (30538051244).jpg|Slender ant (Formicidae, Formicoidea)
Carpenter Bee (Xylocopa inconstans, female) (6021534858).jpg|Female carpenter bee (Apidae, Apoidea)
Chalybion 2019 12 02 2314.jpg|Blue mud-dauber wasp (Sphecidae, Apoidea)
Bethylidae 2019 08 21 13 42 21 8760.jpg|Flat wasp (Bethylidae, Chrysidoidea)
Belonogaster juncea colonialis, manlik, h, Pretoria.jpg|Paper wasp (Vespidae, Vespoidea)
Meria 2020 08 12 2193.jpg|''Meria'' sp. female (Thynnidae, Thynnoidea)
Dasylabris stimulatrix iNat37068107 2019 12 23 2990 W.jpg|''Dasylabris stimulatrix'', a velvet ant (Mutillidae, Pompiloidea)
Pompilidae 2019 05 28 0247.jpg|Spider-hunting wasp (Pompilidae, Pompiloidea)
Scoliid Wasp 2019 06 08 0492.jpg|Campsomerine wasp (Scoliidae, Scolioidea)
&lt;/gallery&gt;&lt;br&gt;

Darwin wasps ([[w:Ichneumonidae|Ichneumonidae]]) and braconids ([[w:Braconidae|Braconidae]]) are also frequently reported. Links to [https://www.waspweb.org/Ichneumonoidea/index.htm African Ichneumonoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=47200 African Ichneumonoidea on iNaturalist].&lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Echthromorpha agrestoria 2019 10 03 5572.jpg|''Echthromorpha agrestoria'' (Ichneumonidae, Ichneumonoidea)
Ichneumonidae inat 88832241.jpg|''Goryphus tricolor'' (Ichneumonidae, Ichneumonoidea)
Braconid Wasp (Archibracon servillei) (11857083186).jpg|Braconine wasp (Braconidae, Ichneumonoidea) 
2019 09 15 12 55 07 Microgastrinae 38425632.jpg|Microgastrine wasp (Braconidae, Ichneumonoidea) 
&lt;/gallery&gt;&lt;br&gt;

Small [[w:Chalcidoidea|chalcidoid]] parasitic wasps (especially those from the families [[w:Pteromalidae|Pteromalidae]], [[w:Chalcididae|Chalcididae]], [[w:Eulophidae|Eulophidae]], [[w:Eurytomidae|Eurytomidae]]) are also common. Links to [https://www.waspweb.org/Chalcidoidea/index.htm African Chalcidoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=69750 African Chalcidoidea on iNaturalist]
&lt;gallery mode=packed heights=200&gt;
Halticoptera 2019 11 26 2173Cedara.jpg|''Halticoptera'' sp. (Pteromalidae, Chalcidoidea)
Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp. (Chalcididae, Chalcidoidea)
Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp. (Eulophidae, Chalcidoidea)
Eurytomidae00.jpg|Mating seed chalcids (Eurytomidae, Chalcidoidea)
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>8xnbzf3pd2yt6pe7vppb1tkugoh5uug</sha1>
    </revision>
    <revision>
      <id>2815536</id>
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      <timestamp>2026-06-13T18:21:16Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Superfamily Apoidea */</comment>
      <origin>2815536</origin>
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      <text bytes="15571" sha1="9d8f09o7hkxotw0xzr40dtf60jain0e" xml:space="preserve">= Superfamily Apoidea =
== African Ampulicidae ==
&lt;gallery mode=packed heights=150&gt;
Ampulicidae 37894270 suncana.jpg
Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg
&lt;/gallery&gt;		
== African Astatidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;		
== African Bembicidae ==
===Subfamily Bembicinae===	
&lt;gallery mode=packed heights=150&gt;
Bembecinus iN 153052948.jpg|''Bembecinus'' sp.
Bembix iN 195581985.jpg|''Bembix'' sp.
Hoplisoides thalia iN 268375748 a.jpg|''Hoplisoides thalia''
&lt;/gallery&gt;		
Tribe Gorytini	
*		''Afrogorytes''
*		''Gorytes''
*		''Harpactus''
*		''Hoplisoides''
*		''Lestiphorus''
		
Tribe Spheciini	
*		''Ammatomus''
*		''Kohlia''
*		''Sphecius''
		
Tribe Handlirschiini	
*		''Handlirschia''
		
Tribe Bembicini	
*		''Bembix''
		
Tribe Stizini	
*		''Bembecinus''
*		''Stizoides''
*		''Stizus''

===Subfamily Nyssoninae===	
*		''Brachystegus''
*		''Hovanysson''
*		''Nysson''
		
===Subfamily Alyssontinae===	
*		''Alysson''
*		''Didineis''

== African Crabronidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;		
== African Pemphredonidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;		
== African Philanthidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;		
== African Psenidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;		
== African Sphecidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;

= Superfamily Chalcidoidea =
 
Chalcidoidea is an incredibly diverse group of wasps with a wide variety of life histories; many are parasitoids, making them ecologically important; some are used for biological control of insect pests. About 22 000 species have been described, but as many as 500 000 are thought to exist worldwide.&lt;ref name=Noyes2019/&gt;

Most chalcidoids are small (body length less than 5 mm), the smallest being males of the mymarid ''Dicopomorpha echmepterygis'', which is the smallest known flying insect (0.13 mm).&lt;ref name=Noyes2019&gt;Noyes, J.S. 2019. Universal Chalcidoidea Database. [http://www.nhm.ac.uk/chalcidoids]&lt;/ref&gt; However, some species may be as long as 20 mm (excluding antennae and ovipositor), such as the pelecinellid ''Doddifoenus wallacei''.&lt;ref name=Krogmann&amp;Burks2009&gt;Krogmann, Lars; Burks, Roger A. (2009-12-31). &quot;Doddifoenus wallacei, a new giant parasitoid wasp of the subfamily Leptofoeninae (Chalcidoidea: Pteromalidae), with a description of its mesosomal skeletal anatomy and a molecular characterization&quot;. Zootaxa. 2194: 21–36. doi:10.5281/zenodo.189452 [https://www.researchgate.net/profile/Lars-Krogmann/publication/250916337 PDF]&lt;/ref&gt; Many have bright metallic colours, hence the name Chalcidoidea or chalcid - the greek word χαλκός (chalcos) means copper or bronze.&lt;ref name=Perseus2023&gt;[http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0073:entry=xalko/s Georg Autenrieth, A Homeric Dictionary]&lt;/ref&gt;

Many new chalcidoid families were proposed in 2022.&lt;ref name=Burks2022&gt;Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. [https://doi.org/10.3897/jhr.94.94263 DOI]&lt;/ref&gt; This was largely based on phylogenetic evidence (both molecular and morphological) accumulated since the families of the Superfamily were outlined in 1993&lt;ref name=Gibson1993&gt;Gibson, G.A.P. (1993) Superfamilies Mymarommatoidea and Chalcidoidea (pp. 570-655). In Goulet, H. &amp; Huber, J. (eds). Hymenoptera of the World: an identification guide to families. Research Branch, Agriculture Canada, Ottawa, Canada, 668 pp. [https://www.researchgate.net/publication/259227143 PDF]&lt;/ref&gt;.


Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. https://doi.org/10.3897/jhr.94.94263

Cruaud, A., Rasplus, J.Y., Zhang, J., Burks, R., Delvare, G., Fusu, L., Gumovsky, A., Huber, J.T., Janšta, P., Mitroiu, M.D. and Noyes, J.S., 2022. The Chalcidoidea bush of life–a massive radiation blurred by mutational saturation. bioRxiv. https://doi.org/10.1101/2022.09.11.507458 https://www.biorxiv.org/content/10.1101/2022.09.11.507458.full.pdf


= Superfamily Ichneumonoidea =

Quicke, D.L., 2015. The braconid and ichneumonid parasitoid wasps: biology, systematics, evolution and ecology. John Wiley &amp; Sons.
https://www.researchgate.net/publication/281662102_The_Braconid_and_Ichneumonid_Parasitoid_Wasps_Biology_Systematics_Evolution_and_Ecology

=== Diplazontinae ===
Quote from Fitton &amp; Rotheray (1982):
&quot;Diplazontines are distinguished easily from other ichneumonids by their general appearance (Fig. 1). The main confirmatory characters are the bifid upper tooth of the mandible (Fig. 2) and the sub-rectangular shape of the first tergite of the gaster (Figs. 1,3,4, 13, 15 and 17). In fact, the diplazontines are so distinct that they can be separated from all other Hymenoptera-Apocrita by possession of the following five characters in combination: (1) antenna of seventeen or more segments (Fig. I); (2) forewing with costal cell obliterated (that is, veins C and Sc + R + Rs contiguous from the wing base to the base of the pterostigma (Fig. 6 ) ; (3) forewing with cross-vein 2 mcu present (Fig. 6); (4) mandible with its upper tooth subdivided (bifid), making the mandible tridentate (Fig. 2); (5) first tergite of gaster sub-rectangular in shape (that is, subequal in width anteriorly and posteriorly) (Figs. 1 , 3 , 4 , 13,15 and 17).&quot;

Fitton, M. G., &amp; Rotheray, G. E. (1982). A key to the European genera of diplazontine ichneumon‐flies, with notes on the British fauna. Systematic Entomology, 7(3), 311-320.

=== Superfamily Pompiloidea ===
Family Mutillidae (Velvet ants)
Family Pompilidae (Spider hunting wasps)
Family Sapygidae (Sapygid wasps)


== Spider-hunting wasps ==
Two wasps from the genus ''Auplopus'' from Europe. Most of the ''Auplopus'' species, along with others from the Tribe Ageniellini, amputate the legs of the spiders they capture; this makes it much easier to move the spider to the nest.
&lt;gallery mode=packed heights=200&gt;
Auplopus carbonarius IMG 1624.jpg
Auplopus carbonarius fg01 20060623 Nied Garten.jpg
&lt;/gallery&gt;
African genera of Ageniellini:
* ''[[Auplopus]]'' &lt;small&gt;Spinola, 1841&lt;/small&gt;
* ''[[Cyemagenia]]'' &lt;small&gt;Arnold, 1946&lt;/small&gt;
* ''[[Dichragenia]]'' &lt;small&gt;Haupt, 1950&lt;/small&gt;
* ''[[Phanagenia]]'' &lt;small&gt;Banks, 1933&lt;/small&gt;   Madagascar
* ''[[Poecilagenia]]'' &lt;small&gt;Haupt, 1926&lt;/small&gt;

== Superfamily Scolioidea ==
=== Family Bradynobaenidae (Bradynobaenid wasps) ===
=== Family Scoliidae (Mammoth wasps) ===

== Superfamily Tiphioidea ==
Family Tiphiidae (Tiphiid wasps)
== Superfamily Thynnoidea ==
Family Thynnidae(Thynnid wasps)
== Superfamily Vespoidea ==
=== Family Rhopalosomatidae (Rhopalosomatid wasps) ===
===Family Vespidae (Paper, Potter &amp; Pollen wasps)===


== '''&lt;big&gt;Hymenoptera&lt;/big&gt;''' ==
About 20 000 described species of [[w:Hymenoptera|Hymenoptera]] (wasps, bees, ants and sawflies) are known from the [[w:Afrotropical|Afrotropical]] region. Estimates of the actual species count for the region range from 100 000 species to as high as 500 000 species.&lt;ref name=WaspWebOverview&gt;[https://www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm van Noort, Simon (2023). WaspWeb: Afrotropical Hymenoptera Initiative. www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm Retrieved 22 February 2023.]&lt;/ref&gt; Nineteen different superfamilies are illustrated on [[User:Alandmanson/Hymenoptera of southern Africa|this gallery]] in Wikimedia Commons.

The huge number of undescribed species means that many species will be extinct before we are even aware of them, as there are too few taxonomists employed to tackle the task of describing all of them in the next few decades.

==Classification==
The Order can be split into two Suborders - The [[w:Symphyta|Symphyta]] (Woodwasps, Horntails, Sawflies) and the [[w:Apocrita|Apocrita]] (Narrow-waisted wasps, ants and bees).&lt;br&gt;

When compared to the temperate regions of the northern hemisphere, the diversity of Symphyta in Africa is relatively poor, although sawflies of the Superfamily [[w:Tenthredinoidea|Tenthredinoidea]] are fairly common in forests and other woody vegetation types.

Africa has a rich diversity of Apocrita. The cladogram shown below&lt;ref name=Peters&gt;{{Cite journal |last1=Peters |first1=Ralph S. |last2=Krogmann |first2=Lars |last3=Mayer |first3=Christoph |last4=Donath |first4=Alexander |last5=Gunkel |first5=Simon |last6=Meusemann |first6=Karen |last7=Kozlov |first7=Alexey |last8=Podsiadlowski |first8=Lars |last9=Petersen |first9=Malte |title=Evolutionary History of the Hymenoptera |journal=Current Biology |volume=27 |issue=7 |pages=1013–1018 |doi=10.1016/j.cub.2017.01.027|pmid=28343967 |year=2017 |doi-access=free }}&lt;/ref&gt; indicates the possible relationships between 11 of the superfamilies that comprise Apocrita; These 11 superfamilies are all represented in Africa. This breakdown is used by iNaturalist.&lt;ref name=inat-Aculeata&gt;https://www.inaturalist.org/taxa/326777-Aculeata&lt;/ref&gt; It is, however, not accepted by all hymenopterists, and may change as more phylogenetic evidence is accumulated.&lt;ref name=waspwebClass&gt;[https://www.waspweb.org/Classification/index.htm van Noort, Simon (2023). WaspWeb: Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants). www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
{{Clade|style=font-size:85%; line-height:85%
|label1=Apocrita
            |1={{Clade
               |label1=[[w:Parasitoida|Parasitoida]]
               |1={{Clade
                  |1={{Clade
                     |1=[[w:Ceraphronoidea|Ceraphronoidea]]
                     |2=[[w:Ichneumonoidea|Ichneumonoidea]]
                     }}
                  |label2=[[w:Proctotrupomorpha|Proctotrupomorpha]]
                  |2={{Clade
                     |1=[[w:Cynipoidea|Cynipoidea]]
                     |2={{Clade
                        |1=[[w:Platygastroidea|Platygastroidea]]
                        |2={{Clade
                           |1=[[w:Chalcidoidea|Chalcidoidea]]
                           |2={{Clade
                              |1=[[w:Diaprioidea|Diaprioidea]]
                              |2=[[w:Proctotrupoidea|Proctotrupoidea]]
                  }} }} }} }} }}
               |2={{Clade
                  |1={{Clade
                     |1=[[w:Evanioidea|Evanioidea]]
                     |2=[[w:Stephanoidea|Stephanoidea]]
                     }}
                  |2={{Clade
                     |1=[[w:Trigonaloidea|Trigonaloidea]]
                     |label2=[[w:Aculeata|Aculeata]]
                     |2={{Clade
                        |1=[[w:Chrysidoidea|Chrysidoidea]]
                        |2={{Clade
                           |1=[[w:Vespoidea|Vespoidea]] (potter, honey and social wasps)
                           |2={{Clade
                              |1={{Clade
                                 |1=[[w:Pompiloidea|Pompiloidea]] (velvet ants, spider wasps and relatives)
                                 |2={{Clade
                                    |1=[[w:Thynnoidea|Thynnoidea]]
                                    |2=[[w:Tiphioidea|Tiphioidea]]
                                 }} }}
                              |2={{Clade
                                 |1=[[w:Scolioidea|Scolioidea]]
                                 |2={{Clade
                                    |1=[[w:Formicoidea|Formicoidea]] (ants)
                                    |2=[[w:Apoidea|Apoidea]] (bees and related wasps)
}} }} }} }} }} }} }} }} }}
&lt;br&gt;

==Some common African Symphyta==
&lt;gallery mode=packed heights=200&gt;
Sawfly, Argidae 2021 12 19 12 39 42.jpg|Sawfly, ''Arge'' sp. (Argidae, Superfamily Tenthredinoidea) 
Athalia 2019 11 21 0815.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Athalia 2019 08 23 8964.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Sirex noctilio F (prof.).JPG|''Sirex noctilio'', a pest in ''Pinus'' plantations (introduced to Africa)&lt;ref name=WaspWebSn&gt;[https://www.waspweb.org/Siricoidea/Siricidae/Sirex/Sirex_noctilio.htm van Noort, Simon (2023). WaspWeb: ''Sirex noctilio''. www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
&lt;/gallery&gt;

== Frequently reported African Apocrita ==
&lt;br&gt;
[[w:Aculeata|Aculeata]] (ants, bees and stinging wasps) are the most commonly observed Hymenoptera in Africa. There are many more photographs of African Aculeata on the web - See [https://www.waspweb.org/Classification/index.htm African Aculeata on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=326777 African Aculeata on iNaturalist] &lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Slender Ant (Tetraponera natalensis) (30538051244).jpg|Slender ant (Formicidae, Formicoidea)
Carpenter Bee (Xylocopa inconstans, female) (6021534858).jpg|Female carpenter bee (Apidae, Apoidea)
Chalybion 2019 12 02 2314.jpg|Blue mud-dauber wasp (Sphecidae, Apoidea)
Bethylidae 2019 08 21 13 42 21 8760.jpg|Flat wasp (Bethylidae, Chrysidoidea)
Belonogaster juncea colonialis, manlik, h, Pretoria.jpg|Paper wasp (Vespidae, Vespoidea)
Meria 2020 08 12 2193.jpg|''Meria'' sp. female (Thynnidae, Thynnoidea)
Dasylabris stimulatrix iNat37068107 2019 12 23 2990 W.jpg|''Dasylabris stimulatrix'', a velvet ant (Mutillidae, Pompiloidea)
Pompilidae 2019 05 28 0247.jpg|Spider-hunting wasp (Pompilidae, Pompiloidea)
Scoliid Wasp 2019 06 08 0492.jpg|Campsomerine wasp (Scoliidae, Scolioidea)
&lt;/gallery&gt;&lt;br&gt;

Darwin wasps ([[w:Ichneumonidae|Ichneumonidae]]) and braconids ([[w:Braconidae|Braconidae]]) are also frequently reported. Links to [https://www.waspweb.org/Ichneumonoidea/index.htm African Ichneumonoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=47200 African Ichneumonoidea on iNaturalist].&lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Echthromorpha agrestoria 2019 10 03 5572.jpg|''Echthromorpha agrestoria'' (Ichneumonidae, Ichneumonoidea)
Ichneumonidae inat 88832241.jpg|''Goryphus tricolor'' (Ichneumonidae, Ichneumonoidea)
Braconid Wasp (Archibracon servillei) (11857083186).jpg|Braconine wasp (Braconidae, Ichneumonoidea) 
2019 09 15 12 55 07 Microgastrinae 38425632.jpg|Microgastrine wasp (Braconidae, Ichneumonoidea) 
&lt;/gallery&gt;&lt;br&gt;

Small [[w:Chalcidoidea|chalcidoid]] parasitic wasps (especially those from the families [[w:Pteromalidae|Pteromalidae]], [[w:Chalcididae|Chalcididae]], [[w:Eulophidae|Eulophidae]], [[w:Eurytomidae|Eurytomidae]]) are also common. Links to [https://www.waspweb.org/Chalcidoidea/index.htm African Chalcidoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=69750 African Chalcidoidea on iNaturalist]
&lt;gallery mode=packed heights=200&gt;
Halticoptera 2019 11 26 2173Cedara.jpg|''Halticoptera'' sp. (Pteromalidae, Chalcidoidea)
Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp. (Chalcididae, Chalcidoidea)
Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp. (Eulophidae, Chalcidoidea)
Eurytomidae00.jpg|Mating seed chalcids (Eurytomidae, Chalcidoidea)
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>9d8f09o7hkxotw0xzr40dtf60jain0e</sha1>
    </revision>
    <revision>
      <id>2815537</id>
      <parentid>2815536</parentid>
      <timestamp>2026-06-13T18:25:30Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* African Astatidae */</comment>
      <origin>2815537</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15631" sha1="0dxtxuvabh8qmuzw3kisjws9j6xn34d" xml:space="preserve">= Superfamily Apoidea =
== African Ampulicidae ==
&lt;gallery mode=packed heights=150&gt;
Ampulicidae 37894270 suncana.jpg
Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg
&lt;/gallery&gt;		
== African Astatidae ==
&lt;gallery mode=packed heights=150&gt;
Astata iN 105162782 Nicola van Berkel.jpg
Astata melanaria.jpg
&lt;/gallery&gt;

== African Bembicidae ==
===Subfamily Bembicinae===	
&lt;gallery mode=packed heights=150&gt;
Bembecinus iN 153052948.jpg|''Bembecinus'' sp.
Bembix iN 195581985.jpg|''Bembix'' sp.
Hoplisoides thalia iN 268375748 a.jpg|''Hoplisoides thalia''
&lt;/gallery&gt;		
Tribe Gorytini	
*		''Afrogorytes''
*		''Gorytes''
*		''Harpactus''
*		''Hoplisoides''
*		''Lestiphorus''
		
Tribe Spheciini	
*		''Ammatomus''
*		''Kohlia''
*		''Sphecius''
		
Tribe Handlirschiini	
*		''Handlirschia''
		
Tribe Bembicini	
*		''Bembix''
		
Tribe Stizini	
*		''Bembecinus''
*		''Stizoides''
*		''Stizus''

===Subfamily Nyssoninae===	
*		''Brachystegus''
*		''Hovanysson''
*		''Nysson''
		
===Subfamily Alyssontinae===	
*		''Alysson''
*		''Didineis''

== African Crabronidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;		
== African Pemphredonidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;		
== African Philanthidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;		
== African Psenidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;		
== African Sphecidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;

= Superfamily Chalcidoidea =
 
Chalcidoidea is an incredibly diverse group of wasps with a wide variety of life histories; many are parasitoids, making them ecologically important; some are used for biological control of insect pests. About 22 000 species have been described, but as many as 500 000 are thought to exist worldwide.&lt;ref name=Noyes2019/&gt;

Most chalcidoids are small (body length less than 5 mm), the smallest being males of the mymarid ''Dicopomorpha echmepterygis'', which is the smallest known flying insect (0.13 mm).&lt;ref name=Noyes2019&gt;Noyes, J.S. 2019. Universal Chalcidoidea Database. [http://www.nhm.ac.uk/chalcidoids]&lt;/ref&gt; However, some species may be as long as 20 mm (excluding antennae and ovipositor), such as the pelecinellid ''Doddifoenus wallacei''.&lt;ref name=Krogmann&amp;Burks2009&gt;Krogmann, Lars; Burks, Roger A. (2009-12-31). &quot;Doddifoenus wallacei, a new giant parasitoid wasp of the subfamily Leptofoeninae (Chalcidoidea: Pteromalidae), with a description of its mesosomal skeletal anatomy and a molecular characterization&quot;. Zootaxa. 2194: 21–36. doi:10.5281/zenodo.189452 [https://www.researchgate.net/profile/Lars-Krogmann/publication/250916337 PDF]&lt;/ref&gt; Many have bright metallic colours, hence the name Chalcidoidea or chalcid - the greek word χαλκός (chalcos) means copper or bronze.&lt;ref name=Perseus2023&gt;[http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0073:entry=xalko/s Georg Autenrieth, A Homeric Dictionary]&lt;/ref&gt;

Many new chalcidoid families were proposed in 2022.&lt;ref name=Burks2022&gt;Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. [https://doi.org/10.3897/jhr.94.94263 DOI]&lt;/ref&gt; This was largely based on phylogenetic evidence (both molecular and morphological) accumulated since the families of the Superfamily were outlined in 1993&lt;ref name=Gibson1993&gt;Gibson, G.A.P. (1993) Superfamilies Mymarommatoidea and Chalcidoidea (pp. 570-655). In Goulet, H. &amp; Huber, J. (eds). Hymenoptera of the World: an identification guide to families. Research Branch, Agriculture Canada, Ottawa, Canada, 668 pp. [https://www.researchgate.net/publication/259227143 PDF]&lt;/ref&gt;.


Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. https://doi.org/10.3897/jhr.94.94263

Cruaud, A., Rasplus, J.Y., Zhang, J., Burks, R., Delvare, G., Fusu, L., Gumovsky, A., Huber, J.T., Janšta, P., Mitroiu, M.D. and Noyes, J.S., 2022. The Chalcidoidea bush of life–a massive radiation blurred by mutational saturation. bioRxiv. https://doi.org/10.1101/2022.09.11.507458 https://www.biorxiv.org/content/10.1101/2022.09.11.507458.full.pdf


= Superfamily Ichneumonoidea =

Quicke, D.L., 2015. The braconid and ichneumonid parasitoid wasps: biology, systematics, evolution and ecology. John Wiley &amp; Sons.
https://www.researchgate.net/publication/281662102_The_Braconid_and_Ichneumonid_Parasitoid_Wasps_Biology_Systematics_Evolution_and_Ecology

=== Diplazontinae ===
Quote from Fitton &amp; Rotheray (1982):
&quot;Diplazontines are distinguished easily from other ichneumonids by their general appearance (Fig. 1). The main confirmatory characters are the bifid upper tooth of the mandible (Fig. 2) and the sub-rectangular shape of the first tergite of the gaster (Figs. 1,3,4, 13, 15 and 17). In fact, the diplazontines are so distinct that they can be separated from all other Hymenoptera-Apocrita by possession of the following five characters in combination: (1) antenna of seventeen or more segments (Fig. I); (2) forewing with costal cell obliterated (that is, veins C and Sc + R + Rs contiguous from the wing base to the base of the pterostigma (Fig. 6 ) ; (3) forewing with cross-vein 2 mcu present (Fig. 6); (4) mandible with its upper tooth subdivided (bifid), making the mandible tridentate (Fig. 2); (5) first tergite of gaster sub-rectangular in shape (that is, subequal in width anteriorly and posteriorly) (Figs. 1 , 3 , 4 , 13,15 and 17).&quot;

Fitton, M. G., &amp; Rotheray, G. E. (1982). A key to the European genera of diplazontine ichneumon‐flies, with notes on the British fauna. Systematic Entomology, 7(3), 311-320.

=== Superfamily Pompiloidea ===
Family Mutillidae (Velvet ants)
Family Pompilidae (Spider hunting wasps)
Family Sapygidae (Sapygid wasps)


== Spider-hunting wasps ==
Two wasps from the genus ''Auplopus'' from Europe. Most of the ''Auplopus'' species, along with others from the Tribe Ageniellini, amputate the legs of the spiders they capture; this makes it much easier to move the spider to the nest.
&lt;gallery mode=packed heights=200&gt;
Auplopus carbonarius IMG 1624.jpg
Auplopus carbonarius fg01 20060623 Nied Garten.jpg
&lt;/gallery&gt;
African genera of Ageniellini:
* ''[[Auplopus]]'' &lt;small&gt;Spinola, 1841&lt;/small&gt;
* ''[[Cyemagenia]]'' &lt;small&gt;Arnold, 1946&lt;/small&gt;
* ''[[Dichragenia]]'' &lt;small&gt;Haupt, 1950&lt;/small&gt;
* ''[[Phanagenia]]'' &lt;small&gt;Banks, 1933&lt;/small&gt;   Madagascar
* ''[[Poecilagenia]]'' &lt;small&gt;Haupt, 1926&lt;/small&gt;

== Superfamily Scolioidea ==
=== Family Bradynobaenidae (Bradynobaenid wasps) ===
=== Family Scoliidae (Mammoth wasps) ===

== Superfamily Tiphioidea ==
Family Tiphiidae (Tiphiid wasps)
== Superfamily Thynnoidea ==
Family Thynnidae(Thynnid wasps)
== Superfamily Vespoidea ==
=== Family Rhopalosomatidae (Rhopalosomatid wasps) ===
===Family Vespidae (Paper, Potter &amp; Pollen wasps)===


== '''&lt;big&gt;Hymenoptera&lt;/big&gt;''' ==
About 20 000 described species of [[w:Hymenoptera|Hymenoptera]] (wasps, bees, ants and sawflies) are known from the [[w:Afrotropical|Afrotropical]] region. Estimates of the actual species count for the region range from 100 000 species to as high as 500 000 species.&lt;ref name=WaspWebOverview&gt;[https://www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm van Noort, Simon (2023). WaspWeb: Afrotropical Hymenoptera Initiative. www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm Retrieved 22 February 2023.]&lt;/ref&gt; Nineteen different superfamilies are illustrated on [[User:Alandmanson/Hymenoptera of southern Africa|this gallery]] in Wikimedia Commons.

The huge number of undescribed species means that many species will be extinct before we are even aware of them, as there are too few taxonomists employed to tackle the task of describing all of them in the next few decades.

==Classification==
The Order can be split into two Suborders - The [[w:Symphyta|Symphyta]] (Woodwasps, Horntails, Sawflies) and the [[w:Apocrita|Apocrita]] (Narrow-waisted wasps, ants and bees).&lt;br&gt;

When compared to the temperate regions of the northern hemisphere, the diversity of Symphyta in Africa is relatively poor, although sawflies of the Superfamily [[w:Tenthredinoidea|Tenthredinoidea]] are fairly common in forests and other woody vegetation types.

Africa has a rich diversity of Apocrita. The cladogram shown below&lt;ref name=Peters&gt;{{Cite journal |last1=Peters |first1=Ralph S. |last2=Krogmann |first2=Lars |last3=Mayer |first3=Christoph |last4=Donath |first4=Alexander |last5=Gunkel |first5=Simon |last6=Meusemann |first6=Karen |last7=Kozlov |first7=Alexey |last8=Podsiadlowski |first8=Lars |last9=Petersen |first9=Malte |title=Evolutionary History of the Hymenoptera |journal=Current Biology |volume=27 |issue=7 |pages=1013–1018 |doi=10.1016/j.cub.2017.01.027|pmid=28343967 |year=2017 |doi-access=free }}&lt;/ref&gt; indicates the possible relationships between 11 of the superfamilies that comprise Apocrita; These 11 superfamilies are all represented in Africa. This breakdown is used by iNaturalist.&lt;ref name=inat-Aculeata&gt;https://www.inaturalist.org/taxa/326777-Aculeata&lt;/ref&gt; It is, however, not accepted by all hymenopterists, and may change as more phylogenetic evidence is accumulated.&lt;ref name=waspwebClass&gt;[https://www.waspweb.org/Classification/index.htm van Noort, Simon (2023). WaspWeb: Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants). www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
{{Clade|style=font-size:85%; line-height:85%
|label1=Apocrita
            |1={{Clade
               |label1=[[w:Parasitoida|Parasitoida]]
               |1={{Clade
                  |1={{Clade
                     |1=[[w:Ceraphronoidea|Ceraphronoidea]]
                     |2=[[w:Ichneumonoidea|Ichneumonoidea]]
                     }}
                  |label2=[[w:Proctotrupomorpha|Proctotrupomorpha]]
                  |2={{Clade
                     |1=[[w:Cynipoidea|Cynipoidea]]
                     |2={{Clade
                        |1=[[w:Platygastroidea|Platygastroidea]]
                        |2={{Clade
                           |1=[[w:Chalcidoidea|Chalcidoidea]]
                           |2={{Clade
                              |1=[[w:Diaprioidea|Diaprioidea]]
                              |2=[[w:Proctotrupoidea|Proctotrupoidea]]
                  }} }} }} }} }}
               |2={{Clade
                  |1={{Clade
                     |1=[[w:Evanioidea|Evanioidea]]
                     |2=[[w:Stephanoidea|Stephanoidea]]
                     }}
                  |2={{Clade
                     |1=[[w:Trigonaloidea|Trigonaloidea]]
                     |label2=[[w:Aculeata|Aculeata]]
                     |2={{Clade
                        |1=[[w:Chrysidoidea|Chrysidoidea]]
                        |2={{Clade
                           |1=[[w:Vespoidea|Vespoidea]] (potter, honey and social wasps)
                           |2={{Clade
                              |1={{Clade
                                 |1=[[w:Pompiloidea|Pompiloidea]] (velvet ants, spider wasps and relatives)
                                 |2={{Clade
                                    |1=[[w:Thynnoidea|Thynnoidea]]
                                    |2=[[w:Tiphioidea|Tiphioidea]]
                                 }} }}
                              |2={{Clade
                                 |1=[[w:Scolioidea|Scolioidea]]
                                 |2={{Clade
                                    |1=[[w:Formicoidea|Formicoidea]] (ants)
                                    |2=[[w:Apoidea|Apoidea]] (bees and related wasps)
}} }} }} }} }} }} }} }} }}
&lt;br&gt;

==Some common African Symphyta==
&lt;gallery mode=packed heights=200&gt;
Sawfly, Argidae 2021 12 19 12 39 42.jpg|Sawfly, ''Arge'' sp. (Argidae, Superfamily Tenthredinoidea) 
Athalia 2019 11 21 0815.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Athalia 2019 08 23 8964.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Sirex noctilio F (prof.).JPG|''Sirex noctilio'', a pest in ''Pinus'' plantations (introduced to Africa)&lt;ref name=WaspWebSn&gt;[https://www.waspweb.org/Siricoidea/Siricidae/Sirex/Sirex_noctilio.htm van Noort, Simon (2023). WaspWeb: ''Sirex noctilio''. www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
&lt;/gallery&gt;

== Frequently reported African Apocrita ==
&lt;br&gt;
[[w:Aculeata|Aculeata]] (ants, bees and stinging wasps) are the most commonly observed Hymenoptera in Africa. There are many more photographs of African Aculeata on the web - See [https://www.waspweb.org/Classification/index.htm African Aculeata on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=326777 African Aculeata on iNaturalist] &lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Slender Ant (Tetraponera natalensis) (30538051244).jpg|Slender ant (Formicidae, Formicoidea)
Carpenter Bee (Xylocopa inconstans, female) (6021534858).jpg|Female carpenter bee (Apidae, Apoidea)
Chalybion 2019 12 02 2314.jpg|Blue mud-dauber wasp (Sphecidae, Apoidea)
Bethylidae 2019 08 21 13 42 21 8760.jpg|Flat wasp (Bethylidae, Chrysidoidea)
Belonogaster juncea colonialis, manlik, h, Pretoria.jpg|Paper wasp (Vespidae, Vespoidea)
Meria 2020 08 12 2193.jpg|''Meria'' sp. female (Thynnidae, Thynnoidea)
Dasylabris stimulatrix iNat37068107 2019 12 23 2990 W.jpg|''Dasylabris stimulatrix'', a velvet ant (Mutillidae, Pompiloidea)
Pompilidae 2019 05 28 0247.jpg|Spider-hunting wasp (Pompilidae, Pompiloidea)
Scoliid Wasp 2019 06 08 0492.jpg|Campsomerine wasp (Scoliidae, Scolioidea)
&lt;/gallery&gt;&lt;br&gt;

Darwin wasps ([[w:Ichneumonidae|Ichneumonidae]]) and braconids ([[w:Braconidae|Braconidae]]) are also frequently reported. Links to [https://www.waspweb.org/Ichneumonoidea/index.htm African Ichneumonoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=47200 African Ichneumonoidea on iNaturalist].&lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Echthromorpha agrestoria 2019 10 03 5572.jpg|''Echthromorpha agrestoria'' (Ichneumonidae, Ichneumonoidea)
Ichneumonidae inat 88832241.jpg|''Goryphus tricolor'' (Ichneumonidae, Ichneumonoidea)
Braconid Wasp (Archibracon servillei) (11857083186).jpg|Braconine wasp (Braconidae, Ichneumonoidea) 
2019 09 15 12 55 07 Microgastrinae 38425632.jpg|Microgastrine wasp (Braconidae, Ichneumonoidea) 
&lt;/gallery&gt;&lt;br&gt;

Small [[w:Chalcidoidea|chalcidoid]] parasitic wasps (especially those from the families [[w:Pteromalidae|Pteromalidae]], [[w:Chalcididae|Chalcididae]], [[w:Eulophidae|Eulophidae]], [[w:Eurytomidae|Eurytomidae]]) are also common. Links to [https://www.waspweb.org/Chalcidoidea/index.htm African Chalcidoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=69750 African Chalcidoidea on iNaturalist]
&lt;gallery mode=packed heights=200&gt;
Halticoptera 2019 11 26 2173Cedara.jpg|''Halticoptera'' sp. (Pteromalidae, Chalcidoidea)
Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp. (Chalcididae, Chalcidoidea)
Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp. (Eulophidae, Chalcidoidea)
Eurytomidae00.jpg|Mating seed chalcids (Eurytomidae, Chalcidoidea)
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>0dxtxuvabh8qmuzw3kisjws9j6xn34d</sha1>
    </revision>
    <revision>
      <id>2815542</id>
      <parentid>2815537</parentid>
      <timestamp>2026-06-13T18:32:41Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* African Crabronidae */</comment>
      <origin>2815542</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15963" sha1="150tec81zl6ia23t8hisvwskib9gnyl" xml:space="preserve">= Superfamily Apoidea =
== African Ampulicidae ==
&lt;gallery mode=packed heights=150&gt;
Ampulicidae 37894270 suncana.jpg
Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg
&lt;/gallery&gt;		
== African Astatidae ==
&lt;gallery mode=packed heights=150&gt;
Astata iN 105162782 Nicola van Berkel.jpg
Astata melanaria.jpg
&lt;/gallery&gt;

== African Bembicidae ==
===Subfamily Bembicinae===	
&lt;gallery mode=packed heights=150&gt;
Bembecinus iN 153052948.jpg|''Bembecinus'' sp.
Bembix iN 195581985.jpg|''Bembix'' sp.
Hoplisoides thalia iN 268375748 a.jpg|''Hoplisoides thalia''
&lt;/gallery&gt;		
Tribe Gorytini	
*		''Afrogorytes''
*		''Gorytes''
*		''Harpactus''
*		''Hoplisoides''
*		''Lestiphorus''
		
Tribe Spheciini	
*		''Ammatomus''
*		''Kohlia''
*		''Sphecius''
		
Tribe Handlirschiini	
*		''Handlirschia''
		
Tribe Bembicini	
*		''Bembix''
		
Tribe Stizini	
*		''Bembecinus''
*		''Stizoides''
*		''Stizus''

===Subfamily Nyssoninae===	
*		''Brachystegus''
*		''Hovanysson''
*		''Nysson''
		
===Subfamily Alyssontinae===	
*		''Alysson''
*		''Didineis''

== African Crabronidae ==
&lt;gallery mode=packed heights=150&gt;
Dasyproctus iN 30029277 a.jpg
Dicranorhina kohli 2023 07 25 iN 176115975.jpg
Liris on Crassula iN 42678436 01.jpg
Palarus Bee Pirate iN 144133368 2022-12-01 01.jpg
Paranysson iN 199673489 1.jpg
Pison iN 144131685 2022-11-30 03.jpg
Tachysphex iN 250449986 2024 10 09 7305.jpg
Tachytes iN 188902572 1964.jpg
Trypoxylon iN 99063113 a.jpg
&lt;/gallery&gt;

== African Pemphredonidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;		
== African Philanthidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;		
== African Psenidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;		
== African Sphecidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;

= Superfamily Chalcidoidea =
 
Chalcidoidea is an incredibly diverse group of wasps with a wide variety of life histories; many are parasitoids, making them ecologically important; some are used for biological control of insect pests. About 22 000 species have been described, but as many as 500 000 are thought to exist worldwide.&lt;ref name=Noyes2019/&gt;

Most chalcidoids are small (body length less than 5 mm), the smallest being males of the mymarid ''Dicopomorpha echmepterygis'', which is the smallest known flying insect (0.13 mm).&lt;ref name=Noyes2019&gt;Noyes, J.S. 2019. Universal Chalcidoidea Database. [http://www.nhm.ac.uk/chalcidoids]&lt;/ref&gt; However, some species may be as long as 20 mm (excluding antennae and ovipositor), such as the pelecinellid ''Doddifoenus wallacei''.&lt;ref name=Krogmann&amp;Burks2009&gt;Krogmann, Lars; Burks, Roger A. (2009-12-31). &quot;Doddifoenus wallacei, a new giant parasitoid wasp of the subfamily Leptofoeninae (Chalcidoidea: Pteromalidae), with a description of its mesosomal skeletal anatomy and a molecular characterization&quot;. Zootaxa. 2194: 21–36. doi:10.5281/zenodo.189452 [https://www.researchgate.net/profile/Lars-Krogmann/publication/250916337 PDF]&lt;/ref&gt; Many have bright metallic colours, hence the name Chalcidoidea or chalcid - the greek word χαλκός (chalcos) means copper or bronze.&lt;ref name=Perseus2023&gt;[http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0073:entry=xalko/s Georg Autenrieth, A Homeric Dictionary]&lt;/ref&gt;

Many new chalcidoid families were proposed in 2022.&lt;ref name=Burks2022&gt;Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. [https://doi.org/10.3897/jhr.94.94263 DOI]&lt;/ref&gt; This was largely based on phylogenetic evidence (both molecular and morphological) accumulated since the families of the Superfamily were outlined in 1993&lt;ref name=Gibson1993&gt;Gibson, G.A.P. (1993) Superfamilies Mymarommatoidea and Chalcidoidea (pp. 570-655). In Goulet, H. &amp; Huber, J. (eds). Hymenoptera of the World: an identification guide to families. Research Branch, Agriculture Canada, Ottawa, Canada, 668 pp. [https://www.researchgate.net/publication/259227143 PDF]&lt;/ref&gt;.


Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. https://doi.org/10.3897/jhr.94.94263

Cruaud, A., Rasplus, J.Y., Zhang, J., Burks, R., Delvare, G., Fusu, L., Gumovsky, A., Huber, J.T., Janšta, P., Mitroiu, M.D. and Noyes, J.S., 2022. The Chalcidoidea bush of life–a massive radiation blurred by mutational saturation. bioRxiv. https://doi.org/10.1101/2022.09.11.507458 https://www.biorxiv.org/content/10.1101/2022.09.11.507458.full.pdf


= Superfamily Ichneumonoidea =

Quicke, D.L., 2015. The braconid and ichneumonid parasitoid wasps: biology, systematics, evolution and ecology. John Wiley &amp; Sons.
https://www.researchgate.net/publication/281662102_The_Braconid_and_Ichneumonid_Parasitoid_Wasps_Biology_Systematics_Evolution_and_Ecology

=== Diplazontinae ===
Quote from Fitton &amp; Rotheray (1982):
&quot;Diplazontines are distinguished easily from other ichneumonids by their general appearance (Fig. 1). The main confirmatory characters are the bifid upper tooth of the mandible (Fig. 2) and the sub-rectangular shape of the first tergite of the gaster (Figs. 1,3,4, 13, 15 and 17). In fact, the diplazontines are so distinct that they can be separated from all other Hymenoptera-Apocrita by possession of the following five characters in combination: (1) antenna of seventeen or more segments (Fig. I); (2) forewing with costal cell obliterated (that is, veins C and Sc + R + Rs contiguous from the wing base to the base of the pterostigma (Fig. 6 ) ; (3) forewing with cross-vein 2 mcu present (Fig. 6); (4) mandible with its upper tooth subdivided (bifid), making the mandible tridentate (Fig. 2); (5) first tergite of gaster sub-rectangular in shape (that is, subequal in width anteriorly and posteriorly) (Figs. 1 , 3 , 4 , 13,15 and 17).&quot;

Fitton, M. G., &amp; Rotheray, G. E. (1982). A key to the European genera of diplazontine ichneumon‐flies, with notes on the British fauna. Systematic Entomology, 7(3), 311-320.

=== Superfamily Pompiloidea ===
Family Mutillidae (Velvet ants)
Family Pompilidae (Spider hunting wasps)
Family Sapygidae (Sapygid wasps)


== Spider-hunting wasps ==
Two wasps from the genus ''Auplopus'' from Europe. Most of the ''Auplopus'' species, along with others from the Tribe Ageniellini, amputate the legs of the spiders they capture; this makes it much easier to move the spider to the nest.
&lt;gallery mode=packed heights=200&gt;
Auplopus carbonarius IMG 1624.jpg
Auplopus carbonarius fg01 20060623 Nied Garten.jpg
&lt;/gallery&gt;
African genera of Ageniellini:
* ''[[Auplopus]]'' &lt;small&gt;Spinola, 1841&lt;/small&gt;
* ''[[Cyemagenia]]'' &lt;small&gt;Arnold, 1946&lt;/small&gt;
* ''[[Dichragenia]]'' &lt;small&gt;Haupt, 1950&lt;/small&gt;
* ''[[Phanagenia]]'' &lt;small&gt;Banks, 1933&lt;/small&gt;   Madagascar
* ''[[Poecilagenia]]'' &lt;small&gt;Haupt, 1926&lt;/small&gt;

== Superfamily Scolioidea ==
=== Family Bradynobaenidae (Bradynobaenid wasps) ===
=== Family Scoliidae (Mammoth wasps) ===

== Superfamily Tiphioidea ==
Family Tiphiidae (Tiphiid wasps)
== Superfamily Thynnoidea ==
Family Thynnidae(Thynnid wasps)
== Superfamily Vespoidea ==
=== Family Rhopalosomatidae (Rhopalosomatid wasps) ===
===Family Vespidae (Paper, Potter &amp; Pollen wasps)===


== '''&lt;big&gt;Hymenoptera&lt;/big&gt;''' ==
About 20 000 described species of [[w:Hymenoptera|Hymenoptera]] (wasps, bees, ants and sawflies) are known from the [[w:Afrotropical|Afrotropical]] region. Estimates of the actual species count for the region range from 100 000 species to as high as 500 000 species.&lt;ref name=WaspWebOverview&gt;[https://www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm van Noort, Simon (2023). WaspWeb: Afrotropical Hymenoptera Initiative. www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm Retrieved 22 February 2023.]&lt;/ref&gt; Nineteen different superfamilies are illustrated on [[User:Alandmanson/Hymenoptera of southern Africa|this gallery]] in Wikimedia Commons.

The huge number of undescribed species means that many species will be extinct before we are even aware of them, as there are too few taxonomists employed to tackle the task of describing all of them in the next few decades.

==Classification==
The Order can be split into two Suborders - The [[w:Symphyta|Symphyta]] (Woodwasps, Horntails, Sawflies) and the [[w:Apocrita|Apocrita]] (Narrow-waisted wasps, ants and bees).&lt;br&gt;

When compared to the temperate regions of the northern hemisphere, the diversity of Symphyta in Africa is relatively poor, although sawflies of the Superfamily [[w:Tenthredinoidea|Tenthredinoidea]] are fairly common in forests and other woody vegetation types.

Africa has a rich diversity of Apocrita. The cladogram shown below&lt;ref name=Peters&gt;{{Cite journal |last1=Peters |first1=Ralph S. |last2=Krogmann |first2=Lars |last3=Mayer |first3=Christoph |last4=Donath |first4=Alexander |last5=Gunkel |first5=Simon |last6=Meusemann |first6=Karen |last7=Kozlov |first7=Alexey |last8=Podsiadlowski |first8=Lars |last9=Petersen |first9=Malte |title=Evolutionary History of the Hymenoptera |journal=Current Biology |volume=27 |issue=7 |pages=1013–1018 |doi=10.1016/j.cub.2017.01.027|pmid=28343967 |year=2017 |doi-access=free }}&lt;/ref&gt; indicates the possible relationships between 11 of the superfamilies that comprise Apocrita; These 11 superfamilies are all represented in Africa. This breakdown is used by iNaturalist.&lt;ref name=inat-Aculeata&gt;https://www.inaturalist.org/taxa/326777-Aculeata&lt;/ref&gt; It is, however, not accepted by all hymenopterists, and may change as more phylogenetic evidence is accumulated.&lt;ref name=waspwebClass&gt;[https://www.waspweb.org/Classification/index.htm van Noort, Simon (2023). WaspWeb: Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants). www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
{{Clade|style=font-size:85%; line-height:85%
|label1=Apocrita
            |1={{Clade
               |label1=[[w:Parasitoida|Parasitoida]]
               |1={{Clade
                  |1={{Clade
                     |1=[[w:Ceraphronoidea|Ceraphronoidea]]
                     |2=[[w:Ichneumonoidea|Ichneumonoidea]]
                     }}
                  |label2=[[w:Proctotrupomorpha|Proctotrupomorpha]]
                  |2={{Clade
                     |1=[[w:Cynipoidea|Cynipoidea]]
                     |2={{Clade
                        |1=[[w:Platygastroidea|Platygastroidea]]
                        |2={{Clade
                           |1=[[w:Chalcidoidea|Chalcidoidea]]
                           |2={{Clade
                              |1=[[w:Diaprioidea|Diaprioidea]]
                              |2=[[w:Proctotrupoidea|Proctotrupoidea]]
                  }} }} }} }} }}
               |2={{Clade
                  |1={{Clade
                     |1=[[w:Evanioidea|Evanioidea]]
                     |2=[[w:Stephanoidea|Stephanoidea]]
                     }}
                  |2={{Clade
                     |1=[[w:Trigonaloidea|Trigonaloidea]]
                     |label2=[[w:Aculeata|Aculeata]]
                     |2={{Clade
                        |1=[[w:Chrysidoidea|Chrysidoidea]]
                        |2={{Clade
                           |1=[[w:Vespoidea|Vespoidea]] (potter, honey and social wasps)
                           |2={{Clade
                              |1={{Clade
                                 |1=[[w:Pompiloidea|Pompiloidea]] (velvet ants, spider wasps and relatives)
                                 |2={{Clade
                                    |1=[[w:Thynnoidea|Thynnoidea]]
                                    |2=[[w:Tiphioidea|Tiphioidea]]
                                 }} }}
                              |2={{Clade
                                 |1=[[w:Scolioidea|Scolioidea]]
                                 |2={{Clade
                                    |1=[[w:Formicoidea|Formicoidea]] (ants)
                                    |2=[[w:Apoidea|Apoidea]] (bees and related wasps)
}} }} }} }} }} }} }} }} }}
&lt;br&gt;

==Some common African Symphyta==
&lt;gallery mode=packed heights=200&gt;
Sawfly, Argidae 2021 12 19 12 39 42.jpg|Sawfly, ''Arge'' sp. (Argidae, Superfamily Tenthredinoidea) 
Athalia 2019 11 21 0815.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Athalia 2019 08 23 8964.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Sirex noctilio F (prof.).JPG|''Sirex noctilio'', a pest in ''Pinus'' plantations (introduced to Africa)&lt;ref name=WaspWebSn&gt;[https://www.waspweb.org/Siricoidea/Siricidae/Sirex/Sirex_noctilio.htm van Noort, Simon (2023). WaspWeb: ''Sirex noctilio''. www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
&lt;/gallery&gt;

== Frequently reported African Apocrita ==
&lt;br&gt;
[[w:Aculeata|Aculeata]] (ants, bees and stinging wasps) are the most commonly observed Hymenoptera in Africa. There are many more photographs of African Aculeata on the web - See [https://www.waspweb.org/Classification/index.htm African Aculeata on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=326777 African Aculeata on iNaturalist] &lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Slender Ant (Tetraponera natalensis) (30538051244).jpg|Slender ant (Formicidae, Formicoidea)
Carpenter Bee (Xylocopa inconstans, female) (6021534858).jpg|Female carpenter bee (Apidae, Apoidea)
Chalybion 2019 12 02 2314.jpg|Blue mud-dauber wasp (Sphecidae, Apoidea)
Bethylidae 2019 08 21 13 42 21 8760.jpg|Flat wasp (Bethylidae, Chrysidoidea)
Belonogaster juncea colonialis, manlik, h, Pretoria.jpg|Paper wasp (Vespidae, Vespoidea)
Meria 2020 08 12 2193.jpg|''Meria'' sp. female (Thynnidae, Thynnoidea)
Dasylabris stimulatrix iNat37068107 2019 12 23 2990 W.jpg|''Dasylabris stimulatrix'', a velvet ant (Mutillidae, Pompiloidea)
Pompilidae 2019 05 28 0247.jpg|Spider-hunting wasp (Pompilidae, Pompiloidea)
Scoliid Wasp 2019 06 08 0492.jpg|Campsomerine wasp (Scoliidae, Scolioidea)
&lt;/gallery&gt;&lt;br&gt;

Darwin wasps ([[w:Ichneumonidae|Ichneumonidae]]) and braconids ([[w:Braconidae|Braconidae]]) are also frequently reported. Links to [https://www.waspweb.org/Ichneumonoidea/index.htm African Ichneumonoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=47200 African Ichneumonoidea on iNaturalist].&lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Echthromorpha agrestoria 2019 10 03 5572.jpg|''Echthromorpha agrestoria'' (Ichneumonidae, Ichneumonoidea)
Ichneumonidae inat 88832241.jpg|''Goryphus tricolor'' (Ichneumonidae, Ichneumonoidea)
Braconid Wasp (Archibracon servillei) (11857083186).jpg|Braconine wasp (Braconidae, Ichneumonoidea) 
2019 09 15 12 55 07 Microgastrinae 38425632.jpg|Microgastrine wasp (Braconidae, Ichneumonoidea) 
&lt;/gallery&gt;&lt;br&gt;

Small [[w:Chalcidoidea|chalcidoid]] parasitic wasps (especially those from the families [[w:Pteromalidae|Pteromalidae]], [[w:Chalcididae|Chalcididae]], [[w:Eulophidae|Eulophidae]], [[w:Eurytomidae|Eurytomidae]]) are also common. Links to [https://www.waspweb.org/Chalcidoidea/index.htm African Chalcidoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=69750 African Chalcidoidea on iNaturalist]
&lt;gallery mode=packed heights=200&gt;
Halticoptera 2019 11 26 2173Cedara.jpg|''Halticoptera'' sp. (Pteromalidae, Chalcidoidea)
Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp. (Chalcididae, Chalcidoidea)
Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp. (Eulophidae, Chalcidoidea)
Eurytomidae00.jpg|Mating seed chalcids (Eurytomidae, Chalcidoidea)
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>150tec81zl6ia23t8hisvwskib9gnyl</sha1>
    </revision>
    <revision>
      <id>2815543</id>
      <parentid>2815542</parentid>
      <timestamp>2026-06-13T18:33:45Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* African Pemphredonidae */</comment>
      <origin>2815543</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16007" sha1="3hlohcokoilimp5xw7jvbzr5n7htquj" xml:space="preserve">= Superfamily Apoidea =
== African Ampulicidae ==
&lt;gallery mode=packed heights=150&gt;
Ampulicidae 37894270 suncana.jpg
Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg
&lt;/gallery&gt;		
== African Astatidae ==
&lt;gallery mode=packed heights=150&gt;
Astata iN 105162782 Nicola van Berkel.jpg
Astata melanaria.jpg
&lt;/gallery&gt;

== African Bembicidae ==
===Subfamily Bembicinae===	
&lt;gallery mode=packed heights=150&gt;
Bembecinus iN 153052948.jpg|''Bembecinus'' sp.
Bembix iN 195581985.jpg|''Bembix'' sp.
Hoplisoides thalia iN 268375748 a.jpg|''Hoplisoides thalia''
&lt;/gallery&gt;		
Tribe Gorytini	
*		''Afrogorytes''
*		''Gorytes''
*		''Harpactus''
*		''Hoplisoides''
*		''Lestiphorus''
		
Tribe Spheciini	
*		''Ammatomus''
*		''Kohlia''
*		''Sphecius''
		
Tribe Handlirschiini	
*		''Handlirschia''
		
Tribe Bembicini	
*		''Bembix''
		
Tribe Stizini	
*		''Bembecinus''
*		''Stizoides''
*		''Stizus''

===Subfamily Nyssoninae===	
*		''Brachystegus''
*		''Hovanysson''
*		''Nysson''
		
===Subfamily Alyssontinae===	
*		''Alysson''
*		''Didineis''

== African Crabronidae ==
&lt;gallery mode=packed heights=150&gt;
Dasyproctus iN 30029277 a.jpg
Dicranorhina kohli 2023 07 25 iN 176115975.jpg
Liris on Crassula iN 42678436 01.jpg
Palarus Bee Pirate iN 144133368 2022-12-01 01.jpg
Paranysson iN 199673489 1.jpg
Pison iN 144131685 2022-11-30 03.jpg
Tachysphex iN 250449986 2024 10 09 7305.jpg
Tachytes iN 188902572 1964.jpg
Trypoxylon iN 99063113 a.jpg
&lt;/gallery&gt;

== African Pemphredonidae ==
&lt;gallery mode=packed heights=150&gt;
Polemistus braunsii iNaturalist 228280708.jpg

&lt;/gallery&gt;

== African Philanthidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;		
== African Psenidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;		
== African Sphecidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;

= Superfamily Chalcidoidea =
 
Chalcidoidea is an incredibly diverse group of wasps with a wide variety of life histories; many are parasitoids, making them ecologically important; some are used for biological control of insect pests. About 22 000 species have been described, but as many as 500 000 are thought to exist worldwide.&lt;ref name=Noyes2019/&gt;

Most chalcidoids are small (body length less than 5 mm), the smallest being males of the mymarid ''Dicopomorpha echmepterygis'', which is the smallest known flying insect (0.13 mm).&lt;ref name=Noyes2019&gt;Noyes, J.S. 2019. Universal Chalcidoidea Database. [http://www.nhm.ac.uk/chalcidoids]&lt;/ref&gt; However, some species may be as long as 20 mm (excluding antennae and ovipositor), such as the pelecinellid ''Doddifoenus wallacei''.&lt;ref name=Krogmann&amp;Burks2009&gt;Krogmann, Lars; Burks, Roger A. (2009-12-31). &quot;Doddifoenus wallacei, a new giant parasitoid wasp of the subfamily Leptofoeninae (Chalcidoidea: Pteromalidae), with a description of its mesosomal skeletal anatomy and a molecular characterization&quot;. Zootaxa. 2194: 21–36. doi:10.5281/zenodo.189452 [https://www.researchgate.net/profile/Lars-Krogmann/publication/250916337 PDF]&lt;/ref&gt; Many have bright metallic colours, hence the name Chalcidoidea or chalcid - the greek word χαλκός (chalcos) means copper or bronze.&lt;ref name=Perseus2023&gt;[http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0073:entry=xalko/s Georg Autenrieth, A Homeric Dictionary]&lt;/ref&gt;

Many new chalcidoid families were proposed in 2022.&lt;ref name=Burks2022&gt;Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. [https://doi.org/10.3897/jhr.94.94263 DOI]&lt;/ref&gt; This was largely based on phylogenetic evidence (both molecular and morphological) accumulated since the families of the Superfamily were outlined in 1993&lt;ref name=Gibson1993&gt;Gibson, G.A.P. (1993) Superfamilies Mymarommatoidea and Chalcidoidea (pp. 570-655). In Goulet, H. &amp; Huber, J. (eds). Hymenoptera of the World: an identification guide to families. Research Branch, Agriculture Canada, Ottawa, Canada, 668 pp. [https://www.researchgate.net/publication/259227143 PDF]&lt;/ref&gt;.


Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. https://doi.org/10.3897/jhr.94.94263

Cruaud, A., Rasplus, J.Y., Zhang, J., Burks, R., Delvare, G., Fusu, L., Gumovsky, A., Huber, J.T., Janšta, P., Mitroiu, M.D. and Noyes, J.S., 2022. The Chalcidoidea bush of life–a massive radiation blurred by mutational saturation. bioRxiv. https://doi.org/10.1101/2022.09.11.507458 https://www.biorxiv.org/content/10.1101/2022.09.11.507458.full.pdf


= Superfamily Ichneumonoidea =

Quicke, D.L., 2015. The braconid and ichneumonid parasitoid wasps: biology, systematics, evolution and ecology. John Wiley &amp; Sons.
https://www.researchgate.net/publication/281662102_The_Braconid_and_Ichneumonid_Parasitoid_Wasps_Biology_Systematics_Evolution_and_Ecology

=== Diplazontinae ===
Quote from Fitton &amp; Rotheray (1982):
&quot;Diplazontines are distinguished easily from other ichneumonids by their general appearance (Fig. 1). The main confirmatory characters are the bifid upper tooth of the mandible (Fig. 2) and the sub-rectangular shape of the first tergite of the gaster (Figs. 1,3,4, 13, 15 and 17). In fact, the diplazontines are so distinct that they can be separated from all other Hymenoptera-Apocrita by possession of the following five characters in combination: (1) antenna of seventeen or more segments (Fig. I); (2) forewing with costal cell obliterated (that is, veins C and Sc + R + Rs contiguous from the wing base to the base of the pterostigma (Fig. 6 ) ; (3) forewing with cross-vein 2 mcu present (Fig. 6); (4) mandible with its upper tooth subdivided (bifid), making the mandible tridentate (Fig. 2); (5) first tergite of gaster sub-rectangular in shape (that is, subequal in width anteriorly and posteriorly) (Figs. 1 , 3 , 4 , 13,15 and 17).&quot;

Fitton, M. G., &amp; Rotheray, G. E. (1982). A key to the European genera of diplazontine ichneumon‐flies, with notes on the British fauna. Systematic Entomology, 7(3), 311-320.

=== Superfamily Pompiloidea ===
Family Mutillidae (Velvet ants)
Family Pompilidae (Spider hunting wasps)
Family Sapygidae (Sapygid wasps)


== Spider-hunting wasps ==
Two wasps from the genus ''Auplopus'' from Europe. Most of the ''Auplopus'' species, along with others from the Tribe Ageniellini, amputate the legs of the spiders they capture; this makes it much easier to move the spider to the nest.
&lt;gallery mode=packed heights=200&gt;
Auplopus carbonarius IMG 1624.jpg
Auplopus carbonarius fg01 20060623 Nied Garten.jpg
&lt;/gallery&gt;
African genera of Ageniellini:
* ''[[Auplopus]]'' &lt;small&gt;Spinola, 1841&lt;/small&gt;
* ''[[Cyemagenia]]'' &lt;small&gt;Arnold, 1946&lt;/small&gt;
* ''[[Dichragenia]]'' &lt;small&gt;Haupt, 1950&lt;/small&gt;
* ''[[Phanagenia]]'' &lt;small&gt;Banks, 1933&lt;/small&gt;   Madagascar
* ''[[Poecilagenia]]'' &lt;small&gt;Haupt, 1926&lt;/small&gt;

== Superfamily Scolioidea ==
=== Family Bradynobaenidae (Bradynobaenid wasps) ===
=== Family Scoliidae (Mammoth wasps) ===

== Superfamily Tiphioidea ==
Family Tiphiidae (Tiphiid wasps)
== Superfamily Thynnoidea ==
Family Thynnidae(Thynnid wasps)
== Superfamily Vespoidea ==
=== Family Rhopalosomatidae (Rhopalosomatid wasps) ===
===Family Vespidae (Paper, Potter &amp; Pollen wasps)===


== '''&lt;big&gt;Hymenoptera&lt;/big&gt;''' ==
About 20 000 described species of [[w:Hymenoptera|Hymenoptera]] (wasps, bees, ants and sawflies) are known from the [[w:Afrotropical|Afrotropical]] region. Estimates of the actual species count for the region range from 100 000 species to as high as 500 000 species.&lt;ref name=WaspWebOverview&gt;[https://www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm van Noort, Simon (2023). WaspWeb: Afrotropical Hymenoptera Initiative. www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm Retrieved 22 February 2023.]&lt;/ref&gt; Nineteen different superfamilies are illustrated on [[User:Alandmanson/Hymenoptera of southern Africa|this gallery]] in Wikimedia Commons.

The huge number of undescribed species means that many species will be extinct before we are even aware of them, as there are too few taxonomists employed to tackle the task of describing all of them in the next few decades.

==Classification==
The Order can be split into two Suborders - The [[w:Symphyta|Symphyta]] (Woodwasps, Horntails, Sawflies) and the [[w:Apocrita|Apocrita]] (Narrow-waisted wasps, ants and bees).&lt;br&gt;

When compared to the temperate regions of the northern hemisphere, the diversity of Symphyta in Africa is relatively poor, although sawflies of the Superfamily [[w:Tenthredinoidea|Tenthredinoidea]] are fairly common in forests and other woody vegetation types.

Africa has a rich diversity of Apocrita. The cladogram shown below&lt;ref name=Peters&gt;{{Cite journal |last1=Peters |first1=Ralph S. |last2=Krogmann |first2=Lars |last3=Mayer |first3=Christoph |last4=Donath |first4=Alexander |last5=Gunkel |first5=Simon |last6=Meusemann |first6=Karen |last7=Kozlov |first7=Alexey |last8=Podsiadlowski |first8=Lars |last9=Petersen |first9=Malte |title=Evolutionary History of the Hymenoptera |journal=Current Biology |volume=27 |issue=7 |pages=1013–1018 |doi=10.1016/j.cub.2017.01.027|pmid=28343967 |year=2017 |doi-access=free }}&lt;/ref&gt; indicates the possible relationships between 11 of the superfamilies that comprise Apocrita; These 11 superfamilies are all represented in Africa. This breakdown is used by iNaturalist.&lt;ref name=inat-Aculeata&gt;https://www.inaturalist.org/taxa/326777-Aculeata&lt;/ref&gt; It is, however, not accepted by all hymenopterists, and may change as more phylogenetic evidence is accumulated.&lt;ref name=waspwebClass&gt;[https://www.waspweb.org/Classification/index.htm van Noort, Simon (2023). WaspWeb: Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants). www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
{{Clade|style=font-size:85%; line-height:85%
|label1=Apocrita
            |1={{Clade
               |label1=[[w:Parasitoida|Parasitoida]]
               |1={{Clade
                  |1={{Clade
                     |1=[[w:Ceraphronoidea|Ceraphronoidea]]
                     |2=[[w:Ichneumonoidea|Ichneumonoidea]]
                     }}
                  |label2=[[w:Proctotrupomorpha|Proctotrupomorpha]]
                  |2={{Clade
                     |1=[[w:Cynipoidea|Cynipoidea]]
                     |2={{Clade
                        |1=[[w:Platygastroidea|Platygastroidea]]
                        |2={{Clade
                           |1=[[w:Chalcidoidea|Chalcidoidea]]
                           |2={{Clade
                              |1=[[w:Diaprioidea|Diaprioidea]]
                              |2=[[w:Proctotrupoidea|Proctotrupoidea]]
                  }} }} }} }} }}
               |2={{Clade
                  |1={{Clade
                     |1=[[w:Evanioidea|Evanioidea]]
                     |2=[[w:Stephanoidea|Stephanoidea]]
                     }}
                  |2={{Clade
                     |1=[[w:Trigonaloidea|Trigonaloidea]]
                     |label2=[[w:Aculeata|Aculeata]]
                     |2={{Clade
                        |1=[[w:Chrysidoidea|Chrysidoidea]]
                        |2={{Clade
                           |1=[[w:Vespoidea|Vespoidea]] (potter, honey and social wasps)
                           |2={{Clade
                              |1={{Clade
                                 |1=[[w:Pompiloidea|Pompiloidea]] (velvet ants, spider wasps and relatives)
                                 |2={{Clade
                                    |1=[[w:Thynnoidea|Thynnoidea]]
                                    |2=[[w:Tiphioidea|Tiphioidea]]
                                 }} }}
                              |2={{Clade
                                 |1=[[w:Scolioidea|Scolioidea]]
                                 |2={{Clade
                                    |1=[[w:Formicoidea|Formicoidea]] (ants)
                                    |2=[[w:Apoidea|Apoidea]] (bees and related wasps)
}} }} }} }} }} }} }} }} }}
&lt;br&gt;

==Some common African Symphyta==
&lt;gallery mode=packed heights=200&gt;
Sawfly, Argidae 2021 12 19 12 39 42.jpg|Sawfly, ''Arge'' sp. (Argidae, Superfamily Tenthredinoidea) 
Athalia 2019 11 21 0815.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Athalia 2019 08 23 8964.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Sirex noctilio F (prof.).JPG|''Sirex noctilio'', a pest in ''Pinus'' plantations (introduced to Africa)&lt;ref name=WaspWebSn&gt;[https://www.waspweb.org/Siricoidea/Siricidae/Sirex/Sirex_noctilio.htm van Noort, Simon (2023). WaspWeb: ''Sirex noctilio''. www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
&lt;/gallery&gt;

== Frequently reported African Apocrita ==
&lt;br&gt;
[[w:Aculeata|Aculeata]] (ants, bees and stinging wasps) are the most commonly observed Hymenoptera in Africa. There are many more photographs of African Aculeata on the web - See [https://www.waspweb.org/Classification/index.htm African Aculeata on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=326777 African Aculeata on iNaturalist] &lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Slender Ant (Tetraponera natalensis) (30538051244).jpg|Slender ant (Formicidae, Formicoidea)
Carpenter Bee (Xylocopa inconstans, female) (6021534858).jpg|Female carpenter bee (Apidae, Apoidea)
Chalybion 2019 12 02 2314.jpg|Blue mud-dauber wasp (Sphecidae, Apoidea)
Bethylidae 2019 08 21 13 42 21 8760.jpg|Flat wasp (Bethylidae, Chrysidoidea)
Belonogaster juncea colonialis, manlik, h, Pretoria.jpg|Paper wasp (Vespidae, Vespoidea)
Meria 2020 08 12 2193.jpg|''Meria'' sp. female (Thynnidae, Thynnoidea)
Dasylabris stimulatrix iNat37068107 2019 12 23 2990 W.jpg|''Dasylabris stimulatrix'', a velvet ant (Mutillidae, Pompiloidea)
Pompilidae 2019 05 28 0247.jpg|Spider-hunting wasp (Pompilidae, Pompiloidea)
Scoliid Wasp 2019 06 08 0492.jpg|Campsomerine wasp (Scoliidae, Scolioidea)
&lt;/gallery&gt;&lt;br&gt;

Darwin wasps ([[w:Ichneumonidae|Ichneumonidae]]) and braconids ([[w:Braconidae|Braconidae]]) are also frequently reported. Links to [https://www.waspweb.org/Ichneumonoidea/index.htm African Ichneumonoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=47200 African Ichneumonoidea on iNaturalist].&lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Echthromorpha agrestoria 2019 10 03 5572.jpg|''Echthromorpha agrestoria'' (Ichneumonidae, Ichneumonoidea)
Ichneumonidae inat 88832241.jpg|''Goryphus tricolor'' (Ichneumonidae, Ichneumonoidea)
Braconid Wasp (Archibracon servillei) (11857083186).jpg|Braconine wasp (Braconidae, Ichneumonoidea) 
2019 09 15 12 55 07 Microgastrinae 38425632.jpg|Microgastrine wasp (Braconidae, Ichneumonoidea) 
&lt;/gallery&gt;&lt;br&gt;

Small [[w:Chalcidoidea|chalcidoid]] parasitic wasps (especially those from the families [[w:Pteromalidae|Pteromalidae]], [[w:Chalcididae|Chalcididae]], [[w:Eulophidae|Eulophidae]], [[w:Eurytomidae|Eurytomidae]]) are also common. Links to [https://www.waspweb.org/Chalcidoidea/index.htm African Chalcidoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=69750 African Chalcidoidea on iNaturalist]
&lt;gallery mode=packed heights=200&gt;
Halticoptera 2019 11 26 2173Cedara.jpg|''Halticoptera'' sp. (Pteromalidae, Chalcidoidea)
Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp. (Chalcididae, Chalcidoidea)
Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp. (Eulophidae, Chalcidoidea)
Eurytomidae00.jpg|Mating seed chalcids (Eurytomidae, Chalcidoidea)
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>3hlohcokoilimp5xw7jvbzr5n7htquj</sha1>
    </revision>
    <revision>
      <id>2815544</id>
      <parentid>2815543</parentid>
      <timestamp>2026-06-13T18:35:06Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* African Philanthidae */</comment>
      <origin>2815544</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16077" sha1="gmvux80o4y99ajdspgbhvg9yne8r1ry" xml:space="preserve">= Superfamily Apoidea =
== African Ampulicidae ==
&lt;gallery mode=packed heights=150&gt;
Ampulicidae 37894270 suncana.jpg
Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg
&lt;/gallery&gt;		
== African Astatidae ==
&lt;gallery mode=packed heights=150&gt;
Astata iN 105162782 Nicola van Berkel.jpg
Astata melanaria.jpg
&lt;/gallery&gt;

== African Bembicidae ==
===Subfamily Bembicinae===	
&lt;gallery mode=packed heights=150&gt;
Bembecinus iN 153052948.jpg|''Bembecinus'' sp.
Bembix iN 195581985.jpg|''Bembix'' sp.
Hoplisoides thalia iN 268375748 a.jpg|''Hoplisoides thalia''
&lt;/gallery&gt;		
Tribe Gorytini	
*		''Afrogorytes''
*		''Gorytes''
*		''Harpactus''
*		''Hoplisoides''
*		''Lestiphorus''
		
Tribe Spheciini	
*		''Ammatomus''
*		''Kohlia''
*		''Sphecius''
		
Tribe Handlirschiini	
*		''Handlirschia''
		
Tribe Bembicini	
*		''Bembix''
		
Tribe Stizini	
*		''Bembecinus''
*		''Stizoides''
*		''Stizus''

===Subfamily Nyssoninae===	
*		''Brachystegus''
*		''Hovanysson''
*		''Nysson''
		
===Subfamily Alyssontinae===	
*		''Alysson''
*		''Didineis''

== African Crabronidae ==
&lt;gallery mode=packed heights=150&gt;
Dasyproctus iN 30029277 a.jpg
Dicranorhina kohli 2023 07 25 iN 176115975.jpg
Liris on Crassula iN 42678436 01.jpg
Palarus Bee Pirate iN 144133368 2022-12-01 01.jpg
Paranysson iN 199673489 1.jpg
Pison iN 144131685 2022-11-30 03.jpg
Tachysphex iN 250449986 2024 10 09 7305.jpg
Tachytes iN 188902572 1964.jpg
Trypoxylon iN 99063113 a.jpg
&lt;/gallery&gt;

== African Pemphredonidae ==
&lt;gallery mode=packed heights=150&gt;
Polemistus braunsii iNaturalist 228280708.jpg

&lt;/gallery&gt;

== African Philanthidae ==
&lt;gallery mode=packed heights=150&gt;
Philanthus triangulum diadema 187037342.jpg
Cerceris 2019 12 02 2310.jpg
&lt;/gallery&gt;

== African Psenidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;		
== African Sphecidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;

= Superfamily Chalcidoidea =
 
Chalcidoidea is an incredibly diverse group of wasps with a wide variety of life histories; many are parasitoids, making them ecologically important; some are used for biological control of insect pests. About 22 000 species have been described, but as many as 500 000 are thought to exist worldwide.&lt;ref name=Noyes2019/&gt;

Most chalcidoids are small (body length less than 5 mm), the smallest being males of the mymarid ''Dicopomorpha echmepterygis'', which is the smallest known flying insect (0.13 mm).&lt;ref name=Noyes2019&gt;Noyes, J.S. 2019. Universal Chalcidoidea Database. [http://www.nhm.ac.uk/chalcidoids]&lt;/ref&gt; However, some species may be as long as 20 mm (excluding antennae and ovipositor), such as the pelecinellid ''Doddifoenus wallacei''.&lt;ref name=Krogmann&amp;Burks2009&gt;Krogmann, Lars; Burks, Roger A. (2009-12-31). &quot;Doddifoenus wallacei, a new giant parasitoid wasp of the subfamily Leptofoeninae (Chalcidoidea: Pteromalidae), with a description of its mesosomal skeletal anatomy and a molecular characterization&quot;. Zootaxa. 2194: 21–36. doi:10.5281/zenodo.189452 [https://www.researchgate.net/profile/Lars-Krogmann/publication/250916337 PDF]&lt;/ref&gt; Many have bright metallic colours, hence the name Chalcidoidea or chalcid - the greek word χαλκός (chalcos) means copper or bronze.&lt;ref name=Perseus2023&gt;[http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0073:entry=xalko/s Georg Autenrieth, A Homeric Dictionary]&lt;/ref&gt;

Many new chalcidoid families were proposed in 2022.&lt;ref name=Burks2022&gt;Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. [https://doi.org/10.3897/jhr.94.94263 DOI]&lt;/ref&gt; This was largely based on phylogenetic evidence (both molecular and morphological) accumulated since the families of the Superfamily were outlined in 1993&lt;ref name=Gibson1993&gt;Gibson, G.A.P. (1993) Superfamilies Mymarommatoidea and Chalcidoidea (pp. 570-655). In Goulet, H. &amp; Huber, J. (eds). Hymenoptera of the World: an identification guide to families. Research Branch, Agriculture Canada, Ottawa, Canada, 668 pp. [https://www.researchgate.net/publication/259227143 PDF]&lt;/ref&gt;.


Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. https://doi.org/10.3897/jhr.94.94263

Cruaud, A., Rasplus, J.Y., Zhang, J., Burks, R., Delvare, G., Fusu, L., Gumovsky, A., Huber, J.T., Janšta, P., Mitroiu, M.D. and Noyes, J.S., 2022. The Chalcidoidea bush of life–a massive radiation blurred by mutational saturation. bioRxiv. https://doi.org/10.1101/2022.09.11.507458 https://www.biorxiv.org/content/10.1101/2022.09.11.507458.full.pdf


= Superfamily Ichneumonoidea =

Quicke, D.L., 2015. The braconid and ichneumonid parasitoid wasps: biology, systematics, evolution and ecology. John Wiley &amp; Sons.
https://www.researchgate.net/publication/281662102_The_Braconid_and_Ichneumonid_Parasitoid_Wasps_Biology_Systematics_Evolution_and_Ecology

=== Diplazontinae ===
Quote from Fitton &amp; Rotheray (1982):
&quot;Diplazontines are distinguished easily from other ichneumonids by their general appearance (Fig. 1). The main confirmatory characters are the bifid upper tooth of the mandible (Fig. 2) and the sub-rectangular shape of the first tergite of the gaster (Figs. 1,3,4, 13, 15 and 17). In fact, the diplazontines are so distinct that they can be separated from all other Hymenoptera-Apocrita by possession of the following five characters in combination: (1) antenna of seventeen or more segments (Fig. I); (2) forewing with costal cell obliterated (that is, veins C and Sc + R + Rs contiguous from the wing base to the base of the pterostigma (Fig. 6 ) ; (3) forewing with cross-vein 2 mcu present (Fig. 6); (4) mandible with its upper tooth subdivided (bifid), making the mandible tridentate (Fig. 2); (5) first tergite of gaster sub-rectangular in shape (that is, subequal in width anteriorly and posteriorly) (Figs. 1 , 3 , 4 , 13,15 and 17).&quot;

Fitton, M. G., &amp; Rotheray, G. E. (1982). A key to the European genera of diplazontine ichneumon‐flies, with notes on the British fauna. Systematic Entomology, 7(3), 311-320.

=== Superfamily Pompiloidea ===
Family Mutillidae (Velvet ants)
Family Pompilidae (Spider hunting wasps)
Family Sapygidae (Sapygid wasps)


== Spider-hunting wasps ==
Two wasps from the genus ''Auplopus'' from Europe. Most of the ''Auplopus'' species, along with others from the Tribe Ageniellini, amputate the legs of the spiders they capture; this makes it much easier to move the spider to the nest.
&lt;gallery mode=packed heights=200&gt;
Auplopus carbonarius IMG 1624.jpg
Auplopus carbonarius fg01 20060623 Nied Garten.jpg
&lt;/gallery&gt;
African genera of Ageniellini:
* ''[[Auplopus]]'' &lt;small&gt;Spinola, 1841&lt;/small&gt;
* ''[[Cyemagenia]]'' &lt;small&gt;Arnold, 1946&lt;/small&gt;
* ''[[Dichragenia]]'' &lt;small&gt;Haupt, 1950&lt;/small&gt;
* ''[[Phanagenia]]'' &lt;small&gt;Banks, 1933&lt;/small&gt;   Madagascar
* ''[[Poecilagenia]]'' &lt;small&gt;Haupt, 1926&lt;/small&gt;

== Superfamily Scolioidea ==
=== Family Bradynobaenidae (Bradynobaenid wasps) ===
=== Family Scoliidae (Mammoth wasps) ===

== Superfamily Tiphioidea ==
Family Tiphiidae (Tiphiid wasps)
== Superfamily Thynnoidea ==
Family Thynnidae(Thynnid wasps)
== Superfamily Vespoidea ==
=== Family Rhopalosomatidae (Rhopalosomatid wasps) ===
===Family Vespidae (Paper, Potter &amp; Pollen wasps)===


== '''&lt;big&gt;Hymenoptera&lt;/big&gt;''' ==
About 20 000 described species of [[w:Hymenoptera|Hymenoptera]] (wasps, bees, ants and sawflies) are known from the [[w:Afrotropical|Afrotropical]] region. Estimates of the actual species count for the region range from 100 000 species to as high as 500 000 species.&lt;ref name=WaspWebOverview&gt;[https://www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm van Noort, Simon (2023). WaspWeb: Afrotropical Hymenoptera Initiative. www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm Retrieved 22 February 2023.]&lt;/ref&gt; Nineteen different superfamilies are illustrated on [[User:Alandmanson/Hymenoptera of southern Africa|this gallery]] in Wikimedia Commons.

The huge number of undescribed species means that many species will be extinct before we are even aware of them, as there are too few taxonomists employed to tackle the task of describing all of them in the next few decades.

==Classification==
The Order can be split into two Suborders - The [[w:Symphyta|Symphyta]] (Woodwasps, Horntails, Sawflies) and the [[w:Apocrita|Apocrita]] (Narrow-waisted wasps, ants and bees).&lt;br&gt;

When compared to the temperate regions of the northern hemisphere, the diversity of Symphyta in Africa is relatively poor, although sawflies of the Superfamily [[w:Tenthredinoidea|Tenthredinoidea]] are fairly common in forests and other woody vegetation types.

Africa has a rich diversity of Apocrita. The cladogram shown below&lt;ref name=Peters&gt;{{Cite journal |last1=Peters |first1=Ralph S. |last2=Krogmann |first2=Lars |last3=Mayer |first3=Christoph |last4=Donath |first4=Alexander |last5=Gunkel |first5=Simon |last6=Meusemann |first6=Karen |last7=Kozlov |first7=Alexey |last8=Podsiadlowski |first8=Lars |last9=Petersen |first9=Malte |title=Evolutionary History of the Hymenoptera |journal=Current Biology |volume=27 |issue=7 |pages=1013–1018 |doi=10.1016/j.cub.2017.01.027|pmid=28343967 |year=2017 |doi-access=free }}&lt;/ref&gt; indicates the possible relationships between 11 of the superfamilies that comprise Apocrita; These 11 superfamilies are all represented in Africa. This breakdown is used by iNaturalist.&lt;ref name=inat-Aculeata&gt;https://www.inaturalist.org/taxa/326777-Aculeata&lt;/ref&gt; It is, however, not accepted by all hymenopterists, and may change as more phylogenetic evidence is accumulated.&lt;ref name=waspwebClass&gt;[https://www.waspweb.org/Classification/index.htm van Noort, Simon (2023). WaspWeb: Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants). www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
{{Clade|style=font-size:85%; line-height:85%
|label1=Apocrita
            |1={{Clade
               |label1=[[w:Parasitoida|Parasitoida]]
               |1={{Clade
                  |1={{Clade
                     |1=[[w:Ceraphronoidea|Ceraphronoidea]]
                     |2=[[w:Ichneumonoidea|Ichneumonoidea]]
                     }}
                  |label2=[[w:Proctotrupomorpha|Proctotrupomorpha]]
                  |2={{Clade
                     |1=[[w:Cynipoidea|Cynipoidea]]
                     |2={{Clade
                        |1=[[w:Platygastroidea|Platygastroidea]]
                        |2={{Clade
                           |1=[[w:Chalcidoidea|Chalcidoidea]]
                           |2={{Clade
                              |1=[[w:Diaprioidea|Diaprioidea]]
                              |2=[[w:Proctotrupoidea|Proctotrupoidea]]
                  }} }} }} }} }}
               |2={{Clade
                  |1={{Clade
                     |1=[[w:Evanioidea|Evanioidea]]
                     |2=[[w:Stephanoidea|Stephanoidea]]
                     }}
                  |2={{Clade
                     |1=[[w:Trigonaloidea|Trigonaloidea]]
                     |label2=[[w:Aculeata|Aculeata]]
                     |2={{Clade
                        |1=[[w:Chrysidoidea|Chrysidoidea]]
                        |2={{Clade
                           |1=[[w:Vespoidea|Vespoidea]] (potter, honey and social wasps)
                           |2={{Clade
                              |1={{Clade
                                 |1=[[w:Pompiloidea|Pompiloidea]] (velvet ants, spider wasps and relatives)
                                 |2={{Clade
                                    |1=[[w:Thynnoidea|Thynnoidea]]
                                    |2=[[w:Tiphioidea|Tiphioidea]]
                                 }} }}
                              |2={{Clade
                                 |1=[[w:Scolioidea|Scolioidea]]
                                 |2={{Clade
                                    |1=[[w:Formicoidea|Formicoidea]] (ants)
                                    |2=[[w:Apoidea|Apoidea]] (bees and related wasps)
}} }} }} }} }} }} }} }} }}
&lt;br&gt;

==Some common African Symphyta==
&lt;gallery mode=packed heights=200&gt;
Sawfly, Argidae 2021 12 19 12 39 42.jpg|Sawfly, ''Arge'' sp. (Argidae, Superfamily Tenthredinoidea) 
Athalia 2019 11 21 0815.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Athalia 2019 08 23 8964.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Sirex noctilio F (prof.).JPG|''Sirex noctilio'', a pest in ''Pinus'' plantations (introduced to Africa)&lt;ref name=WaspWebSn&gt;[https://www.waspweb.org/Siricoidea/Siricidae/Sirex/Sirex_noctilio.htm van Noort, Simon (2023). WaspWeb: ''Sirex noctilio''. www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
&lt;/gallery&gt;

== Frequently reported African Apocrita ==
&lt;br&gt;
[[w:Aculeata|Aculeata]] (ants, bees and stinging wasps) are the most commonly observed Hymenoptera in Africa. There are many more photographs of African Aculeata on the web - See [https://www.waspweb.org/Classification/index.htm African Aculeata on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=326777 African Aculeata on iNaturalist] &lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Slender Ant (Tetraponera natalensis) (30538051244).jpg|Slender ant (Formicidae, Formicoidea)
Carpenter Bee (Xylocopa inconstans, female) (6021534858).jpg|Female carpenter bee (Apidae, Apoidea)
Chalybion 2019 12 02 2314.jpg|Blue mud-dauber wasp (Sphecidae, Apoidea)
Bethylidae 2019 08 21 13 42 21 8760.jpg|Flat wasp (Bethylidae, Chrysidoidea)
Belonogaster juncea colonialis, manlik, h, Pretoria.jpg|Paper wasp (Vespidae, Vespoidea)
Meria 2020 08 12 2193.jpg|''Meria'' sp. female (Thynnidae, Thynnoidea)
Dasylabris stimulatrix iNat37068107 2019 12 23 2990 W.jpg|''Dasylabris stimulatrix'', a velvet ant (Mutillidae, Pompiloidea)
Pompilidae 2019 05 28 0247.jpg|Spider-hunting wasp (Pompilidae, Pompiloidea)
Scoliid Wasp 2019 06 08 0492.jpg|Campsomerine wasp (Scoliidae, Scolioidea)
&lt;/gallery&gt;&lt;br&gt;

Darwin wasps ([[w:Ichneumonidae|Ichneumonidae]]) and braconids ([[w:Braconidae|Braconidae]]) are also frequently reported. Links to [https://www.waspweb.org/Ichneumonoidea/index.htm African Ichneumonoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=47200 African Ichneumonoidea on iNaturalist].&lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Echthromorpha agrestoria 2019 10 03 5572.jpg|''Echthromorpha agrestoria'' (Ichneumonidae, Ichneumonoidea)
Ichneumonidae inat 88832241.jpg|''Goryphus tricolor'' (Ichneumonidae, Ichneumonoidea)
Braconid Wasp (Archibracon servillei) (11857083186).jpg|Braconine wasp (Braconidae, Ichneumonoidea) 
2019 09 15 12 55 07 Microgastrinae 38425632.jpg|Microgastrine wasp (Braconidae, Ichneumonoidea) 
&lt;/gallery&gt;&lt;br&gt;

Small [[w:Chalcidoidea|chalcidoid]] parasitic wasps (especially those from the families [[w:Pteromalidae|Pteromalidae]], [[w:Chalcididae|Chalcididae]], [[w:Eulophidae|Eulophidae]], [[w:Eurytomidae|Eurytomidae]]) are also common. Links to [https://www.waspweb.org/Chalcidoidea/index.htm African Chalcidoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=69750 African Chalcidoidea on iNaturalist]
&lt;gallery mode=packed heights=200&gt;
Halticoptera 2019 11 26 2173Cedara.jpg|''Halticoptera'' sp. (Pteromalidae, Chalcidoidea)
Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp. (Chalcididae, Chalcidoidea)
Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp. (Eulophidae, Chalcidoidea)
Eurytomidae00.jpg|Mating seed chalcids (Eurytomidae, Chalcidoidea)
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>gmvux80o4y99ajdspgbhvg9yne8r1ry</sha1>
    </revision>
    <revision>
      <id>2815545</id>
      <parentid>2815544</parentid>
      <timestamp>2026-06-13T18:42:16Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* African Psenidae */</comment>
      <origin>2815545</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16110" sha1="pym7mwvjtazl2mu6m5rwweam14rctww" xml:space="preserve">= Superfamily Apoidea =
== African Ampulicidae ==
&lt;gallery mode=packed heights=150&gt;
Ampulicidae 37894270 suncana.jpg
Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg
&lt;/gallery&gt;		
== African Astatidae ==
&lt;gallery mode=packed heights=150&gt;
Astata iN 105162782 Nicola van Berkel.jpg
Astata melanaria.jpg
&lt;/gallery&gt;

== African Bembicidae ==
===Subfamily Bembicinae===	
&lt;gallery mode=packed heights=150&gt;
Bembecinus iN 153052948.jpg|''Bembecinus'' sp.
Bembix iN 195581985.jpg|''Bembix'' sp.
Hoplisoides thalia iN 268375748 a.jpg|''Hoplisoides thalia''
&lt;/gallery&gt;		
Tribe Gorytini	
*		''Afrogorytes''
*		''Gorytes''
*		''Harpactus''
*		''Hoplisoides''
*		''Lestiphorus''
		
Tribe Spheciini	
*		''Ammatomus''
*		''Kohlia''
*		''Sphecius''
		
Tribe Handlirschiini	
*		''Handlirschia''
		
Tribe Bembicini	
*		''Bembix''
		
Tribe Stizini	
*		''Bembecinus''
*		''Stizoides''
*		''Stizus''

===Subfamily Nyssoninae===	
*		''Brachystegus''
*		''Hovanysson''
*		''Nysson''
		
===Subfamily Alyssontinae===	
*		''Alysson''
*		''Didineis''

== African Crabronidae ==
&lt;gallery mode=packed heights=150&gt;
Dasyproctus iN 30029277 a.jpg
Dicranorhina kohli 2023 07 25 iN 176115975.jpg
Liris on Crassula iN 42678436 01.jpg
Palarus Bee Pirate iN 144133368 2022-12-01 01.jpg
Paranysson iN 199673489 1.jpg
Pison iN 144131685 2022-11-30 03.jpg
Tachysphex iN 250449986 2024 10 09 7305.jpg
Tachytes iN 188902572 1964.jpg
Trypoxylon iN 99063113 a.jpg
&lt;/gallery&gt;

== African Pemphredonidae ==
&lt;gallery mode=packed heights=150&gt;
Polemistus braunsii iNaturalist 228280708.jpg

&lt;/gallery&gt;

== African Philanthidae ==
&lt;gallery mode=packed heights=150&gt;
Philanthus triangulum diadema 187037342.jpg
Cerceris 2019 12 02 2310.jpg
&lt;/gallery&gt;

== African Psenidae ==
&lt;gallery mode=packed heights=150&gt;
Psenini iN 1022563 i c riddell.jpg

&lt;/gallery&gt;

== African Sphecidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;

= Superfamily Chalcidoidea =
 
Chalcidoidea is an incredibly diverse group of wasps with a wide variety of life histories; many are parasitoids, making them ecologically important; some are used for biological control of insect pests. About 22 000 species have been described, but as many as 500 000 are thought to exist worldwide.&lt;ref name=Noyes2019/&gt;

Most chalcidoids are small (body length less than 5 mm), the smallest being males of the mymarid ''Dicopomorpha echmepterygis'', which is the smallest known flying insect (0.13 mm).&lt;ref name=Noyes2019&gt;Noyes, J.S. 2019. Universal Chalcidoidea Database. [http://www.nhm.ac.uk/chalcidoids]&lt;/ref&gt; However, some species may be as long as 20 mm (excluding antennae and ovipositor), such as the pelecinellid ''Doddifoenus wallacei''.&lt;ref name=Krogmann&amp;Burks2009&gt;Krogmann, Lars; Burks, Roger A. (2009-12-31). &quot;Doddifoenus wallacei, a new giant parasitoid wasp of the subfamily Leptofoeninae (Chalcidoidea: Pteromalidae), with a description of its mesosomal skeletal anatomy and a molecular characterization&quot;. Zootaxa. 2194: 21–36. doi:10.5281/zenodo.189452 [https://www.researchgate.net/profile/Lars-Krogmann/publication/250916337 PDF]&lt;/ref&gt; Many have bright metallic colours, hence the name Chalcidoidea or chalcid - the greek word χαλκός (chalcos) means copper or bronze.&lt;ref name=Perseus2023&gt;[http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0073:entry=xalko/s Georg Autenrieth, A Homeric Dictionary]&lt;/ref&gt;

Many new chalcidoid families were proposed in 2022.&lt;ref name=Burks2022&gt;Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. [https://doi.org/10.3897/jhr.94.94263 DOI]&lt;/ref&gt; This was largely based on phylogenetic evidence (both molecular and morphological) accumulated since the families of the Superfamily were outlined in 1993&lt;ref name=Gibson1993&gt;Gibson, G.A.P. (1993) Superfamilies Mymarommatoidea and Chalcidoidea (pp. 570-655). In Goulet, H. &amp; Huber, J. (eds). Hymenoptera of the World: an identification guide to families. Research Branch, Agriculture Canada, Ottawa, Canada, 668 pp. [https://www.researchgate.net/publication/259227143 PDF]&lt;/ref&gt;.


Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. https://doi.org/10.3897/jhr.94.94263

Cruaud, A., Rasplus, J.Y., Zhang, J., Burks, R., Delvare, G., Fusu, L., Gumovsky, A., Huber, J.T., Janšta, P., Mitroiu, M.D. and Noyes, J.S., 2022. The Chalcidoidea bush of life–a massive radiation blurred by mutational saturation. bioRxiv. https://doi.org/10.1101/2022.09.11.507458 https://www.biorxiv.org/content/10.1101/2022.09.11.507458.full.pdf


= Superfamily Ichneumonoidea =

Quicke, D.L., 2015. The braconid and ichneumonid parasitoid wasps: biology, systematics, evolution and ecology. John Wiley &amp; Sons.
https://www.researchgate.net/publication/281662102_The_Braconid_and_Ichneumonid_Parasitoid_Wasps_Biology_Systematics_Evolution_and_Ecology

=== Diplazontinae ===
Quote from Fitton &amp; Rotheray (1982):
&quot;Diplazontines are distinguished easily from other ichneumonids by their general appearance (Fig. 1). The main confirmatory characters are the bifid upper tooth of the mandible (Fig. 2) and the sub-rectangular shape of the first tergite of the gaster (Figs. 1,3,4, 13, 15 and 17). In fact, the diplazontines are so distinct that they can be separated from all other Hymenoptera-Apocrita by possession of the following five characters in combination: (1) antenna of seventeen or more segments (Fig. I); (2) forewing with costal cell obliterated (that is, veins C and Sc + R + Rs contiguous from the wing base to the base of the pterostigma (Fig. 6 ) ; (3) forewing with cross-vein 2 mcu present (Fig. 6); (4) mandible with its upper tooth subdivided (bifid), making the mandible tridentate (Fig. 2); (5) first tergite of gaster sub-rectangular in shape (that is, subequal in width anteriorly and posteriorly) (Figs. 1 , 3 , 4 , 13,15 and 17).&quot;

Fitton, M. G., &amp; Rotheray, G. E. (1982). A key to the European genera of diplazontine ichneumon‐flies, with notes on the British fauna. Systematic Entomology, 7(3), 311-320.

=== Superfamily Pompiloidea ===
Family Mutillidae (Velvet ants)
Family Pompilidae (Spider hunting wasps)
Family Sapygidae (Sapygid wasps)


== Spider-hunting wasps ==
Two wasps from the genus ''Auplopus'' from Europe. Most of the ''Auplopus'' species, along with others from the Tribe Ageniellini, amputate the legs of the spiders they capture; this makes it much easier to move the spider to the nest.
&lt;gallery mode=packed heights=200&gt;
Auplopus carbonarius IMG 1624.jpg
Auplopus carbonarius fg01 20060623 Nied Garten.jpg
&lt;/gallery&gt;
African genera of Ageniellini:
* ''[[Auplopus]]'' &lt;small&gt;Spinola, 1841&lt;/small&gt;
* ''[[Cyemagenia]]'' &lt;small&gt;Arnold, 1946&lt;/small&gt;
* ''[[Dichragenia]]'' &lt;small&gt;Haupt, 1950&lt;/small&gt;
* ''[[Phanagenia]]'' &lt;small&gt;Banks, 1933&lt;/small&gt;   Madagascar
* ''[[Poecilagenia]]'' &lt;small&gt;Haupt, 1926&lt;/small&gt;

== Superfamily Scolioidea ==
=== Family Bradynobaenidae (Bradynobaenid wasps) ===
=== Family Scoliidae (Mammoth wasps) ===

== Superfamily Tiphioidea ==
Family Tiphiidae (Tiphiid wasps)
== Superfamily Thynnoidea ==
Family Thynnidae(Thynnid wasps)
== Superfamily Vespoidea ==
=== Family Rhopalosomatidae (Rhopalosomatid wasps) ===
===Family Vespidae (Paper, Potter &amp; Pollen wasps)===


== '''&lt;big&gt;Hymenoptera&lt;/big&gt;''' ==
About 20 000 described species of [[w:Hymenoptera|Hymenoptera]] (wasps, bees, ants and sawflies) are known from the [[w:Afrotropical|Afrotropical]] region. Estimates of the actual species count for the region range from 100 000 species to as high as 500 000 species.&lt;ref name=WaspWebOverview&gt;[https://www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm van Noort, Simon (2023). WaspWeb: Afrotropical Hymenoptera Initiative. www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm Retrieved 22 February 2023.]&lt;/ref&gt; Nineteen different superfamilies are illustrated on [[User:Alandmanson/Hymenoptera of southern Africa|this gallery]] in Wikimedia Commons.

The huge number of undescribed species means that many species will be extinct before we are even aware of them, as there are too few taxonomists employed to tackle the task of describing all of them in the next few decades.

==Classification==
The Order can be split into two Suborders - The [[w:Symphyta|Symphyta]] (Woodwasps, Horntails, Sawflies) and the [[w:Apocrita|Apocrita]] (Narrow-waisted wasps, ants and bees).&lt;br&gt;

When compared to the temperate regions of the northern hemisphere, the diversity of Symphyta in Africa is relatively poor, although sawflies of the Superfamily [[w:Tenthredinoidea|Tenthredinoidea]] are fairly common in forests and other woody vegetation types.

Africa has a rich diversity of Apocrita. The cladogram shown below&lt;ref name=Peters&gt;{{Cite journal |last1=Peters |first1=Ralph S. |last2=Krogmann |first2=Lars |last3=Mayer |first3=Christoph |last4=Donath |first4=Alexander |last5=Gunkel |first5=Simon |last6=Meusemann |first6=Karen |last7=Kozlov |first7=Alexey |last8=Podsiadlowski |first8=Lars |last9=Petersen |first9=Malte |title=Evolutionary History of the Hymenoptera |journal=Current Biology |volume=27 |issue=7 |pages=1013–1018 |doi=10.1016/j.cub.2017.01.027|pmid=28343967 |year=2017 |doi-access=free }}&lt;/ref&gt; indicates the possible relationships between 11 of the superfamilies that comprise Apocrita; These 11 superfamilies are all represented in Africa. This breakdown is used by iNaturalist.&lt;ref name=inat-Aculeata&gt;https://www.inaturalist.org/taxa/326777-Aculeata&lt;/ref&gt; It is, however, not accepted by all hymenopterists, and may change as more phylogenetic evidence is accumulated.&lt;ref name=waspwebClass&gt;[https://www.waspweb.org/Classification/index.htm van Noort, Simon (2023). WaspWeb: Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants). www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
{{Clade|style=font-size:85%; line-height:85%
|label1=Apocrita
            |1={{Clade
               |label1=[[w:Parasitoida|Parasitoida]]
               |1={{Clade
                  |1={{Clade
                     |1=[[w:Ceraphronoidea|Ceraphronoidea]]
                     |2=[[w:Ichneumonoidea|Ichneumonoidea]]
                     }}
                  |label2=[[w:Proctotrupomorpha|Proctotrupomorpha]]
                  |2={{Clade
                     |1=[[w:Cynipoidea|Cynipoidea]]
                     |2={{Clade
                        |1=[[w:Platygastroidea|Platygastroidea]]
                        |2={{Clade
                           |1=[[w:Chalcidoidea|Chalcidoidea]]
                           |2={{Clade
                              |1=[[w:Diaprioidea|Diaprioidea]]
                              |2=[[w:Proctotrupoidea|Proctotrupoidea]]
                  }} }} }} }} }}
               |2={{Clade
                  |1={{Clade
                     |1=[[w:Evanioidea|Evanioidea]]
                     |2=[[w:Stephanoidea|Stephanoidea]]
                     }}
                  |2={{Clade
                     |1=[[w:Trigonaloidea|Trigonaloidea]]
                     |label2=[[w:Aculeata|Aculeata]]
                     |2={{Clade
                        |1=[[w:Chrysidoidea|Chrysidoidea]]
                        |2={{Clade
                           |1=[[w:Vespoidea|Vespoidea]] (potter, honey and social wasps)
                           |2={{Clade
                              |1={{Clade
                                 |1=[[w:Pompiloidea|Pompiloidea]] (velvet ants, spider wasps and relatives)
                                 |2={{Clade
                                    |1=[[w:Thynnoidea|Thynnoidea]]
                                    |2=[[w:Tiphioidea|Tiphioidea]]
                                 }} }}
                              |2={{Clade
                                 |1=[[w:Scolioidea|Scolioidea]]
                                 |2={{Clade
                                    |1=[[w:Formicoidea|Formicoidea]] (ants)
                                    |2=[[w:Apoidea|Apoidea]] (bees and related wasps)
}} }} }} }} }} }} }} }} }}
&lt;br&gt;

==Some common African Symphyta==
&lt;gallery mode=packed heights=200&gt;
Sawfly, Argidae 2021 12 19 12 39 42.jpg|Sawfly, ''Arge'' sp. (Argidae, Superfamily Tenthredinoidea) 
Athalia 2019 11 21 0815.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Athalia 2019 08 23 8964.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Sirex noctilio F (prof.).JPG|''Sirex noctilio'', a pest in ''Pinus'' plantations (introduced to Africa)&lt;ref name=WaspWebSn&gt;[https://www.waspweb.org/Siricoidea/Siricidae/Sirex/Sirex_noctilio.htm van Noort, Simon (2023). WaspWeb: ''Sirex noctilio''. www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
&lt;/gallery&gt;

== Frequently reported African Apocrita ==
&lt;br&gt;
[[w:Aculeata|Aculeata]] (ants, bees and stinging wasps) are the most commonly observed Hymenoptera in Africa. There are many more photographs of African Aculeata on the web - See [https://www.waspweb.org/Classification/index.htm African Aculeata on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=326777 African Aculeata on iNaturalist] &lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Slender Ant (Tetraponera natalensis) (30538051244).jpg|Slender ant (Formicidae, Formicoidea)
Carpenter Bee (Xylocopa inconstans, female) (6021534858).jpg|Female carpenter bee (Apidae, Apoidea)
Chalybion 2019 12 02 2314.jpg|Blue mud-dauber wasp (Sphecidae, Apoidea)
Bethylidae 2019 08 21 13 42 21 8760.jpg|Flat wasp (Bethylidae, Chrysidoidea)
Belonogaster juncea colonialis, manlik, h, Pretoria.jpg|Paper wasp (Vespidae, Vespoidea)
Meria 2020 08 12 2193.jpg|''Meria'' sp. female (Thynnidae, Thynnoidea)
Dasylabris stimulatrix iNat37068107 2019 12 23 2990 W.jpg|''Dasylabris stimulatrix'', a velvet ant (Mutillidae, Pompiloidea)
Pompilidae 2019 05 28 0247.jpg|Spider-hunting wasp (Pompilidae, Pompiloidea)
Scoliid Wasp 2019 06 08 0492.jpg|Campsomerine wasp (Scoliidae, Scolioidea)
&lt;/gallery&gt;&lt;br&gt;

Darwin wasps ([[w:Ichneumonidae|Ichneumonidae]]) and braconids ([[w:Braconidae|Braconidae]]) are also frequently reported. Links to [https://www.waspweb.org/Ichneumonoidea/index.htm African Ichneumonoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=47200 African Ichneumonoidea on iNaturalist].&lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Echthromorpha agrestoria 2019 10 03 5572.jpg|''Echthromorpha agrestoria'' (Ichneumonidae, Ichneumonoidea)
Ichneumonidae inat 88832241.jpg|''Goryphus tricolor'' (Ichneumonidae, Ichneumonoidea)
Braconid Wasp (Archibracon servillei) (11857083186).jpg|Braconine wasp (Braconidae, Ichneumonoidea) 
2019 09 15 12 55 07 Microgastrinae 38425632.jpg|Microgastrine wasp (Braconidae, Ichneumonoidea) 
&lt;/gallery&gt;&lt;br&gt;

Small [[w:Chalcidoidea|chalcidoid]] parasitic wasps (especially those from the families [[w:Pteromalidae|Pteromalidae]], [[w:Chalcididae|Chalcididae]], [[w:Eulophidae|Eulophidae]], [[w:Eurytomidae|Eurytomidae]]) are also common. Links to [https://www.waspweb.org/Chalcidoidea/index.htm African Chalcidoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=69750 African Chalcidoidea on iNaturalist]
&lt;gallery mode=packed heights=200&gt;
Halticoptera 2019 11 26 2173Cedara.jpg|''Halticoptera'' sp. (Pteromalidae, Chalcidoidea)
Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp. (Chalcididae, Chalcidoidea)
Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp. (Eulophidae, Chalcidoidea)
Eurytomidae00.jpg|Mating seed chalcids (Eurytomidae, Chalcidoidea)
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>pym7mwvjtazl2mu6m5rwweam14rctww</sha1>
    </revision>
    <revision>
      <id>2815546</id>
      <parentid>2815545</parentid>
      <timestamp>2026-06-13T18:47:58Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* African Crabronidae */</comment>
      <origin>2815546</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16152" sha1="n8j4ifxzbpreigc9rd91yktxwu0hq3o" xml:space="preserve">= Superfamily Apoidea =
== African Ampulicidae ==
&lt;gallery mode=packed heights=150&gt;
Ampulicidae 37894270 suncana.jpg
Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg
&lt;/gallery&gt;		
== African Astatidae ==
&lt;gallery mode=packed heights=150&gt;
Astata iN 105162782 Nicola van Berkel.jpg
Astata melanaria.jpg
&lt;/gallery&gt;

== African Bembicidae ==
===Subfamily Bembicinae===	
&lt;gallery mode=packed heights=150&gt;
Bembecinus iN 153052948.jpg|''Bembecinus'' sp.
Bembix iN 195581985.jpg|''Bembix'' sp.
Hoplisoides thalia iN 268375748 a.jpg|''Hoplisoides thalia''
&lt;/gallery&gt;		
Tribe Gorytini	
*		''Afrogorytes''
*		''Gorytes''
*		''Harpactus''
*		''Hoplisoides''
*		''Lestiphorus''
		
Tribe Spheciini	
*		''Ammatomus''
*		''Kohlia''
*		''Sphecius''
		
Tribe Handlirschiini	
*		''Handlirschia''
		
Tribe Bembicini	
*		''Bembix''
		
Tribe Stizini	
*		''Bembecinus''
*		''Stizoides''
*		''Stizus''

===Subfamily Nyssoninae===	
*		''Brachystegus''
*		''Hovanysson''
*		''Nysson''
		
===Subfamily Alyssontinae===	
*		''Alysson''
*		''Didineis''

== African Crabronidae ==
&lt;gallery mode=packed heights=150&gt;
Dasyproctus iN 30029277 a.jpg
Dicranorhina kohli 2023 07 25 iN 176115975.jpg
Liris on Crassula iN 42678436 01.jpg
Oxybelus iN 250449990 2024 10 09 - 02.jpg
Palarus Bee Pirate iN 144133368 2022-12-01 01.jpg
Paranysson iN 199673489 1.jpg
Pison iN 144131685 2022-11-30 03.jpg
Tachysphex iN 250449986 2024 10 09 7305.jpg
Tachytes iN 188902572 1964.jpg
Trypoxylon iN 99063113 a.jpg
&lt;/gallery&gt;

== African Pemphredonidae ==
&lt;gallery mode=packed heights=150&gt;
Polemistus braunsii iNaturalist 228280708.jpg

&lt;/gallery&gt;

== African Philanthidae ==
&lt;gallery mode=packed heights=150&gt;
Philanthus triangulum diadema 187037342.jpg
Cerceris 2019 12 02 2310.jpg
&lt;/gallery&gt;

== African Psenidae ==
&lt;gallery mode=packed heights=150&gt;
Psenini iN 1022563 i c riddell.jpg

&lt;/gallery&gt;

== African Sphecidae ==
&lt;gallery mode=packed heights=150&gt;


&lt;/gallery&gt;

= Superfamily Chalcidoidea =
 
Chalcidoidea is an incredibly diverse group of wasps with a wide variety of life histories; many are parasitoids, making them ecologically important; some are used for biological control of insect pests. About 22 000 species have been described, but as many as 500 000 are thought to exist worldwide.&lt;ref name=Noyes2019/&gt;

Most chalcidoids are small (body length less than 5 mm), the smallest being males of the mymarid ''Dicopomorpha echmepterygis'', which is the smallest known flying insect (0.13 mm).&lt;ref name=Noyes2019&gt;Noyes, J.S. 2019. Universal Chalcidoidea Database. [http://www.nhm.ac.uk/chalcidoids]&lt;/ref&gt; However, some species may be as long as 20 mm (excluding antennae and ovipositor), such as the pelecinellid ''Doddifoenus wallacei''.&lt;ref name=Krogmann&amp;Burks2009&gt;Krogmann, Lars; Burks, Roger A. (2009-12-31). &quot;Doddifoenus wallacei, a new giant parasitoid wasp of the subfamily Leptofoeninae (Chalcidoidea: Pteromalidae), with a description of its mesosomal skeletal anatomy and a molecular characterization&quot;. Zootaxa. 2194: 21–36. doi:10.5281/zenodo.189452 [https://www.researchgate.net/profile/Lars-Krogmann/publication/250916337 PDF]&lt;/ref&gt; Many have bright metallic colours, hence the name Chalcidoidea or chalcid - the greek word χαλκός (chalcos) means copper or bronze.&lt;ref name=Perseus2023&gt;[http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0073:entry=xalko/s Georg Autenrieth, A Homeric Dictionary]&lt;/ref&gt;

Many new chalcidoid families were proposed in 2022.&lt;ref name=Burks2022&gt;Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. [https://doi.org/10.3897/jhr.94.94263 DOI]&lt;/ref&gt; This was largely based on phylogenetic evidence (both molecular and morphological) accumulated since the families of the Superfamily were outlined in 1993&lt;ref name=Gibson1993&gt;Gibson, G.A.P. (1993) Superfamilies Mymarommatoidea and Chalcidoidea (pp. 570-655). In Goulet, H. &amp; Huber, J. (eds). Hymenoptera of the World: an identification guide to families. Research Branch, Agriculture Canada, Ottawa, Canada, 668 pp. [https://www.researchgate.net/publication/259227143 PDF]&lt;/ref&gt;.


Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. https://doi.org/10.3897/jhr.94.94263

Cruaud, A., Rasplus, J.Y., Zhang, J., Burks, R., Delvare, G., Fusu, L., Gumovsky, A., Huber, J.T., Janšta, P., Mitroiu, M.D. and Noyes, J.S., 2022. The Chalcidoidea bush of life–a massive radiation blurred by mutational saturation. bioRxiv. https://doi.org/10.1101/2022.09.11.507458 https://www.biorxiv.org/content/10.1101/2022.09.11.507458.full.pdf


= Superfamily Ichneumonoidea =

Quicke, D.L., 2015. The braconid and ichneumonid parasitoid wasps: biology, systematics, evolution and ecology. John Wiley &amp; Sons.
https://www.researchgate.net/publication/281662102_The_Braconid_and_Ichneumonid_Parasitoid_Wasps_Biology_Systematics_Evolution_and_Ecology

=== Diplazontinae ===
Quote from Fitton &amp; Rotheray (1982):
&quot;Diplazontines are distinguished easily from other ichneumonids by their general appearance (Fig. 1). The main confirmatory characters are the bifid upper tooth of the mandible (Fig. 2) and the sub-rectangular shape of the first tergite of the gaster (Figs. 1,3,4, 13, 15 and 17). In fact, the diplazontines are so distinct that they can be separated from all other Hymenoptera-Apocrita by possession of the following five characters in combination: (1) antenna of seventeen or more segments (Fig. I); (2) forewing with costal cell obliterated (that is, veins C and Sc + R + Rs contiguous from the wing base to the base of the pterostigma (Fig. 6 ) ; (3) forewing with cross-vein 2 mcu present (Fig. 6); (4) mandible with its upper tooth subdivided (bifid), making the mandible tridentate (Fig. 2); (5) first tergite of gaster sub-rectangular in shape (that is, subequal in width anteriorly and posteriorly) (Figs. 1 , 3 , 4 , 13,15 and 17).&quot;

Fitton, M. G., &amp; Rotheray, G. E. (1982). A key to the European genera of diplazontine ichneumon‐flies, with notes on the British fauna. Systematic Entomology, 7(3), 311-320.

=== Superfamily Pompiloidea ===
Family Mutillidae (Velvet ants)
Family Pompilidae (Spider hunting wasps)
Family Sapygidae (Sapygid wasps)


== Spider-hunting wasps ==
Two wasps from the genus ''Auplopus'' from Europe. Most of the ''Auplopus'' species, along with others from the Tribe Ageniellini, amputate the legs of the spiders they capture; this makes it much easier to move the spider to the nest.
&lt;gallery mode=packed heights=200&gt;
Auplopus carbonarius IMG 1624.jpg
Auplopus carbonarius fg01 20060623 Nied Garten.jpg
&lt;/gallery&gt;
African genera of Ageniellini:
* ''[[Auplopus]]'' &lt;small&gt;Spinola, 1841&lt;/small&gt;
* ''[[Cyemagenia]]'' &lt;small&gt;Arnold, 1946&lt;/small&gt;
* ''[[Dichragenia]]'' &lt;small&gt;Haupt, 1950&lt;/small&gt;
* ''[[Phanagenia]]'' &lt;small&gt;Banks, 1933&lt;/small&gt;   Madagascar
* ''[[Poecilagenia]]'' &lt;small&gt;Haupt, 1926&lt;/small&gt;

== Superfamily Scolioidea ==
=== Family Bradynobaenidae (Bradynobaenid wasps) ===
=== Family Scoliidae (Mammoth wasps) ===

== Superfamily Tiphioidea ==
Family Tiphiidae (Tiphiid wasps)
== Superfamily Thynnoidea ==
Family Thynnidae(Thynnid wasps)
== Superfamily Vespoidea ==
=== Family Rhopalosomatidae (Rhopalosomatid wasps) ===
===Family Vespidae (Paper, Potter &amp; Pollen wasps)===


== '''&lt;big&gt;Hymenoptera&lt;/big&gt;''' ==
About 20 000 described species of [[w:Hymenoptera|Hymenoptera]] (wasps, bees, ants and sawflies) are known from the [[w:Afrotropical|Afrotropical]] region. Estimates of the actual species count for the region range from 100 000 species to as high as 500 000 species.&lt;ref name=WaspWebOverview&gt;[https://www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm van Noort, Simon (2023). WaspWeb: Afrotropical Hymenoptera Initiative. www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm Retrieved 22 February 2023.]&lt;/ref&gt; Nineteen different superfamilies are illustrated on [[User:Alandmanson/Hymenoptera of southern Africa|this gallery]] in Wikimedia Commons.

The huge number of undescribed species means that many species will be extinct before we are even aware of them, as there are too few taxonomists employed to tackle the task of describing all of them in the next few decades.

==Classification==
The Order can be split into two Suborders - The [[w:Symphyta|Symphyta]] (Woodwasps, Horntails, Sawflies) and the [[w:Apocrita|Apocrita]] (Narrow-waisted wasps, ants and bees).&lt;br&gt;

When compared to the temperate regions of the northern hemisphere, the diversity of Symphyta in Africa is relatively poor, although sawflies of the Superfamily [[w:Tenthredinoidea|Tenthredinoidea]] are fairly common in forests and other woody vegetation types.

Africa has a rich diversity of Apocrita. The cladogram shown below&lt;ref name=Peters&gt;{{Cite journal |last1=Peters |first1=Ralph S. |last2=Krogmann |first2=Lars |last3=Mayer |first3=Christoph |last4=Donath |first4=Alexander |last5=Gunkel |first5=Simon |last6=Meusemann |first6=Karen |last7=Kozlov |first7=Alexey |last8=Podsiadlowski |first8=Lars |last9=Petersen |first9=Malte |title=Evolutionary History of the Hymenoptera |journal=Current Biology |volume=27 |issue=7 |pages=1013–1018 |doi=10.1016/j.cub.2017.01.027|pmid=28343967 |year=2017 |doi-access=free }}&lt;/ref&gt; indicates the possible relationships between 11 of the superfamilies that comprise Apocrita; These 11 superfamilies are all represented in Africa. This breakdown is used by iNaturalist.&lt;ref name=inat-Aculeata&gt;https://www.inaturalist.org/taxa/326777-Aculeata&lt;/ref&gt; It is, however, not accepted by all hymenopterists, and may change as more phylogenetic evidence is accumulated.&lt;ref name=waspwebClass&gt;[https://www.waspweb.org/Classification/index.htm van Noort, Simon (2023). WaspWeb: Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants). www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
{{Clade|style=font-size:85%; line-height:85%
|label1=Apocrita
            |1={{Clade
               |label1=[[w:Parasitoida|Parasitoida]]
               |1={{Clade
                  |1={{Clade
                     |1=[[w:Ceraphronoidea|Ceraphronoidea]]
                     |2=[[w:Ichneumonoidea|Ichneumonoidea]]
                     }}
                  |label2=[[w:Proctotrupomorpha|Proctotrupomorpha]]
                  |2={{Clade
                     |1=[[w:Cynipoidea|Cynipoidea]]
                     |2={{Clade
                        |1=[[w:Platygastroidea|Platygastroidea]]
                        |2={{Clade
                           |1=[[w:Chalcidoidea|Chalcidoidea]]
                           |2={{Clade
                              |1=[[w:Diaprioidea|Diaprioidea]]
                              |2=[[w:Proctotrupoidea|Proctotrupoidea]]
                  }} }} }} }} }}
               |2={{Clade
                  |1={{Clade
                     |1=[[w:Evanioidea|Evanioidea]]
                     |2=[[w:Stephanoidea|Stephanoidea]]
                     }}
                  |2={{Clade
                     |1=[[w:Trigonaloidea|Trigonaloidea]]
                     |label2=[[w:Aculeata|Aculeata]]
                     |2={{Clade
                        |1=[[w:Chrysidoidea|Chrysidoidea]]
                        |2={{Clade
                           |1=[[w:Vespoidea|Vespoidea]] (potter, honey and social wasps)
                           |2={{Clade
                              |1={{Clade
                                 |1=[[w:Pompiloidea|Pompiloidea]] (velvet ants, spider wasps and relatives)
                                 |2={{Clade
                                    |1=[[w:Thynnoidea|Thynnoidea]]
                                    |2=[[w:Tiphioidea|Tiphioidea]]
                                 }} }}
                              |2={{Clade
                                 |1=[[w:Scolioidea|Scolioidea]]
                                 |2={{Clade
                                    |1=[[w:Formicoidea|Formicoidea]] (ants)
                                    |2=[[w:Apoidea|Apoidea]] (bees and related wasps)
}} }} }} }} }} }} }} }} }}
&lt;br&gt;

==Some common African Symphyta==
&lt;gallery mode=packed heights=200&gt;
Sawfly, Argidae 2021 12 19 12 39 42.jpg|Sawfly, ''Arge'' sp. (Argidae, Superfamily Tenthredinoidea) 
Athalia 2019 11 21 0815.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Athalia 2019 08 23 8964.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Sirex noctilio F (prof.).JPG|''Sirex noctilio'', a pest in ''Pinus'' plantations (introduced to Africa)&lt;ref name=WaspWebSn&gt;[https://www.waspweb.org/Siricoidea/Siricidae/Sirex/Sirex_noctilio.htm van Noort, Simon (2023). WaspWeb: ''Sirex noctilio''. www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
&lt;/gallery&gt;

== Frequently reported African Apocrita ==
&lt;br&gt;
[[w:Aculeata|Aculeata]] (ants, bees and stinging wasps) are the most commonly observed Hymenoptera in Africa. There are many more photographs of African Aculeata on the web - See [https://www.waspweb.org/Classification/index.htm African Aculeata on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=326777 African Aculeata on iNaturalist] &lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Slender Ant (Tetraponera natalensis) (30538051244).jpg|Slender ant (Formicidae, Formicoidea)
Carpenter Bee (Xylocopa inconstans, female) (6021534858).jpg|Female carpenter bee (Apidae, Apoidea)
Chalybion 2019 12 02 2314.jpg|Blue mud-dauber wasp (Sphecidae, Apoidea)
Bethylidae 2019 08 21 13 42 21 8760.jpg|Flat wasp (Bethylidae, Chrysidoidea)
Belonogaster juncea colonialis, manlik, h, Pretoria.jpg|Paper wasp (Vespidae, Vespoidea)
Meria 2020 08 12 2193.jpg|''Meria'' sp. female (Thynnidae, Thynnoidea)
Dasylabris stimulatrix iNat37068107 2019 12 23 2990 W.jpg|''Dasylabris stimulatrix'', a velvet ant (Mutillidae, Pompiloidea)
Pompilidae 2019 05 28 0247.jpg|Spider-hunting wasp (Pompilidae, Pompiloidea)
Scoliid Wasp 2019 06 08 0492.jpg|Campsomerine wasp (Scoliidae, Scolioidea)
&lt;/gallery&gt;&lt;br&gt;

Darwin wasps ([[w:Ichneumonidae|Ichneumonidae]]) and braconids ([[w:Braconidae|Braconidae]]) are also frequently reported. Links to [https://www.waspweb.org/Ichneumonoidea/index.htm African Ichneumonoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=47200 African Ichneumonoidea on iNaturalist].&lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Echthromorpha agrestoria 2019 10 03 5572.jpg|''Echthromorpha agrestoria'' (Ichneumonidae, Ichneumonoidea)
Ichneumonidae inat 88832241.jpg|''Goryphus tricolor'' (Ichneumonidae, Ichneumonoidea)
Braconid Wasp (Archibracon servillei) (11857083186).jpg|Braconine wasp (Braconidae, Ichneumonoidea) 
2019 09 15 12 55 07 Microgastrinae 38425632.jpg|Microgastrine wasp (Braconidae, Ichneumonoidea) 
&lt;/gallery&gt;&lt;br&gt;

Small [[w:Chalcidoidea|chalcidoid]] parasitic wasps (especially those from the families [[w:Pteromalidae|Pteromalidae]], [[w:Chalcididae|Chalcididae]], [[w:Eulophidae|Eulophidae]], [[w:Eurytomidae|Eurytomidae]]) are also common. Links to [https://www.waspweb.org/Chalcidoidea/index.htm African Chalcidoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=69750 African Chalcidoidea on iNaturalist]
&lt;gallery mode=packed heights=200&gt;
Halticoptera 2019 11 26 2173Cedara.jpg|''Halticoptera'' sp. (Pteromalidae, Chalcidoidea)
Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp. (Chalcididae, Chalcidoidea)
Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp. (Eulophidae, Chalcidoidea)
Eurytomidae00.jpg|Mating seed chalcids (Eurytomidae, Chalcidoidea)
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>n8j4ifxzbpreigc9rd91yktxwu0hq3o</sha1>
    </revision>
    <revision>
      <id>2815553</id>
      <parentid>2815546</parentid>
      <timestamp>2026-06-13T18:53:55Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* African Sphecidae */</comment>
      <origin>2815553</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16443" sha1="tqvi5ak4pia6acqmck1q76yehaasi7w" xml:space="preserve">= Superfamily Apoidea =
== African Ampulicidae ==
&lt;gallery mode=packed heights=150&gt;
Ampulicidae 37894270 suncana.jpg
Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg
&lt;/gallery&gt;		
== African Astatidae ==
&lt;gallery mode=packed heights=150&gt;
Astata iN 105162782 Nicola van Berkel.jpg
Astata melanaria.jpg
&lt;/gallery&gt;

== African Bembicidae ==
===Subfamily Bembicinae===	
&lt;gallery mode=packed heights=150&gt;
Bembecinus iN 153052948.jpg|''Bembecinus'' sp.
Bembix iN 195581985.jpg|''Bembix'' sp.
Hoplisoides thalia iN 268375748 a.jpg|''Hoplisoides thalia''
&lt;/gallery&gt;		
Tribe Gorytini	
*		''Afrogorytes''
*		''Gorytes''
*		''Harpactus''
*		''Hoplisoides''
*		''Lestiphorus''
		
Tribe Spheciini	
*		''Ammatomus''
*		''Kohlia''
*		''Sphecius''
		
Tribe Handlirschiini	
*		''Handlirschia''
		
Tribe Bembicini	
*		''Bembix''
		
Tribe Stizini	
*		''Bembecinus''
*		''Stizoides''
*		''Stizus''

===Subfamily Nyssoninae===	
*		''Brachystegus''
*		''Hovanysson''
*		''Nysson''
		
===Subfamily Alyssontinae===	
*		''Alysson''
*		''Didineis''

== African Crabronidae ==
&lt;gallery mode=packed heights=150&gt;
Dasyproctus iN 30029277 a.jpg
Dicranorhina kohli 2023 07 25 iN 176115975.jpg
Liris on Crassula iN 42678436 01.jpg
Oxybelus iN 250449990 2024 10 09 - 02.jpg
Palarus Bee Pirate iN 144133368 2022-12-01 01.jpg
Paranysson iN 199673489 1.jpg
Pison iN 144131685 2022-11-30 03.jpg
Tachysphex iN 250449986 2024 10 09 7305.jpg
Tachytes iN 188902572 1964.jpg
Trypoxylon iN 99063113 a.jpg
&lt;/gallery&gt;

== African Pemphredonidae ==
&lt;gallery mode=packed heights=150&gt;
Polemistus braunsii iNaturalist 228280708.jpg

&lt;/gallery&gt;

== African Philanthidae ==
&lt;gallery mode=packed heights=150&gt;
Philanthus triangulum diadema 187037342.jpg
Cerceris 2019 12 02 2310.jpg
&lt;/gallery&gt;

== African Psenidae ==
&lt;gallery mode=packed heights=150&gt;
Psenini iN 1022563 i c riddell.jpg

&lt;/gallery&gt;

== African Sphecidae ==
&lt;gallery mode=packed heights=150&gt;
Chalybion 2019 12 02 2314.jpg
Thread-waisted Sand Wasp (Ammophila sp.) carrying a Lienard's Achaea (Achaea lienardi) caterpillar ... (52715218455).jpg
Ammophila ferrugineipes04.jpg
Black Mud-dauber Wasp (Sceliphron spirifex) on Buffalo-Thorn (Ziziphus mucronata) flowers ... (52739846889).jpg
&lt;/gallery&gt;

= Superfamily Chalcidoidea =
 
Chalcidoidea is an incredibly diverse group of wasps with a wide variety of life histories; many are parasitoids, making them ecologically important; some are used for biological control of insect pests. About 22 000 species have been described, but as many as 500 000 are thought to exist worldwide.&lt;ref name=Noyes2019/&gt;

Most chalcidoids are small (body length less than 5 mm), the smallest being males of the mymarid ''Dicopomorpha echmepterygis'', which is the smallest known flying insect (0.13 mm).&lt;ref name=Noyes2019&gt;Noyes, J.S. 2019. Universal Chalcidoidea Database. [http://www.nhm.ac.uk/chalcidoids]&lt;/ref&gt; However, some species may be as long as 20 mm (excluding antennae and ovipositor), such as the pelecinellid ''Doddifoenus wallacei''.&lt;ref name=Krogmann&amp;Burks2009&gt;Krogmann, Lars; Burks, Roger A. (2009-12-31). &quot;Doddifoenus wallacei, a new giant parasitoid wasp of the subfamily Leptofoeninae (Chalcidoidea: Pteromalidae), with a description of its mesosomal skeletal anatomy and a molecular characterization&quot;. Zootaxa. 2194: 21–36. doi:10.5281/zenodo.189452 [https://www.researchgate.net/profile/Lars-Krogmann/publication/250916337 PDF]&lt;/ref&gt; Many have bright metallic colours, hence the name Chalcidoidea or chalcid - the greek word χαλκός (chalcos) means copper or bronze.&lt;ref name=Perseus2023&gt;[http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0073:entry=xalko/s Georg Autenrieth, A Homeric Dictionary]&lt;/ref&gt;

Many new chalcidoid families were proposed in 2022.&lt;ref name=Burks2022&gt;Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. [https://doi.org/10.3897/jhr.94.94263 DOI]&lt;/ref&gt; This was largely based on phylogenetic evidence (both molecular and morphological) accumulated since the families of the Superfamily were outlined in 1993&lt;ref name=Gibson1993&gt;Gibson, G.A.P. (1993) Superfamilies Mymarommatoidea and Chalcidoidea (pp. 570-655). In Goulet, H. &amp; Huber, J. (eds). Hymenoptera of the World: an identification guide to families. Research Branch, Agriculture Canada, Ottawa, Canada, 668 pp. [https://www.researchgate.net/publication/259227143 PDF]&lt;/ref&gt;.


Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. https://doi.org/10.3897/jhr.94.94263

Cruaud, A., Rasplus, J.Y., Zhang, J., Burks, R., Delvare, G., Fusu, L., Gumovsky, A., Huber, J.T., Janšta, P., Mitroiu, M.D. and Noyes, J.S., 2022. The Chalcidoidea bush of life–a massive radiation blurred by mutational saturation. bioRxiv. https://doi.org/10.1101/2022.09.11.507458 https://www.biorxiv.org/content/10.1101/2022.09.11.507458.full.pdf


= Superfamily Ichneumonoidea =

Quicke, D.L., 2015. The braconid and ichneumonid parasitoid wasps: biology, systematics, evolution and ecology. John Wiley &amp; Sons.
https://www.researchgate.net/publication/281662102_The_Braconid_and_Ichneumonid_Parasitoid_Wasps_Biology_Systematics_Evolution_and_Ecology

=== Diplazontinae ===
Quote from Fitton &amp; Rotheray (1982):
&quot;Diplazontines are distinguished easily from other ichneumonids by their general appearance (Fig. 1). The main confirmatory characters are the bifid upper tooth of the mandible (Fig. 2) and the sub-rectangular shape of the first tergite of the gaster (Figs. 1,3,4, 13, 15 and 17). In fact, the diplazontines are so distinct that they can be separated from all other Hymenoptera-Apocrita by possession of the following five characters in combination: (1) antenna of seventeen or more segments (Fig. I); (2) forewing with costal cell obliterated (that is, veins C and Sc + R + Rs contiguous from the wing base to the base of the pterostigma (Fig. 6 ) ; (3) forewing with cross-vein 2 mcu present (Fig. 6); (4) mandible with its upper tooth subdivided (bifid), making the mandible tridentate (Fig. 2); (5) first tergite of gaster sub-rectangular in shape (that is, subequal in width anteriorly and posteriorly) (Figs. 1 , 3 , 4 , 13,15 and 17).&quot;

Fitton, M. G., &amp; Rotheray, G. E. (1982). A key to the European genera of diplazontine ichneumon‐flies, with notes on the British fauna. Systematic Entomology, 7(3), 311-320.

=== Superfamily Pompiloidea ===
Family Mutillidae (Velvet ants)
Family Pompilidae (Spider hunting wasps)
Family Sapygidae (Sapygid wasps)


== Spider-hunting wasps ==
Two wasps from the genus ''Auplopus'' from Europe. Most of the ''Auplopus'' species, along with others from the Tribe Ageniellini, amputate the legs of the spiders they capture; this makes it much easier to move the spider to the nest.
&lt;gallery mode=packed heights=200&gt;
Auplopus carbonarius IMG 1624.jpg
Auplopus carbonarius fg01 20060623 Nied Garten.jpg
&lt;/gallery&gt;
African genera of Ageniellini:
* ''[[Auplopus]]'' &lt;small&gt;Spinola, 1841&lt;/small&gt;
* ''[[Cyemagenia]]'' &lt;small&gt;Arnold, 1946&lt;/small&gt;
* ''[[Dichragenia]]'' &lt;small&gt;Haupt, 1950&lt;/small&gt;
* ''[[Phanagenia]]'' &lt;small&gt;Banks, 1933&lt;/small&gt;   Madagascar
* ''[[Poecilagenia]]'' &lt;small&gt;Haupt, 1926&lt;/small&gt;

== Superfamily Scolioidea ==
=== Family Bradynobaenidae (Bradynobaenid wasps) ===
=== Family Scoliidae (Mammoth wasps) ===

== Superfamily Tiphioidea ==
Family Tiphiidae (Tiphiid wasps)
== Superfamily Thynnoidea ==
Family Thynnidae(Thynnid wasps)
== Superfamily Vespoidea ==
=== Family Rhopalosomatidae (Rhopalosomatid wasps) ===
===Family Vespidae (Paper, Potter &amp; Pollen wasps)===


== '''&lt;big&gt;Hymenoptera&lt;/big&gt;''' ==
About 20 000 described species of [[w:Hymenoptera|Hymenoptera]] (wasps, bees, ants and sawflies) are known from the [[w:Afrotropical|Afrotropical]] region. Estimates of the actual species count for the region range from 100 000 species to as high as 500 000 species.&lt;ref name=WaspWebOverview&gt;[https://www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm van Noort, Simon (2023). WaspWeb: Afrotropical Hymenoptera Initiative. www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm Retrieved 22 February 2023.]&lt;/ref&gt; Nineteen different superfamilies are illustrated on [[User:Alandmanson/Hymenoptera of southern Africa|this gallery]] in Wikimedia Commons.

The huge number of undescribed species means that many species will be extinct before we are even aware of them, as there are too few taxonomists employed to tackle the task of describing all of them in the next few decades.

==Classification==
The Order can be split into two Suborders - The [[w:Symphyta|Symphyta]] (Woodwasps, Horntails, Sawflies) and the [[w:Apocrita|Apocrita]] (Narrow-waisted wasps, ants and bees).&lt;br&gt;

When compared to the temperate regions of the northern hemisphere, the diversity of Symphyta in Africa is relatively poor, although sawflies of the Superfamily [[w:Tenthredinoidea|Tenthredinoidea]] are fairly common in forests and other woody vegetation types.

Africa has a rich diversity of Apocrita. The cladogram shown below&lt;ref name=Peters&gt;{{Cite journal |last1=Peters |first1=Ralph S. |last2=Krogmann |first2=Lars |last3=Mayer |first3=Christoph |last4=Donath |first4=Alexander |last5=Gunkel |first5=Simon |last6=Meusemann |first6=Karen |last7=Kozlov |first7=Alexey |last8=Podsiadlowski |first8=Lars |last9=Petersen |first9=Malte |title=Evolutionary History of the Hymenoptera |journal=Current Biology |volume=27 |issue=7 |pages=1013–1018 |doi=10.1016/j.cub.2017.01.027|pmid=28343967 |year=2017 |doi-access=free }}&lt;/ref&gt; indicates the possible relationships between 11 of the superfamilies that comprise Apocrita; These 11 superfamilies are all represented in Africa. This breakdown is used by iNaturalist.&lt;ref name=inat-Aculeata&gt;https://www.inaturalist.org/taxa/326777-Aculeata&lt;/ref&gt; It is, however, not accepted by all hymenopterists, and may change as more phylogenetic evidence is accumulated.&lt;ref name=waspwebClass&gt;[https://www.waspweb.org/Classification/index.htm van Noort, Simon (2023). WaspWeb: Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants). www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
{{Clade|style=font-size:85%; line-height:85%
|label1=Apocrita
            |1={{Clade
               |label1=[[w:Parasitoida|Parasitoida]]
               |1={{Clade
                  |1={{Clade
                     |1=[[w:Ceraphronoidea|Ceraphronoidea]]
                     |2=[[w:Ichneumonoidea|Ichneumonoidea]]
                     }}
                  |label2=[[w:Proctotrupomorpha|Proctotrupomorpha]]
                  |2={{Clade
                     |1=[[w:Cynipoidea|Cynipoidea]]
                     |2={{Clade
                        |1=[[w:Platygastroidea|Platygastroidea]]
                        |2={{Clade
                           |1=[[w:Chalcidoidea|Chalcidoidea]]
                           |2={{Clade
                              |1=[[w:Diaprioidea|Diaprioidea]]
                              |2=[[w:Proctotrupoidea|Proctotrupoidea]]
                  }} }} }} }} }}
               |2={{Clade
                  |1={{Clade
                     |1=[[w:Evanioidea|Evanioidea]]
                     |2=[[w:Stephanoidea|Stephanoidea]]
                     }}
                  |2={{Clade
                     |1=[[w:Trigonaloidea|Trigonaloidea]]
                     |label2=[[w:Aculeata|Aculeata]]
                     |2={{Clade
                        |1=[[w:Chrysidoidea|Chrysidoidea]]
                        |2={{Clade
                           |1=[[w:Vespoidea|Vespoidea]] (potter, honey and social wasps)
                           |2={{Clade
                              |1={{Clade
                                 |1=[[w:Pompiloidea|Pompiloidea]] (velvet ants, spider wasps and relatives)
                                 |2={{Clade
                                    |1=[[w:Thynnoidea|Thynnoidea]]
                                    |2=[[w:Tiphioidea|Tiphioidea]]
                                 }} }}
                              |2={{Clade
                                 |1=[[w:Scolioidea|Scolioidea]]
                                 |2={{Clade
                                    |1=[[w:Formicoidea|Formicoidea]] (ants)
                                    |2=[[w:Apoidea|Apoidea]] (bees and related wasps)
}} }} }} }} }} }} }} }} }}
&lt;br&gt;

==Some common African Symphyta==
&lt;gallery mode=packed heights=200&gt;
Sawfly, Argidae 2021 12 19 12 39 42.jpg|Sawfly, ''Arge'' sp. (Argidae, Superfamily Tenthredinoidea) 
Athalia 2019 11 21 0815.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Athalia 2019 08 23 8964.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Sirex noctilio F (prof.).JPG|''Sirex noctilio'', a pest in ''Pinus'' plantations (introduced to Africa)&lt;ref name=WaspWebSn&gt;[https://www.waspweb.org/Siricoidea/Siricidae/Sirex/Sirex_noctilio.htm van Noort, Simon (2023). WaspWeb: ''Sirex noctilio''. www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
&lt;/gallery&gt;

== Frequently reported African Apocrita ==
&lt;br&gt;
[[w:Aculeata|Aculeata]] (ants, bees and stinging wasps) are the most commonly observed Hymenoptera in Africa. There are many more photographs of African Aculeata on the web - See [https://www.waspweb.org/Classification/index.htm African Aculeata on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=326777 African Aculeata on iNaturalist] &lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Slender Ant (Tetraponera natalensis) (30538051244).jpg|Slender ant (Formicidae, Formicoidea)
Carpenter Bee (Xylocopa inconstans, female) (6021534858).jpg|Female carpenter bee (Apidae, Apoidea)
Chalybion 2019 12 02 2314.jpg|Blue mud-dauber wasp (Sphecidae, Apoidea)
Bethylidae 2019 08 21 13 42 21 8760.jpg|Flat wasp (Bethylidae, Chrysidoidea)
Belonogaster juncea colonialis, manlik, h, Pretoria.jpg|Paper wasp (Vespidae, Vespoidea)
Meria 2020 08 12 2193.jpg|''Meria'' sp. female (Thynnidae, Thynnoidea)
Dasylabris stimulatrix iNat37068107 2019 12 23 2990 W.jpg|''Dasylabris stimulatrix'', a velvet ant (Mutillidae, Pompiloidea)
Pompilidae 2019 05 28 0247.jpg|Spider-hunting wasp (Pompilidae, Pompiloidea)
Scoliid Wasp 2019 06 08 0492.jpg|Campsomerine wasp (Scoliidae, Scolioidea)
&lt;/gallery&gt;&lt;br&gt;

Darwin wasps ([[w:Ichneumonidae|Ichneumonidae]]) and braconids ([[w:Braconidae|Braconidae]]) are also frequently reported. Links to [https://www.waspweb.org/Ichneumonoidea/index.htm African Ichneumonoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=47200 African Ichneumonoidea on iNaturalist].&lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Echthromorpha agrestoria 2019 10 03 5572.jpg|''Echthromorpha agrestoria'' (Ichneumonidae, Ichneumonoidea)
Ichneumonidae inat 88832241.jpg|''Goryphus tricolor'' (Ichneumonidae, Ichneumonoidea)
Braconid Wasp (Archibracon servillei) (11857083186).jpg|Braconine wasp (Braconidae, Ichneumonoidea) 
2019 09 15 12 55 07 Microgastrinae 38425632.jpg|Microgastrine wasp (Braconidae, Ichneumonoidea) 
&lt;/gallery&gt;&lt;br&gt;

Small [[w:Chalcidoidea|chalcidoid]] parasitic wasps (especially those from the families [[w:Pteromalidae|Pteromalidae]], [[w:Chalcididae|Chalcididae]], [[w:Eulophidae|Eulophidae]], [[w:Eurytomidae|Eurytomidae]]) are also common. Links to [https://www.waspweb.org/Chalcidoidea/index.htm African Chalcidoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=69750 African Chalcidoidea on iNaturalist]
&lt;gallery mode=packed heights=200&gt;
Halticoptera 2019 11 26 2173Cedara.jpg|''Halticoptera'' sp. (Pteromalidae, Chalcidoidea)
Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp. (Chalcididae, Chalcidoidea)
Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp. (Eulophidae, Chalcidoidea)
Eurytomidae00.jpg|Mating seed chalcids (Eurytomidae, Chalcidoidea)
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>tqvi5ak4pia6acqmck1q76yehaasi7w</sha1>
    </revision>
    <revision>
      <id>2815640</id>
      <parentid>2815553</parentid>
      <timestamp>2026-06-14T09:08:34Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Subfamily Bembicinae */</comment>
      <origin>2815640</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16476" sha1="nnt4onxocpi05cv3vn6inqk6akra4k9" xml:space="preserve">= Superfamily Apoidea =
== African Ampulicidae ==
&lt;gallery mode=packed heights=150&gt;
Ampulicidae 37894270 suncana.jpg
Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg
&lt;/gallery&gt;		
== African Astatidae ==
&lt;gallery mode=packed heights=150&gt;
Astata iN 105162782 Nicola van Berkel.jpg
Astata melanaria.jpg
&lt;/gallery&gt;

== African Bembicidae ==
===Subfamily Bembicinae===	
&lt;gallery mode=packed heights=150&gt;
Bembecinus iN 153052948.jpg|''Bembecinus'' sp.
Bembix iN 195581985.jpg|''Bembix'' sp.
Gorytes natalensis 112517046.jpg
Hoplisoides thalia iN 268375748 a.jpg|''Hoplisoides thalia''
&lt;/gallery&gt;		
Tribe Gorytini	
*		''Afrogorytes''
*		''Gorytes''
*		''Harpactus''
*		''Hoplisoides''
*		''Lestiphorus''
		
Tribe Spheciini	
*		''Ammatomus''
*		''Kohlia''
*		''Sphecius''
		
Tribe Handlirschiini	
*		''Handlirschia''
		
Tribe Bembicini	
*		''Bembix''
		
Tribe Stizini	
*		''Bembecinus''
*		''Stizoides''
*		''Stizus''

===Subfamily Nyssoninae===	
*		''Brachystegus''
*		''Hovanysson''
*		''Nysson''
		
===Subfamily Alyssontinae===	
*		''Alysson''
*		''Didineis''

== African Crabronidae ==
&lt;gallery mode=packed heights=150&gt;
Dasyproctus iN 30029277 a.jpg
Dicranorhina kohli 2023 07 25 iN 176115975.jpg
Liris on Crassula iN 42678436 01.jpg
Oxybelus iN 250449990 2024 10 09 - 02.jpg
Palarus Bee Pirate iN 144133368 2022-12-01 01.jpg
Paranysson iN 199673489 1.jpg
Pison iN 144131685 2022-11-30 03.jpg
Tachysphex iN 250449986 2024 10 09 7305.jpg
Tachytes iN 188902572 1964.jpg
Trypoxylon iN 99063113 a.jpg
&lt;/gallery&gt;

== African Pemphredonidae ==
&lt;gallery mode=packed heights=150&gt;
Polemistus braunsii iNaturalist 228280708.jpg

&lt;/gallery&gt;

== African Philanthidae ==
&lt;gallery mode=packed heights=150&gt;
Philanthus triangulum diadema 187037342.jpg
Cerceris 2019 12 02 2310.jpg
&lt;/gallery&gt;

== African Psenidae ==
&lt;gallery mode=packed heights=150&gt;
Psenini iN 1022563 i c riddell.jpg

&lt;/gallery&gt;

== African Sphecidae ==
&lt;gallery mode=packed heights=150&gt;
Chalybion 2019 12 02 2314.jpg
Thread-waisted Sand Wasp (Ammophila sp.) carrying a Lienard's Achaea (Achaea lienardi) caterpillar ... (52715218455).jpg
Ammophila ferrugineipes04.jpg
Black Mud-dauber Wasp (Sceliphron spirifex) on Buffalo-Thorn (Ziziphus mucronata) flowers ... (52739846889).jpg
&lt;/gallery&gt;

= Superfamily Chalcidoidea =
 
Chalcidoidea is an incredibly diverse group of wasps with a wide variety of life histories; many are parasitoids, making them ecologically important; some are used for biological control of insect pests. About 22 000 species have been described, but as many as 500 000 are thought to exist worldwide.&lt;ref name=Noyes2019/&gt;

Most chalcidoids are small (body length less than 5 mm), the smallest being males of the mymarid ''Dicopomorpha echmepterygis'', which is the smallest known flying insect (0.13 mm).&lt;ref name=Noyes2019&gt;Noyes, J.S. 2019. Universal Chalcidoidea Database. [http://www.nhm.ac.uk/chalcidoids]&lt;/ref&gt; However, some species may be as long as 20 mm (excluding antennae and ovipositor), such as the pelecinellid ''Doddifoenus wallacei''.&lt;ref name=Krogmann&amp;Burks2009&gt;Krogmann, Lars; Burks, Roger A. (2009-12-31). &quot;Doddifoenus wallacei, a new giant parasitoid wasp of the subfamily Leptofoeninae (Chalcidoidea: Pteromalidae), with a description of its mesosomal skeletal anatomy and a molecular characterization&quot;. Zootaxa. 2194: 21–36. doi:10.5281/zenodo.189452 [https://www.researchgate.net/profile/Lars-Krogmann/publication/250916337 PDF]&lt;/ref&gt; Many have bright metallic colours, hence the name Chalcidoidea or chalcid - the greek word χαλκός (chalcos) means copper or bronze.&lt;ref name=Perseus2023&gt;[http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0073:entry=xalko/s Georg Autenrieth, A Homeric Dictionary]&lt;/ref&gt;

Many new chalcidoid families were proposed in 2022.&lt;ref name=Burks2022&gt;Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. [https://doi.org/10.3897/jhr.94.94263 DOI]&lt;/ref&gt; This was largely based on phylogenetic evidence (both molecular and morphological) accumulated since the families of the Superfamily were outlined in 1993&lt;ref name=Gibson1993&gt;Gibson, G.A.P. (1993) Superfamilies Mymarommatoidea and Chalcidoidea (pp. 570-655). In Goulet, H. &amp; Huber, J. (eds). Hymenoptera of the World: an identification guide to families. Research Branch, Agriculture Canada, Ottawa, Canada, 668 pp. [https://www.researchgate.net/publication/259227143 PDF]&lt;/ref&gt;.


Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. https://doi.org/10.3897/jhr.94.94263

Cruaud, A., Rasplus, J.Y., Zhang, J., Burks, R., Delvare, G., Fusu, L., Gumovsky, A., Huber, J.T., Janšta, P., Mitroiu, M.D. and Noyes, J.S., 2022. The Chalcidoidea bush of life–a massive radiation blurred by mutational saturation. bioRxiv. https://doi.org/10.1101/2022.09.11.507458 https://www.biorxiv.org/content/10.1101/2022.09.11.507458.full.pdf


= Superfamily Ichneumonoidea =

Quicke, D.L., 2015. The braconid and ichneumonid parasitoid wasps: biology, systematics, evolution and ecology. John Wiley &amp; Sons.
https://www.researchgate.net/publication/281662102_The_Braconid_and_Ichneumonid_Parasitoid_Wasps_Biology_Systematics_Evolution_and_Ecology

=== Diplazontinae ===
Quote from Fitton &amp; Rotheray (1982):
&quot;Diplazontines are distinguished easily from other ichneumonids by their general appearance (Fig. 1). The main confirmatory characters are the bifid upper tooth of the mandible (Fig. 2) and the sub-rectangular shape of the first tergite of the gaster (Figs. 1,3,4, 13, 15 and 17). In fact, the diplazontines are so distinct that they can be separated from all other Hymenoptera-Apocrita by possession of the following five characters in combination: (1) antenna of seventeen or more segments (Fig. I); (2) forewing with costal cell obliterated (that is, veins C and Sc + R + Rs contiguous from the wing base to the base of the pterostigma (Fig. 6 ) ; (3) forewing with cross-vein 2 mcu present (Fig. 6); (4) mandible with its upper tooth subdivided (bifid), making the mandible tridentate (Fig. 2); (5) first tergite of gaster sub-rectangular in shape (that is, subequal in width anteriorly and posteriorly) (Figs. 1 , 3 , 4 , 13,15 and 17).&quot;

Fitton, M. G., &amp; Rotheray, G. E. (1982). A key to the European genera of diplazontine ichneumon‐flies, with notes on the British fauna. Systematic Entomology, 7(3), 311-320.

=== Superfamily Pompiloidea ===
Family Mutillidae (Velvet ants)
Family Pompilidae (Spider hunting wasps)
Family Sapygidae (Sapygid wasps)


== Spider-hunting wasps ==
Two wasps from the genus ''Auplopus'' from Europe. Most of the ''Auplopus'' species, along with others from the Tribe Ageniellini, amputate the legs of the spiders they capture; this makes it much easier to move the spider to the nest.
&lt;gallery mode=packed heights=200&gt;
Auplopus carbonarius IMG 1624.jpg
Auplopus carbonarius fg01 20060623 Nied Garten.jpg
&lt;/gallery&gt;
African genera of Ageniellini:
* ''[[Auplopus]]'' &lt;small&gt;Spinola, 1841&lt;/small&gt;
* ''[[Cyemagenia]]'' &lt;small&gt;Arnold, 1946&lt;/small&gt;
* ''[[Dichragenia]]'' &lt;small&gt;Haupt, 1950&lt;/small&gt;
* ''[[Phanagenia]]'' &lt;small&gt;Banks, 1933&lt;/small&gt;   Madagascar
* ''[[Poecilagenia]]'' &lt;small&gt;Haupt, 1926&lt;/small&gt;

== Superfamily Scolioidea ==
=== Family Bradynobaenidae (Bradynobaenid wasps) ===
=== Family Scoliidae (Mammoth wasps) ===

== Superfamily Tiphioidea ==
Family Tiphiidae (Tiphiid wasps)
== Superfamily Thynnoidea ==
Family Thynnidae(Thynnid wasps)
== Superfamily Vespoidea ==
=== Family Rhopalosomatidae (Rhopalosomatid wasps) ===
===Family Vespidae (Paper, Potter &amp; Pollen wasps)===


== '''&lt;big&gt;Hymenoptera&lt;/big&gt;''' ==
About 20 000 described species of [[w:Hymenoptera|Hymenoptera]] (wasps, bees, ants and sawflies) are known from the [[w:Afrotropical|Afrotropical]] region. Estimates of the actual species count for the region range from 100 000 species to as high as 500 000 species.&lt;ref name=WaspWebOverview&gt;[https://www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm van Noort, Simon (2023). WaspWeb: Afrotropical Hymenoptera Initiative. www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm Retrieved 22 February 2023.]&lt;/ref&gt; Nineteen different superfamilies are illustrated on [[User:Alandmanson/Hymenoptera of southern Africa|this gallery]] in Wikimedia Commons.

The huge number of undescribed species means that many species will be extinct before we are even aware of them, as there are too few taxonomists employed to tackle the task of describing all of them in the next few decades.

==Classification==
The Order can be split into two Suborders - The [[w:Symphyta|Symphyta]] (Woodwasps, Horntails, Sawflies) and the [[w:Apocrita|Apocrita]] (Narrow-waisted wasps, ants and bees).&lt;br&gt;

When compared to the temperate regions of the northern hemisphere, the diversity of Symphyta in Africa is relatively poor, although sawflies of the Superfamily [[w:Tenthredinoidea|Tenthredinoidea]] are fairly common in forests and other woody vegetation types.

Africa has a rich diversity of Apocrita. The cladogram shown below&lt;ref name=Peters&gt;{{Cite journal |last1=Peters |first1=Ralph S. |last2=Krogmann |first2=Lars |last3=Mayer |first3=Christoph |last4=Donath |first4=Alexander |last5=Gunkel |first5=Simon |last6=Meusemann |first6=Karen |last7=Kozlov |first7=Alexey |last8=Podsiadlowski |first8=Lars |last9=Petersen |first9=Malte |title=Evolutionary History of the Hymenoptera |journal=Current Biology |volume=27 |issue=7 |pages=1013–1018 |doi=10.1016/j.cub.2017.01.027|pmid=28343967 |year=2017 |doi-access=free }}&lt;/ref&gt; indicates the possible relationships between 11 of the superfamilies that comprise Apocrita; These 11 superfamilies are all represented in Africa. This breakdown is used by iNaturalist.&lt;ref name=inat-Aculeata&gt;https://www.inaturalist.org/taxa/326777-Aculeata&lt;/ref&gt; It is, however, not accepted by all hymenopterists, and may change as more phylogenetic evidence is accumulated.&lt;ref name=waspwebClass&gt;[https://www.waspweb.org/Classification/index.htm van Noort, Simon (2023). WaspWeb: Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants). www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
{{Clade|style=font-size:85%; line-height:85%
|label1=Apocrita
            |1={{Clade
               |label1=[[w:Parasitoida|Parasitoida]]
               |1={{Clade
                  |1={{Clade
                     |1=[[w:Ceraphronoidea|Ceraphronoidea]]
                     |2=[[w:Ichneumonoidea|Ichneumonoidea]]
                     }}
                  |label2=[[w:Proctotrupomorpha|Proctotrupomorpha]]
                  |2={{Clade
                     |1=[[w:Cynipoidea|Cynipoidea]]
                     |2={{Clade
                        |1=[[w:Platygastroidea|Platygastroidea]]
                        |2={{Clade
                           |1=[[w:Chalcidoidea|Chalcidoidea]]
                           |2={{Clade
                              |1=[[w:Diaprioidea|Diaprioidea]]
                              |2=[[w:Proctotrupoidea|Proctotrupoidea]]
                  }} }} }} }} }}
               |2={{Clade
                  |1={{Clade
                     |1=[[w:Evanioidea|Evanioidea]]
                     |2=[[w:Stephanoidea|Stephanoidea]]
                     }}
                  |2={{Clade
                     |1=[[w:Trigonaloidea|Trigonaloidea]]
                     |label2=[[w:Aculeata|Aculeata]]
                     |2={{Clade
                        |1=[[w:Chrysidoidea|Chrysidoidea]]
                        |2={{Clade
                           |1=[[w:Vespoidea|Vespoidea]] (potter, honey and social wasps)
                           |2={{Clade
                              |1={{Clade
                                 |1=[[w:Pompiloidea|Pompiloidea]] (velvet ants, spider wasps and relatives)
                                 |2={{Clade
                                    |1=[[w:Thynnoidea|Thynnoidea]]
                                    |2=[[w:Tiphioidea|Tiphioidea]]
                                 }} }}
                              |2={{Clade
                                 |1=[[w:Scolioidea|Scolioidea]]
                                 |2={{Clade
                                    |1=[[w:Formicoidea|Formicoidea]] (ants)
                                    |2=[[w:Apoidea|Apoidea]] (bees and related wasps)
}} }} }} }} }} }} }} }} }}
&lt;br&gt;

==Some common African Symphyta==
&lt;gallery mode=packed heights=200&gt;
Sawfly, Argidae 2021 12 19 12 39 42.jpg|Sawfly, ''Arge'' sp. (Argidae, Superfamily Tenthredinoidea) 
Athalia 2019 11 21 0815.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Athalia 2019 08 23 8964.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Sirex noctilio F (prof.).JPG|''Sirex noctilio'', a pest in ''Pinus'' plantations (introduced to Africa)&lt;ref name=WaspWebSn&gt;[https://www.waspweb.org/Siricoidea/Siricidae/Sirex/Sirex_noctilio.htm van Noort, Simon (2023). WaspWeb: ''Sirex noctilio''. www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
&lt;/gallery&gt;

== Frequently reported African Apocrita ==
&lt;br&gt;
[[w:Aculeata|Aculeata]] (ants, bees and stinging wasps) are the most commonly observed Hymenoptera in Africa. There are many more photographs of African Aculeata on the web - See [https://www.waspweb.org/Classification/index.htm African Aculeata on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=326777 African Aculeata on iNaturalist] &lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Slender Ant (Tetraponera natalensis) (30538051244).jpg|Slender ant (Formicidae, Formicoidea)
Carpenter Bee (Xylocopa inconstans, female) (6021534858).jpg|Female carpenter bee (Apidae, Apoidea)
Chalybion 2019 12 02 2314.jpg|Blue mud-dauber wasp (Sphecidae, Apoidea)
Bethylidae 2019 08 21 13 42 21 8760.jpg|Flat wasp (Bethylidae, Chrysidoidea)
Belonogaster juncea colonialis, manlik, h, Pretoria.jpg|Paper wasp (Vespidae, Vespoidea)
Meria 2020 08 12 2193.jpg|''Meria'' sp. female (Thynnidae, Thynnoidea)
Dasylabris stimulatrix iNat37068107 2019 12 23 2990 W.jpg|''Dasylabris stimulatrix'', a velvet ant (Mutillidae, Pompiloidea)
Pompilidae 2019 05 28 0247.jpg|Spider-hunting wasp (Pompilidae, Pompiloidea)
Scoliid Wasp 2019 06 08 0492.jpg|Campsomerine wasp (Scoliidae, Scolioidea)
&lt;/gallery&gt;&lt;br&gt;

Darwin wasps ([[w:Ichneumonidae|Ichneumonidae]]) and braconids ([[w:Braconidae|Braconidae]]) are also frequently reported. Links to [https://www.waspweb.org/Ichneumonoidea/index.htm African Ichneumonoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=47200 African Ichneumonoidea on iNaturalist].&lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Echthromorpha agrestoria 2019 10 03 5572.jpg|''Echthromorpha agrestoria'' (Ichneumonidae, Ichneumonoidea)
Ichneumonidae inat 88832241.jpg|''Goryphus tricolor'' (Ichneumonidae, Ichneumonoidea)
Braconid Wasp (Archibracon servillei) (11857083186).jpg|Braconine wasp (Braconidae, Ichneumonoidea) 
2019 09 15 12 55 07 Microgastrinae 38425632.jpg|Microgastrine wasp (Braconidae, Ichneumonoidea) 
&lt;/gallery&gt;&lt;br&gt;

Small [[w:Chalcidoidea|chalcidoid]] parasitic wasps (especially those from the families [[w:Pteromalidae|Pteromalidae]], [[w:Chalcididae|Chalcididae]], [[w:Eulophidae|Eulophidae]], [[w:Eurytomidae|Eurytomidae]]) are also common. Links to [https://www.waspweb.org/Chalcidoidea/index.htm African Chalcidoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=69750 African Chalcidoidea on iNaturalist]
&lt;gallery mode=packed heights=200&gt;
Halticoptera 2019 11 26 2173Cedara.jpg|''Halticoptera'' sp. (Pteromalidae, Chalcidoidea)
Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp. (Chalcididae, Chalcidoidea)
Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp. (Eulophidae, Chalcidoidea)
Eurytomidae00.jpg|Mating seed chalcids (Eurytomidae, Chalcidoidea)
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>nnt4onxocpi05cv3vn6inqk6akra4k9</sha1>
    </revision>
    <revision>
      <id>2815641</id>
      <parentid>2815640</parentid>
      <timestamp>2026-06-14T09:15:12Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* African Ampulicidae */</comment>
      <origin>2815641</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16525" sha1="g0ussevbpipk1t2hfmfm6mly8u7gkta" xml:space="preserve">= Superfamily Apoidea =
== African Ampulicidae ==
&lt;gallery mode=packed heights=150&gt;
Ampulicidae 37894270 suncana.jpg|''Ampulex apicalis''
Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg|''Dolichurus'' cf ''basuto''
&lt;/gallery&gt;

== African Astatidae ==
&lt;gallery mode=packed heights=150&gt;
Astata iN 105162782 Nicola van Berkel.jpg
Astata melanaria.jpg
&lt;/gallery&gt;

== African Bembicidae ==
===Subfamily Bembicinae===	
&lt;gallery mode=packed heights=150&gt;
Bembecinus iN 153052948.jpg|''Bembecinus'' sp.
Bembix iN 195581985.jpg|''Bembix'' sp.
Gorytes natalensis 112517046.jpg
Hoplisoides thalia iN 268375748 a.jpg|''Hoplisoides thalia''
&lt;/gallery&gt;		
Tribe Gorytini	
*		''Afrogorytes''
*		''Gorytes''
*		''Harpactus''
*		''Hoplisoides''
*		''Lestiphorus''
		
Tribe Spheciini	
*		''Ammatomus''
*		''Kohlia''
*		''Sphecius''
		
Tribe Handlirschiini	
*		''Handlirschia''
		
Tribe Bembicini	
*		''Bembix''
		
Tribe Stizini	
*		''Bembecinus''
*		''Stizoides''
*		''Stizus''

===Subfamily Nyssoninae===	
*		''Brachystegus''
*		''Hovanysson''
*		''Nysson''
		
===Subfamily Alyssontinae===	
*		''Alysson''
*		''Didineis''

== African Crabronidae ==
&lt;gallery mode=packed heights=150&gt;
Dasyproctus iN 30029277 a.jpg
Dicranorhina kohli 2023 07 25 iN 176115975.jpg
Liris on Crassula iN 42678436 01.jpg
Oxybelus iN 250449990 2024 10 09 - 02.jpg
Palarus Bee Pirate iN 144133368 2022-12-01 01.jpg
Paranysson iN 199673489 1.jpg
Pison iN 144131685 2022-11-30 03.jpg
Tachysphex iN 250449986 2024 10 09 7305.jpg
Tachytes iN 188902572 1964.jpg
Trypoxylon iN 99063113 a.jpg
&lt;/gallery&gt;

== African Pemphredonidae ==
&lt;gallery mode=packed heights=150&gt;
Polemistus braunsii iNaturalist 228280708.jpg

&lt;/gallery&gt;

== African Philanthidae ==
&lt;gallery mode=packed heights=150&gt;
Philanthus triangulum diadema 187037342.jpg
Cerceris 2019 12 02 2310.jpg
&lt;/gallery&gt;

== African Psenidae ==
&lt;gallery mode=packed heights=150&gt;
Psenini iN 1022563 i c riddell.jpg

&lt;/gallery&gt;

== African Sphecidae ==
&lt;gallery mode=packed heights=150&gt;
Chalybion 2019 12 02 2314.jpg
Thread-waisted Sand Wasp (Ammophila sp.) carrying a Lienard's Achaea (Achaea lienardi) caterpillar ... (52715218455).jpg
Ammophila ferrugineipes04.jpg
Black Mud-dauber Wasp (Sceliphron spirifex) on Buffalo-Thorn (Ziziphus mucronata) flowers ... (52739846889).jpg
&lt;/gallery&gt;

= Superfamily Chalcidoidea =
 
Chalcidoidea is an incredibly diverse group of wasps with a wide variety of life histories; many are parasitoids, making them ecologically important; some are used for biological control of insect pests. About 22 000 species have been described, but as many as 500 000 are thought to exist worldwide.&lt;ref name=Noyes2019/&gt;

Most chalcidoids are small (body length less than 5 mm), the smallest being males of the mymarid ''Dicopomorpha echmepterygis'', which is the smallest known flying insect (0.13 mm).&lt;ref name=Noyes2019&gt;Noyes, J.S. 2019. Universal Chalcidoidea Database. [http://www.nhm.ac.uk/chalcidoids]&lt;/ref&gt; However, some species may be as long as 20 mm (excluding antennae and ovipositor), such as the pelecinellid ''Doddifoenus wallacei''.&lt;ref name=Krogmann&amp;Burks2009&gt;Krogmann, Lars; Burks, Roger A. (2009-12-31). &quot;Doddifoenus wallacei, a new giant parasitoid wasp of the subfamily Leptofoeninae (Chalcidoidea: Pteromalidae), with a description of its mesosomal skeletal anatomy and a molecular characterization&quot;. Zootaxa. 2194: 21–36. doi:10.5281/zenodo.189452 [https://www.researchgate.net/profile/Lars-Krogmann/publication/250916337 PDF]&lt;/ref&gt; Many have bright metallic colours, hence the name Chalcidoidea or chalcid - the greek word χαλκός (chalcos) means copper or bronze.&lt;ref name=Perseus2023&gt;[http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0073:entry=xalko/s Georg Autenrieth, A Homeric Dictionary]&lt;/ref&gt;

Many new chalcidoid families were proposed in 2022.&lt;ref name=Burks2022&gt;Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. [https://doi.org/10.3897/jhr.94.94263 DOI]&lt;/ref&gt; This was largely based on phylogenetic evidence (both molecular and morphological) accumulated since the families of the Superfamily were outlined in 1993&lt;ref name=Gibson1993&gt;Gibson, G.A.P. (1993) Superfamilies Mymarommatoidea and Chalcidoidea (pp. 570-655). In Goulet, H. &amp; Huber, J. (eds). Hymenoptera of the World: an identification guide to families. Research Branch, Agriculture Canada, Ottawa, Canada, 668 pp. [https://www.researchgate.net/publication/259227143 PDF]&lt;/ref&gt;.


Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. https://doi.org/10.3897/jhr.94.94263

Cruaud, A., Rasplus, J.Y., Zhang, J., Burks, R., Delvare, G., Fusu, L., Gumovsky, A., Huber, J.T., Janšta, P., Mitroiu, M.D. and Noyes, J.S., 2022. The Chalcidoidea bush of life–a massive radiation blurred by mutational saturation. bioRxiv. https://doi.org/10.1101/2022.09.11.507458 https://www.biorxiv.org/content/10.1101/2022.09.11.507458.full.pdf


= Superfamily Ichneumonoidea =

Quicke, D.L., 2015. The braconid and ichneumonid parasitoid wasps: biology, systematics, evolution and ecology. John Wiley &amp; Sons.
https://www.researchgate.net/publication/281662102_The_Braconid_and_Ichneumonid_Parasitoid_Wasps_Biology_Systematics_Evolution_and_Ecology

=== Diplazontinae ===
Quote from Fitton &amp; Rotheray (1982):
&quot;Diplazontines are distinguished easily from other ichneumonids by their general appearance (Fig. 1). The main confirmatory characters are the bifid upper tooth of the mandible (Fig. 2) and the sub-rectangular shape of the first tergite of the gaster (Figs. 1,3,4, 13, 15 and 17). In fact, the diplazontines are so distinct that they can be separated from all other Hymenoptera-Apocrita by possession of the following five characters in combination: (1) antenna of seventeen or more segments (Fig. I); (2) forewing with costal cell obliterated (that is, veins C and Sc + R + Rs contiguous from the wing base to the base of the pterostigma (Fig. 6 ) ; (3) forewing with cross-vein 2 mcu present (Fig. 6); (4) mandible with its upper tooth subdivided (bifid), making the mandible tridentate (Fig. 2); (5) first tergite of gaster sub-rectangular in shape (that is, subequal in width anteriorly and posteriorly) (Figs. 1 , 3 , 4 , 13,15 and 17).&quot;

Fitton, M. G., &amp; Rotheray, G. E. (1982). A key to the European genera of diplazontine ichneumon‐flies, with notes on the British fauna. Systematic Entomology, 7(3), 311-320.

=== Superfamily Pompiloidea ===
Family Mutillidae (Velvet ants)
Family Pompilidae (Spider hunting wasps)
Family Sapygidae (Sapygid wasps)


== Spider-hunting wasps ==
Two wasps from the genus ''Auplopus'' from Europe. Most of the ''Auplopus'' species, along with others from the Tribe Ageniellini, amputate the legs of the spiders they capture; this makes it much easier to move the spider to the nest.
&lt;gallery mode=packed heights=200&gt;
Auplopus carbonarius IMG 1624.jpg
Auplopus carbonarius fg01 20060623 Nied Garten.jpg
&lt;/gallery&gt;
African genera of Ageniellini:
* ''[[Auplopus]]'' &lt;small&gt;Spinola, 1841&lt;/small&gt;
* ''[[Cyemagenia]]'' &lt;small&gt;Arnold, 1946&lt;/small&gt;
* ''[[Dichragenia]]'' &lt;small&gt;Haupt, 1950&lt;/small&gt;
* ''[[Phanagenia]]'' &lt;small&gt;Banks, 1933&lt;/small&gt;   Madagascar
* ''[[Poecilagenia]]'' &lt;small&gt;Haupt, 1926&lt;/small&gt;

== Superfamily Scolioidea ==
=== Family Bradynobaenidae (Bradynobaenid wasps) ===
=== Family Scoliidae (Mammoth wasps) ===

== Superfamily Tiphioidea ==
Family Tiphiidae (Tiphiid wasps)
== Superfamily Thynnoidea ==
Family Thynnidae(Thynnid wasps)
== Superfamily Vespoidea ==
=== Family Rhopalosomatidae (Rhopalosomatid wasps) ===
===Family Vespidae (Paper, Potter &amp; Pollen wasps)===


== '''&lt;big&gt;Hymenoptera&lt;/big&gt;''' ==
About 20 000 described species of [[w:Hymenoptera|Hymenoptera]] (wasps, bees, ants and sawflies) are known from the [[w:Afrotropical|Afrotropical]] region. Estimates of the actual species count for the region range from 100 000 species to as high as 500 000 species.&lt;ref name=WaspWebOverview&gt;[https://www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm van Noort, Simon (2023). WaspWeb: Afrotropical Hymenoptera Initiative. www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm Retrieved 22 February 2023.]&lt;/ref&gt; Nineteen different superfamilies are illustrated on [[User:Alandmanson/Hymenoptera of southern Africa|this gallery]] in Wikimedia Commons.

The huge number of undescribed species means that many species will be extinct before we are even aware of them, as there are too few taxonomists employed to tackle the task of describing all of them in the next few decades.

==Classification==
The Order can be split into two Suborders - The [[w:Symphyta|Symphyta]] (Woodwasps, Horntails, Sawflies) and the [[w:Apocrita|Apocrita]] (Narrow-waisted wasps, ants and bees).&lt;br&gt;

When compared to the temperate regions of the northern hemisphere, the diversity of Symphyta in Africa is relatively poor, although sawflies of the Superfamily [[w:Tenthredinoidea|Tenthredinoidea]] are fairly common in forests and other woody vegetation types.

Africa has a rich diversity of Apocrita. The cladogram shown below&lt;ref name=Peters&gt;{{Cite journal |last1=Peters |first1=Ralph S. |last2=Krogmann |first2=Lars |last3=Mayer |first3=Christoph |last4=Donath |first4=Alexander |last5=Gunkel |first5=Simon |last6=Meusemann |first6=Karen |last7=Kozlov |first7=Alexey |last8=Podsiadlowski |first8=Lars |last9=Petersen |first9=Malte |title=Evolutionary History of the Hymenoptera |journal=Current Biology |volume=27 |issue=7 |pages=1013–1018 |doi=10.1016/j.cub.2017.01.027|pmid=28343967 |year=2017 |doi-access=free }}&lt;/ref&gt; indicates the possible relationships between 11 of the superfamilies that comprise Apocrita; These 11 superfamilies are all represented in Africa. This breakdown is used by iNaturalist.&lt;ref name=inat-Aculeata&gt;https://www.inaturalist.org/taxa/326777-Aculeata&lt;/ref&gt; It is, however, not accepted by all hymenopterists, and may change as more phylogenetic evidence is accumulated.&lt;ref name=waspwebClass&gt;[https://www.waspweb.org/Classification/index.htm van Noort, Simon (2023). WaspWeb: Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants). www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
{{Clade|style=font-size:85%; line-height:85%
|label1=Apocrita
            |1={{Clade
               |label1=[[w:Parasitoida|Parasitoida]]
               |1={{Clade
                  |1={{Clade
                     |1=[[w:Ceraphronoidea|Ceraphronoidea]]
                     |2=[[w:Ichneumonoidea|Ichneumonoidea]]
                     }}
                  |label2=[[w:Proctotrupomorpha|Proctotrupomorpha]]
                  |2={{Clade
                     |1=[[w:Cynipoidea|Cynipoidea]]
                     |2={{Clade
                        |1=[[w:Platygastroidea|Platygastroidea]]
                        |2={{Clade
                           |1=[[w:Chalcidoidea|Chalcidoidea]]
                           |2={{Clade
                              |1=[[w:Diaprioidea|Diaprioidea]]
                              |2=[[w:Proctotrupoidea|Proctotrupoidea]]
                  }} }} }} }} }}
               |2={{Clade
                  |1={{Clade
                     |1=[[w:Evanioidea|Evanioidea]]
                     |2=[[w:Stephanoidea|Stephanoidea]]
                     }}
                  |2={{Clade
                     |1=[[w:Trigonaloidea|Trigonaloidea]]
                     |label2=[[w:Aculeata|Aculeata]]
                     |2={{Clade
                        |1=[[w:Chrysidoidea|Chrysidoidea]]
                        |2={{Clade
                           |1=[[w:Vespoidea|Vespoidea]] (potter, honey and social wasps)
                           |2={{Clade
                              |1={{Clade
                                 |1=[[w:Pompiloidea|Pompiloidea]] (velvet ants, spider wasps and relatives)
                                 |2={{Clade
                                    |1=[[w:Thynnoidea|Thynnoidea]]
                                    |2=[[w:Tiphioidea|Tiphioidea]]
                                 }} }}
                              |2={{Clade
                                 |1=[[w:Scolioidea|Scolioidea]]
                                 |2={{Clade
                                    |1=[[w:Formicoidea|Formicoidea]] (ants)
                                    |2=[[w:Apoidea|Apoidea]] (bees and related wasps)
}} }} }} }} }} }} }} }} }}
&lt;br&gt;

==Some common African Symphyta==
&lt;gallery mode=packed heights=200&gt;
Sawfly, Argidae 2021 12 19 12 39 42.jpg|Sawfly, ''Arge'' sp. (Argidae, Superfamily Tenthredinoidea) 
Athalia 2019 11 21 0815.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Athalia 2019 08 23 8964.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Sirex noctilio F (prof.).JPG|''Sirex noctilio'', a pest in ''Pinus'' plantations (introduced to Africa)&lt;ref name=WaspWebSn&gt;[https://www.waspweb.org/Siricoidea/Siricidae/Sirex/Sirex_noctilio.htm van Noort, Simon (2023). WaspWeb: ''Sirex noctilio''. www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
&lt;/gallery&gt;

== Frequently reported African Apocrita ==
&lt;br&gt;
[[w:Aculeata|Aculeata]] (ants, bees and stinging wasps) are the most commonly observed Hymenoptera in Africa. There are many more photographs of African Aculeata on the web - See [https://www.waspweb.org/Classification/index.htm African Aculeata on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=326777 African Aculeata on iNaturalist] &lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Slender Ant (Tetraponera natalensis) (30538051244).jpg|Slender ant (Formicidae, Formicoidea)
Carpenter Bee (Xylocopa inconstans, female) (6021534858).jpg|Female carpenter bee (Apidae, Apoidea)
Chalybion 2019 12 02 2314.jpg|Blue mud-dauber wasp (Sphecidae, Apoidea)
Bethylidae 2019 08 21 13 42 21 8760.jpg|Flat wasp (Bethylidae, Chrysidoidea)
Belonogaster juncea colonialis, manlik, h, Pretoria.jpg|Paper wasp (Vespidae, Vespoidea)
Meria 2020 08 12 2193.jpg|''Meria'' sp. female (Thynnidae, Thynnoidea)
Dasylabris stimulatrix iNat37068107 2019 12 23 2990 W.jpg|''Dasylabris stimulatrix'', a velvet ant (Mutillidae, Pompiloidea)
Pompilidae 2019 05 28 0247.jpg|Spider-hunting wasp (Pompilidae, Pompiloidea)
Scoliid Wasp 2019 06 08 0492.jpg|Campsomerine wasp (Scoliidae, Scolioidea)
&lt;/gallery&gt;&lt;br&gt;

Darwin wasps ([[w:Ichneumonidae|Ichneumonidae]]) and braconids ([[w:Braconidae|Braconidae]]) are also frequently reported. Links to [https://www.waspweb.org/Ichneumonoidea/index.htm African Ichneumonoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=47200 African Ichneumonoidea on iNaturalist].&lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Echthromorpha agrestoria 2019 10 03 5572.jpg|''Echthromorpha agrestoria'' (Ichneumonidae, Ichneumonoidea)
Ichneumonidae inat 88832241.jpg|''Goryphus tricolor'' (Ichneumonidae, Ichneumonoidea)
Braconid Wasp (Archibracon servillei) (11857083186).jpg|Braconine wasp (Braconidae, Ichneumonoidea) 
2019 09 15 12 55 07 Microgastrinae 38425632.jpg|Microgastrine wasp (Braconidae, Ichneumonoidea) 
&lt;/gallery&gt;&lt;br&gt;

Small [[w:Chalcidoidea|chalcidoid]] parasitic wasps (especially those from the families [[w:Pteromalidae|Pteromalidae]], [[w:Chalcididae|Chalcididae]], [[w:Eulophidae|Eulophidae]], [[w:Eurytomidae|Eurytomidae]]) are also common. Links to [https://www.waspweb.org/Chalcidoidea/index.htm African Chalcidoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=69750 African Chalcidoidea on iNaturalist]
&lt;gallery mode=packed heights=200&gt;
Halticoptera 2019 11 26 2173Cedara.jpg|''Halticoptera'' sp. (Pteromalidae, Chalcidoidea)
Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp. (Chalcididae, Chalcidoidea)
Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp. (Eulophidae, Chalcidoidea)
Eurytomidae00.jpg|Mating seed chalcids (Eurytomidae, Chalcidoidea)
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>g0ussevbpipk1t2hfmfm6mly8u7gkta</sha1>
    </revision>
    <revision>
      <id>2815642</id>
      <parentid>2815641</parentid>
      <timestamp>2026-06-14T09:16:31Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* African Astatidae */</comment>
      <origin>2815642</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16561" sha1="enuy6ubpxkdcv0dtoqihlmpqwrfw63a" xml:space="preserve">= Superfamily Apoidea =
== African Ampulicidae ==
&lt;gallery mode=packed heights=150&gt;
Ampulicidae 37894270 suncana.jpg|''Ampulex apicalis''
Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg|''Dolichurus'' cf ''basuto''
&lt;/gallery&gt;

== African Astatidae ==
&lt;gallery mode=packed heights=150&gt;
Astata iN 105162782 Nicola van Berkel.jpg|''Astata'' sp.
Astata melanaria.jpg|''Astata melanaria''
&lt;/gallery&gt;

== African Bembicidae ==
===Subfamily Bembicinae===	
&lt;gallery mode=packed heights=150&gt;
Bembecinus iN 153052948.jpg|''Bembecinus'' sp.
Bembix iN 195581985.jpg|''Bembix'' sp.
Gorytes natalensis 112517046.jpg
Hoplisoides thalia iN 268375748 a.jpg|''Hoplisoides thalia''
&lt;/gallery&gt;		
Tribe Gorytini	
*		''Afrogorytes''
*		''Gorytes''
*		''Harpactus''
*		''Hoplisoides''
*		''Lestiphorus''
		
Tribe Spheciini	
*		''Ammatomus''
*		''Kohlia''
*		''Sphecius''
		
Tribe Handlirschiini	
*		''Handlirschia''
		
Tribe Bembicini	
*		''Bembix''
		
Tribe Stizini	
*		''Bembecinus''
*		''Stizoides''
*		''Stizus''

===Subfamily Nyssoninae===	
*		''Brachystegus''
*		''Hovanysson''
*		''Nysson''
		
===Subfamily Alyssontinae===	
*		''Alysson''
*		''Didineis''

== African Crabronidae ==
&lt;gallery mode=packed heights=150&gt;
Dasyproctus iN 30029277 a.jpg
Dicranorhina kohli 2023 07 25 iN 176115975.jpg
Liris on Crassula iN 42678436 01.jpg
Oxybelus iN 250449990 2024 10 09 - 02.jpg
Palarus Bee Pirate iN 144133368 2022-12-01 01.jpg
Paranysson iN 199673489 1.jpg
Pison iN 144131685 2022-11-30 03.jpg
Tachysphex iN 250449986 2024 10 09 7305.jpg
Tachytes iN 188902572 1964.jpg
Trypoxylon iN 99063113 a.jpg
&lt;/gallery&gt;

== African Pemphredonidae ==
&lt;gallery mode=packed heights=150&gt;
Polemistus braunsii iNaturalist 228280708.jpg

&lt;/gallery&gt;

== African Philanthidae ==
&lt;gallery mode=packed heights=150&gt;
Philanthus triangulum diadema 187037342.jpg
Cerceris 2019 12 02 2310.jpg
&lt;/gallery&gt;

== African Psenidae ==
&lt;gallery mode=packed heights=150&gt;
Psenini iN 1022563 i c riddell.jpg

&lt;/gallery&gt;

== African Sphecidae ==
&lt;gallery mode=packed heights=150&gt;
Chalybion 2019 12 02 2314.jpg
Thread-waisted Sand Wasp (Ammophila sp.) carrying a Lienard's Achaea (Achaea lienardi) caterpillar ... (52715218455).jpg
Ammophila ferrugineipes04.jpg
Black Mud-dauber Wasp (Sceliphron spirifex) on Buffalo-Thorn (Ziziphus mucronata) flowers ... (52739846889).jpg
&lt;/gallery&gt;

= Superfamily Chalcidoidea =
 
Chalcidoidea is an incredibly diverse group of wasps with a wide variety of life histories; many are parasitoids, making them ecologically important; some are used for biological control of insect pests. About 22 000 species have been described, but as many as 500 000 are thought to exist worldwide.&lt;ref name=Noyes2019/&gt;

Most chalcidoids are small (body length less than 5 mm), the smallest being males of the mymarid ''Dicopomorpha echmepterygis'', which is the smallest known flying insect (0.13 mm).&lt;ref name=Noyes2019&gt;Noyes, J.S. 2019. Universal Chalcidoidea Database. [http://www.nhm.ac.uk/chalcidoids]&lt;/ref&gt; However, some species may be as long as 20 mm (excluding antennae and ovipositor), such as the pelecinellid ''Doddifoenus wallacei''.&lt;ref name=Krogmann&amp;Burks2009&gt;Krogmann, Lars; Burks, Roger A. (2009-12-31). &quot;Doddifoenus wallacei, a new giant parasitoid wasp of the subfamily Leptofoeninae (Chalcidoidea: Pteromalidae), with a description of its mesosomal skeletal anatomy and a molecular characterization&quot;. Zootaxa. 2194: 21–36. doi:10.5281/zenodo.189452 [https://www.researchgate.net/profile/Lars-Krogmann/publication/250916337 PDF]&lt;/ref&gt; Many have bright metallic colours, hence the name Chalcidoidea or chalcid - the greek word χαλκός (chalcos) means copper or bronze.&lt;ref name=Perseus2023&gt;[http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0073:entry=xalko/s Georg Autenrieth, A Homeric Dictionary]&lt;/ref&gt;

Many new chalcidoid families were proposed in 2022.&lt;ref name=Burks2022&gt;Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. [https://doi.org/10.3897/jhr.94.94263 DOI]&lt;/ref&gt; This was largely based on phylogenetic evidence (both molecular and morphological) accumulated since the families of the Superfamily were outlined in 1993&lt;ref name=Gibson1993&gt;Gibson, G.A.P. (1993) Superfamilies Mymarommatoidea and Chalcidoidea (pp. 570-655). In Goulet, H. &amp; Huber, J. (eds). Hymenoptera of the World: an identification guide to families. Research Branch, Agriculture Canada, Ottawa, Canada, 668 pp. [https://www.researchgate.net/publication/259227143 PDF]&lt;/ref&gt;.


Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. https://doi.org/10.3897/jhr.94.94263

Cruaud, A., Rasplus, J.Y., Zhang, J., Burks, R., Delvare, G., Fusu, L., Gumovsky, A., Huber, J.T., Janšta, P., Mitroiu, M.D. and Noyes, J.S., 2022. The Chalcidoidea bush of life–a massive radiation blurred by mutational saturation. bioRxiv. https://doi.org/10.1101/2022.09.11.507458 https://www.biorxiv.org/content/10.1101/2022.09.11.507458.full.pdf


= Superfamily Ichneumonoidea =

Quicke, D.L., 2015. The braconid and ichneumonid parasitoid wasps: biology, systematics, evolution and ecology. John Wiley &amp; Sons.
https://www.researchgate.net/publication/281662102_The_Braconid_and_Ichneumonid_Parasitoid_Wasps_Biology_Systematics_Evolution_and_Ecology

=== Diplazontinae ===
Quote from Fitton &amp; Rotheray (1982):
&quot;Diplazontines are distinguished easily from other ichneumonids by their general appearance (Fig. 1). The main confirmatory characters are the bifid upper tooth of the mandible (Fig. 2) and the sub-rectangular shape of the first tergite of the gaster (Figs. 1,3,4, 13, 15 and 17). In fact, the diplazontines are so distinct that they can be separated from all other Hymenoptera-Apocrita by possession of the following five characters in combination: (1) antenna of seventeen or more segments (Fig. I); (2) forewing with costal cell obliterated (that is, veins C and Sc + R + Rs contiguous from the wing base to the base of the pterostigma (Fig. 6 ) ; (3) forewing with cross-vein 2 mcu present (Fig. 6); (4) mandible with its upper tooth subdivided (bifid), making the mandible tridentate (Fig. 2); (5) first tergite of gaster sub-rectangular in shape (that is, subequal in width anteriorly and posteriorly) (Figs. 1 , 3 , 4 , 13,15 and 17).&quot;

Fitton, M. G., &amp; Rotheray, G. E. (1982). A key to the European genera of diplazontine ichneumon‐flies, with notes on the British fauna. Systematic Entomology, 7(3), 311-320.

=== Superfamily Pompiloidea ===
Family Mutillidae (Velvet ants)
Family Pompilidae (Spider hunting wasps)
Family Sapygidae (Sapygid wasps)


== Spider-hunting wasps ==
Two wasps from the genus ''Auplopus'' from Europe. Most of the ''Auplopus'' species, along with others from the Tribe Ageniellini, amputate the legs of the spiders they capture; this makes it much easier to move the spider to the nest.
&lt;gallery mode=packed heights=200&gt;
Auplopus carbonarius IMG 1624.jpg
Auplopus carbonarius fg01 20060623 Nied Garten.jpg
&lt;/gallery&gt;
African genera of Ageniellini:
* ''[[Auplopus]]'' &lt;small&gt;Spinola, 1841&lt;/small&gt;
* ''[[Cyemagenia]]'' &lt;small&gt;Arnold, 1946&lt;/small&gt;
* ''[[Dichragenia]]'' &lt;small&gt;Haupt, 1950&lt;/small&gt;
* ''[[Phanagenia]]'' &lt;small&gt;Banks, 1933&lt;/small&gt;   Madagascar
* ''[[Poecilagenia]]'' &lt;small&gt;Haupt, 1926&lt;/small&gt;

== Superfamily Scolioidea ==
=== Family Bradynobaenidae (Bradynobaenid wasps) ===
=== Family Scoliidae (Mammoth wasps) ===

== Superfamily Tiphioidea ==
Family Tiphiidae (Tiphiid wasps)
== Superfamily Thynnoidea ==
Family Thynnidae(Thynnid wasps)
== Superfamily Vespoidea ==
=== Family Rhopalosomatidae (Rhopalosomatid wasps) ===
===Family Vespidae (Paper, Potter &amp; Pollen wasps)===


== '''&lt;big&gt;Hymenoptera&lt;/big&gt;''' ==
About 20 000 described species of [[w:Hymenoptera|Hymenoptera]] (wasps, bees, ants and sawflies) are known from the [[w:Afrotropical|Afrotropical]] region. Estimates of the actual species count for the region range from 100 000 species to as high as 500 000 species.&lt;ref name=WaspWebOverview&gt;[https://www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm van Noort, Simon (2023). WaspWeb: Afrotropical Hymenoptera Initiative. www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm Retrieved 22 February 2023.]&lt;/ref&gt; Nineteen different superfamilies are illustrated on [[User:Alandmanson/Hymenoptera of southern Africa|this gallery]] in Wikimedia Commons.

The huge number of undescribed species means that many species will be extinct before we are even aware of them, as there are too few taxonomists employed to tackle the task of describing all of them in the next few decades.

==Classification==
The Order can be split into two Suborders - The [[w:Symphyta|Symphyta]] (Woodwasps, Horntails, Sawflies) and the [[w:Apocrita|Apocrita]] (Narrow-waisted wasps, ants and bees).&lt;br&gt;

When compared to the temperate regions of the northern hemisphere, the diversity of Symphyta in Africa is relatively poor, although sawflies of the Superfamily [[w:Tenthredinoidea|Tenthredinoidea]] are fairly common in forests and other woody vegetation types.

Africa has a rich diversity of Apocrita. The cladogram shown below&lt;ref name=Peters&gt;{{Cite journal |last1=Peters |first1=Ralph S. |last2=Krogmann |first2=Lars |last3=Mayer |first3=Christoph |last4=Donath |first4=Alexander |last5=Gunkel |first5=Simon |last6=Meusemann |first6=Karen |last7=Kozlov |first7=Alexey |last8=Podsiadlowski |first8=Lars |last9=Petersen |first9=Malte |title=Evolutionary History of the Hymenoptera |journal=Current Biology |volume=27 |issue=7 |pages=1013–1018 |doi=10.1016/j.cub.2017.01.027|pmid=28343967 |year=2017 |doi-access=free }}&lt;/ref&gt; indicates the possible relationships between 11 of the superfamilies that comprise Apocrita; These 11 superfamilies are all represented in Africa. This breakdown is used by iNaturalist.&lt;ref name=inat-Aculeata&gt;https://www.inaturalist.org/taxa/326777-Aculeata&lt;/ref&gt; It is, however, not accepted by all hymenopterists, and may change as more phylogenetic evidence is accumulated.&lt;ref name=waspwebClass&gt;[https://www.waspweb.org/Classification/index.htm van Noort, Simon (2023). WaspWeb: Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants). www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
{{Clade|style=font-size:85%; line-height:85%
|label1=Apocrita
            |1={{Clade
               |label1=[[w:Parasitoida|Parasitoida]]
               |1={{Clade
                  |1={{Clade
                     |1=[[w:Ceraphronoidea|Ceraphronoidea]]
                     |2=[[w:Ichneumonoidea|Ichneumonoidea]]
                     }}
                  |label2=[[w:Proctotrupomorpha|Proctotrupomorpha]]
                  |2={{Clade
                     |1=[[w:Cynipoidea|Cynipoidea]]
                     |2={{Clade
                        |1=[[w:Platygastroidea|Platygastroidea]]
                        |2={{Clade
                           |1=[[w:Chalcidoidea|Chalcidoidea]]
                           |2={{Clade
                              |1=[[w:Diaprioidea|Diaprioidea]]
                              |2=[[w:Proctotrupoidea|Proctotrupoidea]]
                  }} }} }} }} }}
               |2={{Clade
                  |1={{Clade
                     |1=[[w:Evanioidea|Evanioidea]]
                     |2=[[w:Stephanoidea|Stephanoidea]]
                     }}
                  |2={{Clade
                     |1=[[w:Trigonaloidea|Trigonaloidea]]
                     |label2=[[w:Aculeata|Aculeata]]
                     |2={{Clade
                        |1=[[w:Chrysidoidea|Chrysidoidea]]
                        |2={{Clade
                           |1=[[w:Vespoidea|Vespoidea]] (potter, honey and social wasps)
                           |2={{Clade
                              |1={{Clade
                                 |1=[[w:Pompiloidea|Pompiloidea]] (velvet ants, spider wasps and relatives)
                                 |2={{Clade
                                    |1=[[w:Thynnoidea|Thynnoidea]]
                                    |2=[[w:Tiphioidea|Tiphioidea]]
                                 }} }}
                              |2={{Clade
                                 |1=[[w:Scolioidea|Scolioidea]]
                                 |2={{Clade
                                    |1=[[w:Formicoidea|Formicoidea]] (ants)
                                    |2=[[w:Apoidea|Apoidea]] (bees and related wasps)
}} }} }} }} }} }} }} }} }}
&lt;br&gt;

==Some common African Symphyta==
&lt;gallery mode=packed heights=200&gt;
Sawfly, Argidae 2021 12 19 12 39 42.jpg|Sawfly, ''Arge'' sp. (Argidae, Superfamily Tenthredinoidea) 
Athalia 2019 11 21 0815.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Athalia 2019 08 23 8964.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Sirex noctilio F (prof.).JPG|''Sirex noctilio'', a pest in ''Pinus'' plantations (introduced to Africa)&lt;ref name=WaspWebSn&gt;[https://www.waspweb.org/Siricoidea/Siricidae/Sirex/Sirex_noctilio.htm van Noort, Simon (2023). WaspWeb: ''Sirex noctilio''. www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
&lt;/gallery&gt;

== Frequently reported African Apocrita ==
&lt;br&gt;
[[w:Aculeata|Aculeata]] (ants, bees and stinging wasps) are the most commonly observed Hymenoptera in Africa. There are many more photographs of African Aculeata on the web - See [https://www.waspweb.org/Classification/index.htm African Aculeata on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=326777 African Aculeata on iNaturalist] &lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Slender Ant (Tetraponera natalensis) (30538051244).jpg|Slender ant (Formicidae, Formicoidea)
Carpenter Bee (Xylocopa inconstans, female) (6021534858).jpg|Female carpenter bee (Apidae, Apoidea)
Chalybion 2019 12 02 2314.jpg|Blue mud-dauber wasp (Sphecidae, Apoidea)
Bethylidae 2019 08 21 13 42 21 8760.jpg|Flat wasp (Bethylidae, Chrysidoidea)
Belonogaster juncea colonialis, manlik, h, Pretoria.jpg|Paper wasp (Vespidae, Vespoidea)
Meria 2020 08 12 2193.jpg|''Meria'' sp. female (Thynnidae, Thynnoidea)
Dasylabris stimulatrix iNat37068107 2019 12 23 2990 W.jpg|''Dasylabris stimulatrix'', a velvet ant (Mutillidae, Pompiloidea)
Pompilidae 2019 05 28 0247.jpg|Spider-hunting wasp (Pompilidae, Pompiloidea)
Scoliid Wasp 2019 06 08 0492.jpg|Campsomerine wasp (Scoliidae, Scolioidea)
&lt;/gallery&gt;&lt;br&gt;

Darwin wasps ([[w:Ichneumonidae|Ichneumonidae]]) and braconids ([[w:Braconidae|Braconidae]]) are also frequently reported. Links to [https://www.waspweb.org/Ichneumonoidea/index.htm African Ichneumonoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=47200 African Ichneumonoidea on iNaturalist].&lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Echthromorpha agrestoria 2019 10 03 5572.jpg|''Echthromorpha agrestoria'' (Ichneumonidae, Ichneumonoidea)
Ichneumonidae inat 88832241.jpg|''Goryphus tricolor'' (Ichneumonidae, Ichneumonoidea)
Braconid Wasp (Archibracon servillei) (11857083186).jpg|Braconine wasp (Braconidae, Ichneumonoidea) 
2019 09 15 12 55 07 Microgastrinae 38425632.jpg|Microgastrine wasp (Braconidae, Ichneumonoidea) 
&lt;/gallery&gt;&lt;br&gt;

Small [[w:Chalcidoidea|chalcidoid]] parasitic wasps (especially those from the families [[w:Pteromalidae|Pteromalidae]], [[w:Chalcididae|Chalcididae]], [[w:Eulophidae|Eulophidae]], [[w:Eurytomidae|Eurytomidae]]) are also common. Links to [https://www.waspweb.org/Chalcidoidea/index.htm African Chalcidoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=69750 African Chalcidoidea on iNaturalist]
&lt;gallery mode=packed heights=200&gt;
Halticoptera 2019 11 26 2173Cedara.jpg|''Halticoptera'' sp. (Pteromalidae, Chalcidoidea)
Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp. (Chalcididae, Chalcidoidea)
Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp. (Eulophidae, Chalcidoidea)
Eurytomidae00.jpg|Mating seed chalcids (Eurytomidae, Chalcidoidea)
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>enuy6ubpxkdcv0dtoqihlmpqwrfw63a</sha1>
    </revision>
    <revision>
      <id>2815643</id>
      <parentid>2815642</parentid>
      <timestamp>2026-06-14T09:17:04Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Subfamily Bembicinae */</comment>
      <origin>2815643</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16584" sha1="gx20ic2y298qamck7nltzt3fo32482e" xml:space="preserve">= Superfamily Apoidea =
== African Ampulicidae ==
&lt;gallery mode=packed heights=150&gt;
Ampulicidae 37894270 suncana.jpg|''Ampulex apicalis''
Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg|''Dolichurus'' cf ''basuto''
&lt;/gallery&gt;

== African Astatidae ==
&lt;gallery mode=packed heights=150&gt;
Astata iN 105162782 Nicola van Berkel.jpg|''Astata'' sp.
Astata melanaria.jpg|''Astata melanaria''
&lt;/gallery&gt;

== African Bembicidae ==
===Subfamily Bembicinae===	
&lt;gallery mode=packed heights=150&gt;
Bembecinus iN 153052948.jpg|''Bembecinus'' sp.
Bembix iN 195581985.jpg|''Bembix'' sp.
Gorytes natalensis 112517046.jpg|''Gorytes natalensis''
Hoplisoides thalia iN 268375748 a.jpg|''Hoplisoides thalia''
&lt;/gallery&gt;		
Tribe Gorytini	
*		''Afrogorytes''
*		''Gorytes''
*		''Harpactus''
*		''Hoplisoides''
*		''Lestiphorus''
		
Tribe Spheciini	
*		''Ammatomus''
*		''Kohlia''
*		''Sphecius''
		
Tribe Handlirschiini	
*		''Handlirschia''
		
Tribe Bembicini	
*		''Bembix''
		
Tribe Stizini	
*		''Bembecinus''
*		''Stizoides''
*		''Stizus''

===Subfamily Nyssoninae===	
*		''Brachystegus''
*		''Hovanysson''
*		''Nysson''
		
===Subfamily Alyssontinae===	
*		''Alysson''
*		''Didineis''

== African Crabronidae ==
&lt;gallery mode=packed heights=150&gt;
Dasyproctus iN 30029277 a.jpg
Dicranorhina kohli 2023 07 25 iN 176115975.jpg
Liris on Crassula iN 42678436 01.jpg
Oxybelus iN 250449990 2024 10 09 - 02.jpg
Palarus Bee Pirate iN 144133368 2022-12-01 01.jpg
Paranysson iN 199673489 1.jpg
Pison iN 144131685 2022-11-30 03.jpg
Tachysphex iN 250449986 2024 10 09 7305.jpg
Tachytes iN 188902572 1964.jpg
Trypoxylon iN 99063113 a.jpg
&lt;/gallery&gt;

== African Pemphredonidae ==
&lt;gallery mode=packed heights=150&gt;
Polemistus braunsii iNaturalist 228280708.jpg

&lt;/gallery&gt;

== African Philanthidae ==
&lt;gallery mode=packed heights=150&gt;
Philanthus triangulum diadema 187037342.jpg
Cerceris 2019 12 02 2310.jpg
&lt;/gallery&gt;

== African Psenidae ==
&lt;gallery mode=packed heights=150&gt;
Psenini iN 1022563 i c riddell.jpg

&lt;/gallery&gt;

== African Sphecidae ==
&lt;gallery mode=packed heights=150&gt;
Chalybion 2019 12 02 2314.jpg
Thread-waisted Sand Wasp (Ammophila sp.) carrying a Lienard's Achaea (Achaea lienardi) caterpillar ... (52715218455).jpg
Ammophila ferrugineipes04.jpg
Black Mud-dauber Wasp (Sceliphron spirifex) on Buffalo-Thorn (Ziziphus mucronata) flowers ... (52739846889).jpg
&lt;/gallery&gt;

= Superfamily Chalcidoidea =
 
Chalcidoidea is an incredibly diverse group of wasps with a wide variety of life histories; many are parasitoids, making them ecologically important; some are used for biological control of insect pests. About 22 000 species have been described, but as many as 500 000 are thought to exist worldwide.&lt;ref name=Noyes2019/&gt;

Most chalcidoids are small (body length less than 5 mm), the smallest being males of the mymarid ''Dicopomorpha echmepterygis'', which is the smallest known flying insect (0.13 mm).&lt;ref name=Noyes2019&gt;Noyes, J.S. 2019. Universal Chalcidoidea Database. [http://www.nhm.ac.uk/chalcidoids]&lt;/ref&gt; However, some species may be as long as 20 mm (excluding antennae and ovipositor), such as the pelecinellid ''Doddifoenus wallacei''.&lt;ref name=Krogmann&amp;Burks2009&gt;Krogmann, Lars; Burks, Roger A. (2009-12-31). &quot;Doddifoenus wallacei, a new giant parasitoid wasp of the subfamily Leptofoeninae (Chalcidoidea: Pteromalidae), with a description of its mesosomal skeletal anatomy and a molecular characterization&quot;. Zootaxa. 2194: 21–36. doi:10.5281/zenodo.189452 [https://www.researchgate.net/profile/Lars-Krogmann/publication/250916337 PDF]&lt;/ref&gt; Many have bright metallic colours, hence the name Chalcidoidea or chalcid - the greek word χαλκός (chalcos) means copper or bronze.&lt;ref name=Perseus2023&gt;[http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0073:entry=xalko/s Georg Autenrieth, A Homeric Dictionary]&lt;/ref&gt;

Many new chalcidoid families were proposed in 2022.&lt;ref name=Burks2022&gt;Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. [https://doi.org/10.3897/jhr.94.94263 DOI]&lt;/ref&gt; This was largely based on phylogenetic evidence (both molecular and morphological) accumulated since the families of the Superfamily were outlined in 1993&lt;ref name=Gibson1993&gt;Gibson, G.A.P. (1993) Superfamilies Mymarommatoidea and Chalcidoidea (pp. 570-655). In Goulet, H. &amp; Huber, J. (eds). Hymenoptera of the World: an identification guide to families. Research Branch, Agriculture Canada, Ottawa, Canada, 668 pp. [https://www.researchgate.net/publication/259227143 PDF]&lt;/ref&gt;.


Burks, R., Mitroiu, M.D., Fusu, L., Heraty, J.M., Janšta, P., Heydon, S., Papilloud, N.D.S., Peters, R.S., Tselikh, E.V., Woolley, J.B. and van Noort, S., (2022).﻿ From hell’s heart I stab at thee! A determined approach towards a monophyletic Pteromalidae and reclassification of Chalcidoidea (Hymenoptera). Journal of Hymenoptera Research, 94, pp.13-88. https://doi.org/10.3897/jhr.94.94263

Cruaud, A., Rasplus, J.Y., Zhang, J., Burks, R., Delvare, G., Fusu, L., Gumovsky, A., Huber, J.T., Janšta, P., Mitroiu, M.D. and Noyes, J.S., 2022. The Chalcidoidea bush of life–a massive radiation blurred by mutational saturation. bioRxiv. https://doi.org/10.1101/2022.09.11.507458 https://www.biorxiv.org/content/10.1101/2022.09.11.507458.full.pdf


= Superfamily Ichneumonoidea =

Quicke, D.L., 2015. The braconid and ichneumonid parasitoid wasps: biology, systematics, evolution and ecology. John Wiley &amp; Sons.
https://www.researchgate.net/publication/281662102_The_Braconid_and_Ichneumonid_Parasitoid_Wasps_Biology_Systematics_Evolution_and_Ecology

=== Diplazontinae ===
Quote from Fitton &amp; Rotheray (1982):
&quot;Diplazontines are distinguished easily from other ichneumonids by their general appearance (Fig. 1). The main confirmatory characters are the bifid upper tooth of the mandible (Fig. 2) and the sub-rectangular shape of the first tergite of the gaster (Figs. 1,3,4, 13, 15 and 17). In fact, the diplazontines are so distinct that they can be separated from all other Hymenoptera-Apocrita by possession of the following five characters in combination: (1) antenna of seventeen or more segments (Fig. I); (2) forewing with costal cell obliterated (that is, veins C and Sc + R + Rs contiguous from the wing base to the base of the pterostigma (Fig. 6 ) ; (3) forewing with cross-vein 2 mcu present (Fig. 6); (4) mandible with its upper tooth subdivided (bifid), making the mandible tridentate (Fig. 2); (5) first tergite of gaster sub-rectangular in shape (that is, subequal in width anteriorly and posteriorly) (Figs. 1 , 3 , 4 , 13,15 and 17).&quot;

Fitton, M. G., &amp; Rotheray, G. E. (1982). A key to the European genera of diplazontine ichneumon‐flies, with notes on the British fauna. Systematic Entomology, 7(3), 311-320.

=== Superfamily Pompiloidea ===
Family Mutillidae (Velvet ants)
Family Pompilidae (Spider hunting wasps)
Family Sapygidae (Sapygid wasps)


== Spider-hunting wasps ==
Two wasps from the genus ''Auplopus'' from Europe. Most of the ''Auplopus'' species, along with others from the Tribe Ageniellini, amputate the legs of the spiders they capture; this makes it much easier to move the spider to the nest.
&lt;gallery mode=packed heights=200&gt;
Auplopus carbonarius IMG 1624.jpg
Auplopus carbonarius fg01 20060623 Nied Garten.jpg
&lt;/gallery&gt;
African genera of Ageniellini:
* ''[[Auplopus]]'' &lt;small&gt;Spinola, 1841&lt;/small&gt;
* ''[[Cyemagenia]]'' &lt;small&gt;Arnold, 1946&lt;/small&gt;
* ''[[Dichragenia]]'' &lt;small&gt;Haupt, 1950&lt;/small&gt;
* ''[[Phanagenia]]'' &lt;small&gt;Banks, 1933&lt;/small&gt;   Madagascar
* ''[[Poecilagenia]]'' &lt;small&gt;Haupt, 1926&lt;/small&gt;

== Superfamily Scolioidea ==
=== Family Bradynobaenidae (Bradynobaenid wasps) ===
=== Family Scoliidae (Mammoth wasps) ===

== Superfamily Tiphioidea ==
Family Tiphiidae (Tiphiid wasps)
== Superfamily Thynnoidea ==
Family Thynnidae(Thynnid wasps)
== Superfamily Vespoidea ==
=== Family Rhopalosomatidae (Rhopalosomatid wasps) ===
===Family Vespidae (Paper, Potter &amp; Pollen wasps)===


== '''&lt;big&gt;Hymenoptera&lt;/big&gt;''' ==
About 20 000 described species of [[w:Hymenoptera|Hymenoptera]] (wasps, bees, ants and sawflies) are known from the [[w:Afrotropical|Afrotropical]] region. Estimates of the actual species count for the region range from 100 000 species to as high as 500 000 species.&lt;ref name=WaspWebOverview&gt;[https://www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm van Noort, Simon (2023). WaspWeb: Afrotropical Hymenoptera Initiative. www.waspweb.org/Afrotropical_Hymenoptera_book/Background_and_Motivation.htm Retrieved 22 February 2023.]&lt;/ref&gt; Nineteen different superfamilies are illustrated on [[User:Alandmanson/Hymenoptera of southern Africa|this gallery]] in Wikimedia Commons.

The huge number of undescribed species means that many species will be extinct before we are even aware of them, as there are too few taxonomists employed to tackle the task of describing all of them in the next few decades.

==Classification==
The Order can be split into two Suborders - The [[w:Symphyta|Symphyta]] (Woodwasps, Horntails, Sawflies) and the [[w:Apocrita|Apocrita]] (Narrow-waisted wasps, ants and bees).&lt;br&gt;

When compared to the temperate regions of the northern hemisphere, the diversity of Symphyta in Africa is relatively poor, although sawflies of the Superfamily [[w:Tenthredinoidea|Tenthredinoidea]] are fairly common in forests and other woody vegetation types.

Africa has a rich diversity of Apocrita. The cladogram shown below&lt;ref name=Peters&gt;{{Cite journal |last1=Peters |first1=Ralph S. |last2=Krogmann |first2=Lars |last3=Mayer |first3=Christoph |last4=Donath |first4=Alexander |last5=Gunkel |first5=Simon |last6=Meusemann |first6=Karen |last7=Kozlov |first7=Alexey |last8=Podsiadlowski |first8=Lars |last9=Petersen |first9=Malte |title=Evolutionary History of the Hymenoptera |journal=Current Biology |volume=27 |issue=7 |pages=1013–1018 |doi=10.1016/j.cub.2017.01.027|pmid=28343967 |year=2017 |doi-access=free }}&lt;/ref&gt; indicates the possible relationships between 11 of the superfamilies that comprise Apocrita; These 11 superfamilies are all represented in Africa. This breakdown is used by iNaturalist.&lt;ref name=inat-Aculeata&gt;https://www.inaturalist.org/taxa/326777-Aculeata&lt;/ref&gt; It is, however, not accepted by all hymenopterists, and may change as more phylogenetic evidence is accumulated.&lt;ref name=waspwebClass&gt;[https://www.waspweb.org/Classification/index.htm van Noort, Simon (2023). WaspWeb: Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants). www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
{{Clade|style=font-size:85%; line-height:85%
|label1=Apocrita
            |1={{Clade
               |label1=[[w:Parasitoida|Parasitoida]]
               |1={{Clade
                  |1={{Clade
                     |1=[[w:Ceraphronoidea|Ceraphronoidea]]
                     |2=[[w:Ichneumonoidea|Ichneumonoidea]]
                     }}
                  |label2=[[w:Proctotrupomorpha|Proctotrupomorpha]]
                  |2={{Clade
                     |1=[[w:Cynipoidea|Cynipoidea]]
                     |2={{Clade
                        |1=[[w:Platygastroidea|Platygastroidea]]
                        |2={{Clade
                           |1=[[w:Chalcidoidea|Chalcidoidea]]
                           |2={{Clade
                              |1=[[w:Diaprioidea|Diaprioidea]]
                              |2=[[w:Proctotrupoidea|Proctotrupoidea]]
                  }} }} }} }} }}
               |2={{Clade
                  |1={{Clade
                     |1=[[w:Evanioidea|Evanioidea]]
                     |2=[[w:Stephanoidea|Stephanoidea]]
                     }}
                  |2={{Clade
                     |1=[[w:Trigonaloidea|Trigonaloidea]]
                     |label2=[[w:Aculeata|Aculeata]]
                     |2={{Clade
                        |1=[[w:Chrysidoidea|Chrysidoidea]]
                        |2={{Clade
                           |1=[[w:Vespoidea|Vespoidea]] (potter, honey and social wasps)
                           |2={{Clade
                              |1={{Clade
                                 |1=[[w:Pompiloidea|Pompiloidea]] (velvet ants, spider wasps and relatives)
                                 |2={{Clade
                                    |1=[[w:Thynnoidea|Thynnoidea]]
                                    |2=[[w:Tiphioidea|Tiphioidea]]
                                 }} }}
                              |2={{Clade
                                 |1=[[w:Scolioidea|Scolioidea]]
                                 |2={{Clade
                                    |1=[[w:Formicoidea|Formicoidea]] (ants)
                                    |2=[[w:Apoidea|Apoidea]] (bees and related wasps)
}} }} }} }} }} }} }} }} }}
&lt;br&gt;

==Some common African Symphyta==
&lt;gallery mode=packed heights=200&gt;
Sawfly, Argidae 2021 12 19 12 39 42.jpg|Sawfly, ''Arge'' sp. (Argidae, Superfamily Tenthredinoidea) 
Athalia 2019 11 21 0815.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Athalia 2019 08 23 8964.jpg|Sawfly, ''Athalia'' sp. (Tenthredinidae, Superfamily Tenthredinoidea) 
Sirex noctilio F (prof.).JPG|''Sirex noctilio'', a pest in ''Pinus'' plantations (introduced to Africa)&lt;ref name=WaspWebSn&gt;[https://www.waspweb.org/Siricoidea/Siricidae/Sirex/Sirex_noctilio.htm van Noort, Simon (2023). WaspWeb: ''Sirex noctilio''. www.waspweb.org/Classification/index.htm Retrieved 22 February 2023.]&lt;/ref&gt;
&lt;/gallery&gt;

== Frequently reported African Apocrita ==
&lt;br&gt;
[[w:Aculeata|Aculeata]] (ants, bees and stinging wasps) are the most commonly observed Hymenoptera in Africa. There are many more photographs of African Aculeata on the web - See [https://www.waspweb.org/Classification/index.htm African Aculeata on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=326777 African Aculeata on iNaturalist] &lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Slender Ant (Tetraponera natalensis) (30538051244).jpg|Slender ant (Formicidae, Formicoidea)
Carpenter Bee (Xylocopa inconstans, female) (6021534858).jpg|Female carpenter bee (Apidae, Apoidea)
Chalybion 2019 12 02 2314.jpg|Blue mud-dauber wasp (Sphecidae, Apoidea)
Bethylidae 2019 08 21 13 42 21 8760.jpg|Flat wasp (Bethylidae, Chrysidoidea)
Belonogaster juncea colonialis, manlik, h, Pretoria.jpg|Paper wasp (Vespidae, Vespoidea)
Meria 2020 08 12 2193.jpg|''Meria'' sp. female (Thynnidae, Thynnoidea)
Dasylabris stimulatrix iNat37068107 2019 12 23 2990 W.jpg|''Dasylabris stimulatrix'', a velvet ant (Mutillidae, Pompiloidea)
Pompilidae 2019 05 28 0247.jpg|Spider-hunting wasp (Pompilidae, Pompiloidea)
Scoliid Wasp 2019 06 08 0492.jpg|Campsomerine wasp (Scoliidae, Scolioidea)
&lt;/gallery&gt;&lt;br&gt;

Darwin wasps ([[w:Ichneumonidae|Ichneumonidae]]) and braconids ([[w:Braconidae|Braconidae]]) are also frequently reported. Links to [https://www.waspweb.org/Ichneumonoidea/index.htm African Ichneumonoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=47200 African Ichneumonoidea on iNaturalist].&lt;br&gt;
&lt;gallery mode=packed heights=200&gt;
Echthromorpha agrestoria 2019 10 03 5572.jpg|''Echthromorpha agrestoria'' (Ichneumonidae, Ichneumonoidea)
Ichneumonidae inat 88832241.jpg|''Goryphus tricolor'' (Ichneumonidae, Ichneumonoidea)
Braconid Wasp (Archibracon servillei) (11857083186).jpg|Braconine wasp (Braconidae, Ichneumonoidea) 
2019 09 15 12 55 07 Microgastrinae 38425632.jpg|Microgastrine wasp (Braconidae, Ichneumonoidea) 
&lt;/gallery&gt;&lt;br&gt;

Small [[w:Chalcidoidea|chalcidoid]] parasitic wasps (especially those from the families [[w:Pteromalidae|Pteromalidae]], [[w:Chalcididae|Chalcididae]], [[w:Eulophidae|Eulophidae]], [[w:Eurytomidae|Eurytomidae]]) are also common. Links to [https://www.waspweb.org/Chalcidoidea/index.htm African Chalcidoidea on WaspWeb] and [https://www.inaturalist.org/observations?place_id=97392&amp;taxon_id=69750 African Chalcidoidea on iNaturalist]
&lt;gallery mode=packed heights=200&gt;
Halticoptera 2019 11 26 2173Cedara.jpg|''Halticoptera'' sp. (Pteromalidae, Chalcidoidea)
Brachymeria 2019 10 09 17 26 38 2520.jpg|''Brachymeria'' sp. (Chalcididae, Chalcidoidea)
Eulophidae Inat 38422095 b.jpg|''Hemiptarsenus'' sp. (Eulophidae, Chalcidoidea)
Eurytomidae00.jpg|Mating seed chalcids (Eurytomidae, Chalcidoidea)
&lt;/gallery&gt;

==References==
{{reflist}}</text>
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    </revision>
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    <title>Motivation and emotion/Book/2026/Eco-emotions</title>
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{{title|Eco-emotions:&lt;br&gt;Subtitle goes here}}

This template provides tips for the [[Motivation and emotion/Assessment/Topic|topic development]] exercise.&lt;br&gt;Gradually remove these suggestions as the chapter develops.&lt;br&gt;Also consult the [[Motivation and emotion/Assessment/Chapter|book chapter guidelines]].
__TOC__


==Overview==
[[File:A picture is worth a thousand words.jpg|right|thumb|150px|'''Figure 1'''. Explore the topic, then develop a structure for the book chapter.]]
Ecological Emotions - an emotional sense reactive to the interconnected changing of our natural and social environment. 

There will be a focus on the eco-emotional response to the changing social climate and the influence of social media the promote change targeting the human ecological emotive response. The social issue of Gender Inequality will be used as an ongoing example with reference to both current and historical feminist movements.   


''\Imagine'' ... a [[#Case studies|scenario or case study]] which illustrates the problem and engages reader interest. 
Consider including an image (see Figure 1). The scenario could be presented in a [[#Feature box|feature box]].

The Overview is typically 180 to 330 words.

Eco-emotions; the advocate for change

{{RoundBoxTop|theme=3}}
'''Focus questions:'''


{{RoundBoxBottom}}

==Main headings==

* Why should we care? Indirect issues on a personal basis. 
*
* '#MeToo'
* Factors limiting Action: (Online Caution, Cultural Differences)

* Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]
* Sub-headings can also be used, but avoid having sections with only one sub-heading

==Key points==
* Why should we care? Indirect issues on a personal basis:
* what are ecological emotions?
* how to deal with ecological emotions?
* Example: Suffragettes 
*
*
* Revolution: Sexual revolution, why protest?
* What are the consequences of acting. 
* social / cultural context. 
*
* #MeToo:
* Influence of Social Media on choosing to act/speak.                                                     - diffusion of responsibility and social influence 
* What influenced people to come forward?
* Act global not local?                                                                                                                     - Why not to the attention of local authorities?                                                                   - Anonymous 

* Gender Inequality issues historically: Birth Control, Sexual Revolution etc., and how these eco-emotions were more/less apparent, yet, always present referring to psychological theory. I will use a historical movement as a case study. 


*Provide at least 3 bullet-points per heading, including for the Overview and Conclusion
* Include key citations

==Learning features==
* Interactive learning features bring online book chapters to life and can be embedded throughout the chapter.

* Collective Action Theory - belief that through collective action ones group could make change. 

===Case studies===

* Bystander Effect on Climate Change as example. 
* Suffragettes 

* Case studies describe real-world examples of concepts in action.
* Case studies can be real or fictional.
* A case study could be split into multiple sections throughout a chapter to illustrate different theories or stages.
* It is often helpful to present case studies using [[#Feature boxes|feature boxes]].

===Feature boxes===
* Important content can be highlighted in a feature box. But don't overuse feature boxes, otherwise they lose their effect. There are several  ways of creating boxes. Recommended: [[Help:Pretty boxes|Pretty boxes]]).
* Consider using feature boxes for:
** Focus questions
** [[#Case studies|Case studies]] or examples
** Quiz questions
** Take-home messages

 A very simple box can be created by using a space at the start of the line

{{RoundBoxTop|theme=3}}
;Feature box example
* Shaded background
* Coloured border
* Change the theme number for different colours
{{RoundBoxBottom}}

===Figures===
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* Use figures to illustrate concepts, add interest, and to serve as examples
* Figures can show photos, diagrams, graphs, etcetera
* Figures can be embedded throughout the chapter, including the Overview section
* Figures should be captioned (using '''Figure #.''' and a description). Captions explain the relevance of the image to the text/
* [[commons:|Wikimedia Commons]] provides a library of embeddable images
* Images can also be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Refer to each figure at least once in the main text (e.g., see Figure 2)

===Links===
* When key words are introduced, use [[Help:Links|interwiki links]]
* These links can go to:
** Wikipedia (e.g., [[w:Sigmund Freud|Sigmund Freud]] wrote about (e.g., [[w:Dreams|dreams]]) or
** Related book chapters (e.g., if your are struggling, you might be interested to read the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]])

===Tables===
* Use tables to organise and summarise information
* As with [[#Figures|figures]], tables should be captioned (e.g., see Table 1)
* Refer to each table at least once in the main text (e.g., see Table 1)
* [[Motivation and emotion/Wikiversity/Tables|Example 3 x 3 tables]] which could be adapted

'''Table 1.''' Descriptive Caption Which Explains The Table and its Relevant to the Text - Johari Window Model

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

===Quizzes===
* Using one or two review questions per major section is usually better than a long quiz at the end
* Quiz conceptual understanding, rather than trivia
* Don't make quizzes too hard
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{Quizzes are an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long quizzes are a good idea:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* The Conclusion is arguably the most important section
* The Conclusion is typically 150 to 330 words
* What are the take-home messages likely to be?
* It should be possible for someone to only read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the problem and what is known based on psychological science
{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
Provide up to 6 [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example:
* [[Motivation and emotion/Book/2021/Cognitive dissonance and motivation|Cognitive dissonance and motivation]] (Book chapter, 2021)
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[w:David McClelland|David McClelland]] (Wikipedia)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)
{{tip|Suggestions for this section:
* Present in alphabetical order
* Use [[w:Letter case#Sentence casing|sentence casing]]
* Include the source in parentheses
}}

==References==
List cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

APA style example:

Kałwak, W., &amp; Weihgold, V. (2022). The Relationality of Ecological Emotions: An   Interdisciplinary Critique of Individual Resilience as Psychology's Response to the Climate Crisis. ''Frontiers in psychology'', ''13'', 823620. &lt;nowiki&gt;https://doi.org/10.3389/fpsyg.2022.823620&lt;/nowiki&gt;

==External links==
Provide up to 6 [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]]. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)
{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Present in alphabetical order
* Include the source in parentheses after the link
}}
[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[File:A picture is worth a thousand words.jpg|right|thumb|150px|'''Figure 1'''. Explore the topic, then develop a structure for the book chapter.]]
''Imagine'' ... a [[#Case studies|scenario or case study]] which illustrates the problem and engages reader interest. 
Consider including an image (see Figure 1). The scenario could be presented in a [[#Feature box|feature box]].

The Overview is typically 180 to 330 words.

{{tip|Suggestions for this section:
* Engage the reader with a scenario, example, or case study, and an accompanying image
* Explain the problem and why it is important
* Outline how psychological science can help
* Present focus questions
}}

{{RoundBoxTop|theme=3}}
'''Focus questions:'''
Break the problem (see the sub-title) down into three to five focus questions. Focus questions could also be used as top-level headings.
* What is the first focus question?
* What is the second focus question? 
* What is the third focus question?

Ask ''[[w:Open-ended question|open-ended]]'' focus questions. For example:
* ''Is there'' a relationship between motivation and success? (closed-ended) {{sad}}
* ''What is the'' relationship between motivation and success? (open-ended) {{smile}}
{{RoundBoxBottom}}


I am using the issue of Gender Inequality, particularly in Western Society, as an ongoing example of how eco emotions are used, and acknowledged to motivate change. 

The exact chapter I’m writing will discuss this in direct reference to modern western society, and the prevalence of eco-emotions to act, using the recent ‘MeToo’ movement, to discuss the influence and use of social media to promote world issues, igniting an ecological emotional response, motivating others to act (what made them come forward), however small, even from a phone. 

Following, the rest of the book chapter headings would discuss why people respond to in-direct issues on a personal basis using psychological theory. 

The other sub headings would refer to briefly the Gender Inequality issues historically, such as the Suffragette Movement, Birth Control, Sexual Revolution etc, and how these eco-emotions were more/less apparent, yet, always present referring to psychological theory. I will use a historical movement as a case study. 

Further subheadings would cover the influence and prominence of eco-emotions and if eco-emotions on Feminism differ in cultures. 

Prior conclusion, I will discuss the limitations of eco-emotions to motivate change, based on cultural and political limitations to act on ecological-emotions (eg government / political restrictions / resources / repercussions) 

Attribution Theory 

==Main headings==
* Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]
* Sub-headings can also be used, but avoid having sections with only one sub-heading

==Key points==
* Provide at least 3 bullet-points per heading, including for the Overview and Conclusion
* Include key citations

==Learning features==
* Interactive learning features bring online book chapters to life and can be embedded throughout the chapter.

===Case studies===
* Case studies describe real-world examples of concepts in action.
* Case studies can be real or fictional.
* A case study could be split into multiple sections throughout a chapter to illustrate different theories or stages.
* It is often helpful to present case studies using [[#Feature boxes|feature boxes]].

===Feature boxes===
* Important content can be highlighted in a feature box. But don't overuse feature boxes, otherwise they lose their effect. There are several  ways of creating boxes. Recommended: [[Help:Pretty boxes|Pretty boxes]]).
* Consider using feature boxes for:
** Focus questions
** [[#Case studies|Case studies]] or examples
** Quiz questions
** Take-home messages

 A very simple box can be created by using a space at the start of the line

{{RoundBoxTop|theme=3}}
;Feature box example
* Shaded background
* Coloured border
* Change the theme number for different colours
{{RoundBoxBottom}}

===Figures===
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* Use figures to illustrate concepts, add interest, and to serve as examples
* Figures can show photos, diagrams, graphs, etcetera
* Figures can be embedded throughout the chapter, including the Overview section
* Figures should be captioned (using '''Figure #.''' and a description). Captions explain the relevance of the image to the text/
* [[commons:|Wikimedia Commons]] provides a library of embeddable images
* Images can also be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Refer to each figure at least once in the main text (e.g., see Figure 2)

===Links===
* When key words are introduced, use [[Help:Links|interwiki links]]
* These links can go to:
** Wikipedia (e.g., [[w:Sigmund Freud|Sigmund Freud]] wrote about (e.g., [[w:Dreams|dreams]]) or
** Related book chapters (e.g., if your are struggling, you might be interested to read the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]])

===Tables===
* Use tables to organise and summarise information
* As with [[#Figures|figures]], tables should be captioned (e.g., see Table 1)
* Refer to each table at least once in the main text (e.g., see Table 1)
* [[Motivation and emotion/Wikiversity/Tables|Example 3 x 3 tables]] which could be adapted

'''Table 1.''' Descriptive Caption Which Explains The Table and its Relevant to the Text - Johari Window Model

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

===Quizzes===
* Using one or two review questions per major section is usually better than a long quiz at the end
* Quiz conceptual understanding, rather than trivia
* Don't make quizzes too hard
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{Quizzes are an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long quizzes are a good idea:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* The Conclusion is arguably the most important section
* The Conclusion is typically 150 to 330 words
* What are the take-home messages likely to be?
* It should be possible for someone to only read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the problem and what is known based on psychological science
{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
Provide up to 6 [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example:
* [[Motivation and emotion/Book/2021/Cognitive dissonance and motivation|Cognitive dissonance and motivation]] (Book chapter, 2021)
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[w:David McClelland|David McClelland]] (Wikipedia)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)
{{tip|Suggestions for this section:
* Present in alphabetical order
* Use [[w:Letter case#Sentence casing|sentence casing]]
* Include the source in parentheses
}}

==References==
List cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

APA style example:
{{Hanging indent|1=

Mercer-Mapstone, L., Dvorakova, S. L., Matthews, K. E., Abbot, S., Cheng, B., Felten, P., Knorr, K., Marquis, E., Shammas, R., &amp; Swaim, K. (2017). A systematic literature review of students as partners in higher education. ''International Journal for Students as Partners'', ''1''(1). https://doi.org/10.15173/ijsap.v1i1.3119

Rosenberg, B. D., &amp; Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091

Sears, C. R., Boyce, M. A., Boon, S. D., Goghari, V. M., Irwin, K., &amp; Boyes, M. (2017). Predictors of student satisfaction in a large psychology undergraduate program. ''Canadian Psychology/Psychologie Canadienne'', ''58''(2), 148–160. https://doi.org/10.1037/cap0000082
}}
{{tip|Suggestions for this section:
* Important aspects for APA style include:
** Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]. Use &quot;Edit source&quot;: &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case except first letter and proper names, ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** Provide the full doi as a URL and working hyperlink
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** Citing sources that weren't read or consulted
}}

==External links==
Provide up to 6 [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]]. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)
{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Present in alphabetical order
* Include the source in parentheses after the link
}}
[[Category:{{#titleparts:{{PAGENAME}}|3}}]]

{{MECR3|1=https://yourlinkgoeshere.com}}
&lt;div align=center&gt;Replace this link once the multimedia presentation has been published.&lt;br&gt;

This template provides tips for the [[Motivation and emotion/Assessment/Topic|topic development]] exercise.&lt;br&gt;Gradually remove these suggestions as the chapter develops.&lt;br&gt;Also consult the [[Motivation and emotion/Assessment/Chapter|book chapter guidelines]].&lt;/div&gt;
__TOC__


==Overview==
[[File:A picture is worth a thousand words.jpg|right|thumb|150px|'''Figure 1'''. Explore the topic, then develop a structure for the book chapter.]]
''Imagine'' ... a [[#Case studies|scenario or case study]] which illustrates the problem and engages reader interest. 
Consider including an image (see Figure 1). The scenario could be presented in a [[#Feature box|feature box]].

The Overview is typically 180 to 330 words.

{{tip|Suggestions for this section:
* Engage the reader with a scenario, example, or case study, and an accompanying image
* Explain the problem and why it is important
* Outline how psychological science can help
* Present focus questions
}}

{{RoundBoxTop|theme=3}}
'''Focus questions:'''


{{RoundBoxBottom}}

==Main headings==
* Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]
* Sub-headings can also be used, but avoid having sections with only one sub-heading

==Key points==
* Provide at least 3 bullet-points per heading, including for the Overview and Conclusion
* Include key citations

==Learning features==
* Interactive learning features bring online book chapters to life and can be embedded throughout the chapter.

===Case studies===
* Case studies describe real-world examples of concepts in action.
* Case studies can be real or fictional.
* A case study could be split into multiple sections throughout a chapter to illustrate different theories or stages.
* It is often helpful to present case studies using [[#Feature boxes|feature boxes]].

===Feature boxes===
* Important content can be highlighted in a feature box. But don't overuse feature boxes, otherwise they lose their effect. There are several  ways of creating boxes. Recommended: [[Help:Pretty boxes|Pretty boxes]]).
* Consider using feature boxes for:
** Focus questions
** [[#Case studies|Case studies]] or examples
** Quiz questions
** Take-home messages

 A very simple box can be created by using a space at the start of the line

{{RoundBoxTop|theme=3}}
;Feature box example
* Shaded background
* Coloured border
* Change the theme number for different colours
{{RoundBoxBottom}}

===Figures===
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* Use figures to illustrate concepts, add interest, and to serve as examples
* Figures can show photos, diagrams, graphs, etcetera
* Figures can be embedded throughout the chapter, including the Overview section
* Figures should be captioned (using '''Figure #.''' and a description). Captions explain the relevance of the image to the text/
* [[commons:|Wikimedia Commons]] provides a library of embeddable images
* Images can also be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Refer to each figure at least once in the main text (e.g., see Figure 2)

===Links===
* When key words are introduced, use [[Help:Links|interwiki links]]
* These links can go to:
** Wikipedia (e.g., [[w:Sigmund Freud|Sigmund Freud]] wrote about (e.g., [[w:Dreams|dreams]]) or
** Related book chapters (e.g., if your are struggling, you might be interested to read the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]])

===Tables===
* Use tables to organise and summarise information
* As with [[#Figures|figures]], tables should be captioned (e.g., see Table 1)
* Refer to each table at least once in the main text (e.g., see Table 1)
* [[Motivation and emotion/Wikiversity/Tables|Example 3 x 3 tables]] which could be adapted

'''Table 1.''' Descriptive Caption Which Explains The Table and its Relevant to the Text - Johari Window Model

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

===Quizzes===
* Using one or two review questions per major section is usually better than a long quiz at the end
* Quiz conceptual understanding, rather than trivia
* Don't make quizzes too hard
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{Quizzes are an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long quizzes are a good idea:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* The Conclusion is arguably the most important section
* The Conclusion is typically 150 to 330 words
* What are the take-home messages likely to be?
* It should be possible for someone to only read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the problem and what is known based on psychological science
{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
Provide up to 6 [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example:
* [[Motivation and emotion/Book/2021/Cognitive dissonance and motivation|Cognitive dissonance and motivation]] (Book chapter, 2021)
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[w:David McClelland|David McClelland]] (Wikipedia)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)
{{tip|Suggestions for this section:
* Present in alphabetical order
* Use [[w:Letter case#Sentence casing|sentence casing]]
* Include the source in parentheses
}}

==References==
List cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

APA style example:
{{Hanging indent|1=

Mercer-Mapstone, L., Dvorakova, S. L., Matthews, K. E., Abbot, S., Cheng, B., Felten, P., Knorr, K., Marquis, E., Shammas, R., &amp; Swaim, K. (2017). A systematic literature review of students as partners in higher education. ''International Journal for Students as Partners'', ''1''(1). https://doi.org/10.15173/ijsap.v1i1.3119

Rosenberg, B. D., &amp; Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091

Sears, C. R., Boyce, M. A., Boon, S. D., Goghari, V. M., Irwin, K., &amp; Boyes, M. (2017). Predictors of student satisfaction in a large psychology undergraduate program. ''Canadian Psychology/Psychologie Canadienne'', ''58''(2), 148–160. https://doi.org/10.1037/cap0000082
}}
{{tip|Suggestions for this section:
* Important aspects for APA style include:
** Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]. Use &quot;Edit source&quot;: &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case except first letter and proper names, ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** Provide the full doi as a URL and working hyperlink
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** Citing sources that weren't read or consulted
}}

==External links==
Provide up to 6 [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]]. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)
{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Present in alphabetical order
* Include the source in parentheses after the link
}}
[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[File:A picture is worth a thousand words.jpg|right|thumb|150px|'''Figure 1'''. Explore the topic, then develop a structure for the book chapter.]]
''Imagine'' ... a [[#Case studies|scenario or case study]] which illustrates the problem and engages reader interest. 
Consider including an image (see Figure 1). The scenario could be presented in a [[#Feature box|feature box]].

The Overview is typically 180 to 330 words.

{{tip|Suggestions for this section:
* Engage the reader with a scenario, example, or case study, and an accompanying image
* Explain the problem and why it is important
* Outline how psychological science can help
* Present focus questions
}}

{{RoundBoxTop|theme=3}}
'''Focus questions:'''
Break the problem (see the sub-title) down into three to five focus questions. Focus questions could also be used as top-level headings.
* What is the first focus question?
* What is the second focus question? 
* What is the third focus question?

Ask ''[[w:Open-ended question|open-ended]]'' focus questions. For example:
* ''Is there'' a relationship between motivation and success? (closed-ended) {{sad}}
* ''What is the'' relationship between motivation and success? (open-ended) {{smile}}
{{RoundBoxBottom}}

==Main headings==
* Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]
* Sub-headings can also be used, but avoid having sections with only one sub-heading

==Key points==
* Provide at least 3 bullet-points per heading, including for the Overview and Conclusion
* Include key citations

==Learning features==
* Interactive learning features bring online book chapters to life and can be embedded throughout the chapter.

===Case studies===
* Case studies describe real-world examples of concepts in action.
* Case studies can be real or fictional.
* A case study could be split into multiple sections throughout a chapter to illustrate different theories or stages.
* It is often helpful to present case studies using [[#Feature boxes|feature boxes]].

===Feature boxes===
* Important content can be highlighted in a feature box. But don't overuse feature boxes, otherwise they lose their effect. There are several  ways of creating boxes. Recommended: [[Help:Pretty boxes|Pretty boxes]]).
* Consider using feature boxes for:
** Focus questions
** [[#Case studies|Case studies]] or examples
** Quiz questions
** Take-home messages

 A very simple box can be created by using a space at the start of the line

{{RoundBoxTop|theme=3}}
;Feature box example
* Shaded background
* Coloured border
* Change the theme number for different colours
{{RoundBoxBottom}}

===Figures===
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* Use figures to illustrate concepts, add interest, and to serve as examples
* Figures can show photos, diagrams, graphs, etcetera
* Figures can be embedded throughout the chapter, including the Overview section
* Figures should be captioned (using '''Figure #.''' and a description). Captions explain the relevance of the image to the text/
* [[commons:|Wikimedia Commons]] provides a library of embeddable images
* Images can also be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Refer to each figure at least once in the main text (e.g., see Figure 2)

===Links===
* When key words are introduced, use [[Help:Links|interwiki links]]
* These links can go to:
** Wikipedia (e.g., [[w:Sigmund Freud|Sigmund Freud]] wrote about (e.g., [[w:Dreams|dreams]]) or
** Related book chapters (e.g., if your are struggling, you might be interested to read the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]])

===Tables===
* Use tables to organise and summarise information
* As with [[#Figures|figures]], tables should be captioned (e.g., see Table 1)
* Refer to each table at least once in the main text (e.g., see Table 1)
* [[Motivation and emotion/Wikiversity/Tables|Example 3 x 3 tables]] which could be adapted

'''Table 1.''' Descriptive Caption Which Explains The Table and its Relevant to the Text - Johari Window Model

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
|-
! !! Known to self !! Not known to self
|-
| '''Known to others''' || Open area || Blind spot
|-
| '''Not known to others''' || Hidden area || Unknown
|}

===Quizzes===
* Using one or two review questions per major section is usually better than a long quiz at the end
* Quiz conceptual understanding, rather than trivia
* Don't make quizzes too hard
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{Quizzes are an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long quizzes are a good idea:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

==Conclusion==
* The Conclusion is arguably the most important section
* The Conclusion is typically 150 to 330 words
* What are the take-home messages likely to be?
* It should be possible for someone to only read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the problem and what is known based on psychological science
{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
}}

==See also==
Provide up to 6 [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example:
* [[Motivation and emotion/Book/2021/Cognitive dissonance and motivation|Cognitive dissonance and motivation]] (Book chapter, 2021)
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[w:David McClelland|David McClelland]] (Wikipedia)
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)
{{tip|Suggestions for this section:
* Present in alphabetical order
* Use [[w:Letter case#Sentence casing|sentence casing]]
* Include the source in parentheses
}}

==References==
List cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

APA style example:
{{Hanging indent|1=

Mercer-Mapstone, L., Dvorakova, S. L., Matthews, K. E., Abbot, S., Cheng, B., Felten, P., Knorr, K., Marquis, E., Shammas, R., &amp; Swaim, K. (2017). A systematic literature review of students as partners in higher education. ''International Journal for Students as Partners'', ''1''(1). https://doi.org/10.15173/ijsap.v1i1.3119

Rosenberg, B. D., &amp; Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091

Sears, C. R., Boyce, M. A., Boon, S. D., Goghari, V. M., Irwin, K., &amp; Boyes, M. (2017). Predictors of student satisfaction in a large psychology undergraduate program. ''Canadian Psychology/Psychologie Canadienne'', ''58''(2), 148–160. https://doi.org/10.1037/cap0000082
}}
{{tip|Suggestions for this section:
* Important aspects for APA style include:
** Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]. Use &quot;Edit source&quot;: &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
** Title of work in lower case except first letter and proper names, ending in a full-stop
** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop
** Provide the full doi as a URL and working hyperlink
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** Citing sources that weren't read or consulted
}}

==External links==
Provide up to 6 [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]]. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)
{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Present in alphabetical order
* Include the source in parentheses after the link
}}
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[[Category:Class Lectures]]

I've asked ChatGPT about the Critical Theory on the Telegraph and it gave me the eight ways on how Critical Theory works with it. Before the telegraph was used by the highest power to communicate with other countries or businesses. Nobody could've had use of the telegraph until later on where the telephone was invented. Critical Theory even analyzed how it affected working conditions and employment patterns of telegraph operators. It also analyzed that the telegraph helped us understand a new language with Morse Code.https://chat.openai.com/share/8891bb96-ec2f-45fb-a82a-d6150c000622
{{:Digital_Media_and_Information_in_Society/Theoretical_Frameworks}}
{{:Digital_Media_and_Information_in_Society/Theoretical_Frameworks_Early_Electronics_Telegraphic_Telephonic_Photographic_Phonographic}}
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== Quantum Computing and Information ==
SecurityQuantum computing is an emerging field that has the potential to disrupt many areas of technology. It also presents a serious threat to all forms of information security systems. This page will describe how quantum computers can undermine the security of information and actions being taken to minimize/mitigate those threats. The Threats Quantum Computers Pose to Information SecurityCurrent encryption techniques rely on mathematical problems that cannot be solved in any reasonable extraordinary amount of time by classical (non-quantum) computers. Quantum computers are capable of conducting specific computations that are exponentially [faster than traditional computers and will directly undermine public key cryptography. Some of the main risks include:

1. Under mining the security of public key encryptographic methods Most secure communications today are based on public key cryptographic (PKC) algorithms such as the following: RSA (Rivest-Shamir-Adleman)Elliptic Curve Cryptography (ECC)Diffie-Hellman Key ExchangeThese PKC systems maintain security because they are based on either the difficultly in - factoring a large product of two prime numbers to recover the original numbers (RSA) or the difficulty in solving the discrete logarithm problem (Diffie-Hellman, ECC). In other words public key cryptography relies on a problem that a classical (non-quantum) computer cannot solve in an unreasonable amount of time. Shor's algorithm demonstrates that it is possible for a practical quantum computer to solve either of these problems efficiently, meaning that PKC algorithms based on these methods will become irrelevant once a large-scale quantum computer is practical.

2. Weaknesses Induced in Symmetric EncryptionWhile symmetric encryption approaches (i.e., AES - Advanced Encryption Standard) have some known resistance to quantum-based attacks, relative to public key cryptography; it should be remembered that Grover’s algorithm used in quantum computers allows an attacker to attack symmetric encryption systems in a way that, in effect, only cuts the strength of the encryption technique in half (in other words, only half the strength against a quantum-based attack). A simple hypothetical example would be:AES-256 would only be providing the same security as AES-128 against a quantum-based attack.This may effectively require the organization to double the length of their encryption keys to maintain an equivalent symmetric encryption security level.

3. Breaking of Digital SignaturesDigital signatures are important for establishing identities on the internet, securing transactions, and authenticating software. Breaking the cryptographic algorithms for digital signatures can lead to a number of broad consequences:Unauthorized modification of software.Identity theft. Monetizing the vulnerability of the blockchain.

4. &quot;Harvest Now, Decrypt Later&quot; AttacksAlthough effective quantum computers are not available just yet, an attacker can still intercept and store encrypted communications today to decrypt when a quantum computer is available. So, if viable quantum computers become attainable, secrets that were previously considered safe for long-term confidentiality may be compromised.5. Weakening Password Hashing AlgorithmsMany systems rely upon storing passwords as cryptographic hashes (e.g., SHA-256 or bcrypt). Quantum computers may weaken hashed passwords from stolen databases, significantly reducing the effort required for password cracking attempts.

5. Countering Quantum Threats: In response to potential risks from quantum computing, researchers have been focused on post-quantum cryptography (PQC) – encryption mechanisms that will be secure against quantum attacks. For example, there are Lattice-based cryptography (which relies on hard problems in high-dimensional lattice);Code based cryptography (which relies on error-correcting codes);Multivariate-quadratic cryptography (which relies on systems of multivariate polynomial equations). For their part, organizations such as the National Institute of Standards and Technology have also expanded their efforts to help facilitate a process of standardizing new encryption mechanisms for security going forward.  In conclusion, quantum computing represents a serious challenge to the current system of cryptographic security. The broad extent will not be realized for years to come. It is prudent to take active steps, including ways to phase in and implement post-quantum cryptography, or increase the length of encryption keys, to safeguard information that is sensitive.

=== Sources ===

* [https://csrc.nist.gov/Projects/post-quantum-cryptography Post-Quantum Cryptography Project]
* [[wikipedia:Shor's_algorithm|Shor's Algorithm Explained]]
* [[wikipedia:Grover's_algorithm|Grover’s]] [[wikipedia:Grover's_algorithm|Algorithm Overview]]
* [https://www.paloaltonetworks.com/cyberpedia/what-is-quantum-computings-threat-to-cybersecurity#:~:text=According%20to%20the%20Global%20Risk,old%20encryption%20methods%20potentially%20ineffective. &quot;What is Quantum Computing's Threat to Cyber Security&quot;]

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== Quantum Computing and Information ==
SecurityQuantum computing is an emerging field that has the potential to disrupt many areas of technology. It also presents a serious threat to all forms of information security systems. This page will describe how quantum computers can undermine the security of information and actions being taken to minimize/mitigate those threats. The Threats Quantum Computers Pose to Information SecurityCurrent encryption techniques rely on mathematical problems that cannot be solved in any reasonable extraordinary amount of time by classical (non-quantum) computers. [[Quantum computing|Quantum computers are capable of conducting specific computations that are exponentially [faster than traditional computers and will directly undermine public key cryptography.]] Some of the main risks include:

== Undermining the Security of Public Key Cryptographic Methods ==
Under mining the security of public key encryptographic methods Most secure communications today are based on public key cryptographic (PKC) algorithms such as the following: RSA (Rivest-Shamir-Adleman)Elliptic Curve Cryptography (ECC)Diffie-Hellman Key ExchangeThese PKC systems maintain security because they are based on either the difficultly in - factoring a large product of two prime numbers to recover the original numbers (RSA) or the difficulty in solving the discrete logarithm problem (Diffie-Hellman, ECC). In other words public key cryptography relies on a problem that a classical (non-quantum) computer cannot solve in an unreasonable amount of time. [https://www.classiq.io/insights/shors-algorithm-explained Shor's algorithm demonstrates that it is possible for a practical quantum computer to solve either of these problems efficiently], meaning that PKC algorithms based on these methods will become irrelevant once a large-scale quantum computer is practical.

== Weaknesses Induced in Symmetric Encryption ==
Weaknesses Induced in Symmetric Encryption While symmetric encryption approaches (i.e., AES - Advanced Encryption Standard) have some known resistance to quantum-based attacks, relative to public key cryptography; it should be remembered that [https://learning.quantum.ibm.com/course/fundamentals-of-quantum-algorithms/grovers-algorithm Grover’s algorithm used in quantum computers allows an attacker to attack symmetric encryption systems in a way that, in effect, only cuts the strength of the encryption technique in half (in other words, only half the strength against a quantum-based attack).] A simple hypothetical example would be:AES-256 would only be providing the same security as AES-128 against a quantum-based attack.This may effectively require the organization to double the length of their encryption keys to maintain an equivalent symmetric encryption security level.

== Breaking of Digital Signatures ==
Breaking of Digital Signatures Digital signatures are important for establishing identities on the internet, securing transactions, and authenticating software. Breaking the cryptographic algorithms for digital signatures can lead to a number of broad consequences:Unauthorized modification of software.Identity theft. Monetizing the vulnerability of the blockchain.

== Harvest and Decrypt Method ==
&quot;Harvest Now, Decrypt Later&quot; Attacks Although effective quantum computers are not available just yet, an attacker can still intercept and store encrypted communications today to decrypt when a quantum computer is available. So, [https://www.keyfactor.com/blog/harvest-now-decrypt-later-a-new-form-of-attack/ if viable quantum computers become attainable, secrets that were previously considered safe for long-term confidentiality may be compromised.]Weakening Password Hashing AlgorithmsMany systems rely upon storing passwords as cryptographic hashes (e.g., SHA-256 or bcrypt). Quantum computers may weaken hashed passwords from stolen databases, significantly reducing the effort required for password cracking attempts.

== Countering Quantum Threats ==
In response to potential risks from quantum computing, researchers have been focused on post-quantum cryptography (PQC) – encryption mechanisms that will be secure against quantum attacks. For example, there are Lattice-based cryptography (which relies on hard problems in high-dimensional lattice);Code based cryptography (which relies on error-correcting codes);Multivariate-quadratic cryptography (which relies on systems of multivariate polynomial equations). For their part, organizations such as the National Institute of Standards and Technology have also expanded their efforts to help facilitate a process of standardizing new encryption mechanisms for security going forward.  In conclusion, quantum computing represents a serious challenge to the current system of cryptographic security. The broad extent will not be realized for years to come. It is prudent to take active steps, including ways to phase in and implement post-quantum cryptography, or increase the length of encryption keys, to safeguard information that is sensitive.

== Sources ==
“Quantum Cryptography - Shor’s Algorithm Explained.” ''RSS'', Classiq Technologies, 19 July 2022, www.classiq.io/insights/shors-algorithm-explained.

“Grover’s Algorithm.” ''Grover’s Algorithm | IBM Quantum Learning'', learning.quantum.ibm.com/course/fundamentals-of-quantum-algorithms/grovers-algorithm. Accessed 6 Apr. 2025.

“Harvest Now, Decrypt Later: A New Form of Attack.” ''Keyfactor'', 26 Nov. 2024, www.keyfactor.com/blog/harvest-now-decrypt-later-a-new-form-of-attack/.

Computer Security Division, Information Technology Laboratory. “Post-Quantum Cryptography: CSRC.” ''CSRC'', csrc.nist.gov/projects/post-quantum-cryptography. Accessed 6 Apr. 2025.

“What Is Quantum Computing’s Threat to Cybersecurity?” ''Palo Alto Networks'', www.paloaltonetworks.com/cyberpedia/what-is-quantum-computings-threat-to-cybersecurity#:~:text=According%20to%20the%20Global%20Risk,old%20encryption%20methods%20potentially%20ineffective. Accessed 6 Apr. 2025.{{BookCat}}

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[[File:Sorbus torminalis Trunk and canopy.jpg|thumb|310px|The intracanopy of a Wild Service Tree, i.e. &lt;small&gt;''Torminalis glaberrima'' (Gand.) Sennikov &amp; Kurtto, ''Memoranda Soc. Fauna Fl. Fenn.'' 93: 32 (2017).&lt;/small&gt;]]&lt;br /&gt;
 
Most of my wiki contributions are made to [[:species:Main Page|Wikispecies]] where I'm an administrator, bureaucrat and interface admin,&lt;small&gt;&lt;sup&gt;[https://species.wikimedia.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Tommy_Kronkvist (verify)]&lt;/sup&gt;&lt;/small&gt; to the Swedish Wikimedia Chapter [[WMSE:|Wikimedia Sverige]] (WMSE) where I'm an administrator,&lt;small&gt;&lt;sup&gt;(&lt;span class=&quot;plainlinks&quot;&gt;[https://se.wikimedia.org/w/index.php?title=Special:Användare&amp;limit=1&amp;username=Tommy_Kronkvist verify]&lt;/span&gt;)&lt;/sup&gt;&lt;/small&gt; and as administrator and interface administrator at the Swedish version of [[wikivoyage:sv:Huvudsida|Wikivoyage]].&lt;small&gt;&lt;sup&gt;(&lt;span class=&quot;plainlinks&quot;&gt;[https://sv.wikivoyage.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Tommy_Kronkvist verify]&lt;/span&gt;)&lt;/sup&gt;&lt;/small&gt;

So far (June 14, 2026), I've made just over 393,300 edits to 153 of the Wikimedia sister projects&amp;nbsp;– the majority of them to Wikispecies and Wikidata. My global account information for all of Wikimedia can be found [[meta:Special:CentralAuth/Tommy Kronkvist|here]].

Swedish is my mother tongue&amp;nbsp;– even though I was born in Finland&amp;nbsp;– but I feel comfortable speaking and writing English and to some extent in German as well. Odd as it may seem, unfortunately I can't speak any Finnish even though I went to school there for a few years prior to moving to Sweden (see [[w:Swedish-speaking population of Finland|Swedish-speaking population of Finland]] in Wikipedia). I've lived all over Sweden but nowadays reside in Uppsala, the fourth biggest city and former capital of Sweden.

I'm only the fourth generation named &quot;Kronkvist&quot;. My family name consists of two parts: ''kron'' – a short form of the Swedish word ''krona'' meaning 'crown', as in coronation crown or tree crown – and ''kvist'', meaning 'bough' or 'twig'. Hence the name ''Kronkvist'' refers to a twig in the canopy of a forest. I'm the fourth generation of Kronkvist's. Prior to that our family name was ''Mattus'': an oeconym meaning &quot;Matthew's Farm&quot;, dating back to at least 1637.
{{Clear}}
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{{Original research}}
{{To be peer reviewed}}


== 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:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

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:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

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 &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== PDT transformation (probability reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''PDT transform''' &lt;math&gt;\widetilde{P}=\mathrm{PDT}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; 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:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The PDT-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

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:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

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:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive PDT transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the PDT transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;. 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:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative PDT transformation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

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:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the PDT update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

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=&quot;wikitable&quot;
! 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:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under iterative PDT dynamics:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

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=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! 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=&quot;wikitable&quot;
|+ 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:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; 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:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

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=&quot;wikitable&quot;
! 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:

&lt;math&gt;
D(x)
=
\exp\!\left(
\sum_k
\lambda_k
\exp\!\left(
-\frac{(x-x_k)^2}{2\sigma_k^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;x_k&lt;/math&gt; are the locations of the dilation peaks;
* &lt;math&gt;\lambda_k&gt;0&lt;/math&gt; control the amplification strength of each peak;
* &lt;math&gt;\sigma_k&lt;/math&gt; 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:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

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:

&lt;pre&gt;
Broad initial distribution
        ↓
Multiple localized amplifications
        ↓
Competing concentration regions
        ↓
Emergent multimodal probability structure
&lt;/pre&gt;

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:

&lt;math&gt;
D_n(x)
=
\exp\!\left(
\sigma \eta_n(x)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\eta_n(x)&lt;/math&gt; is a random field or stochastic fluctuation at iteration &lt;math&gt;n&lt;/math&gt;;
* &lt;math&gt;\sigma&gt;0&lt;/math&gt; 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:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D_n(x)P_n(x)
}{
\sum_y D_n(y)P_n(y)
}
&lt;/math&gt;

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:

&lt;pre&gt;
Broad initial distribution
        ↓
Random localized amplification
        ↓
Fluctuating concentration regions
        ↓
Dynamic probabilistic structure
&lt;/pre&gt;

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=&quot;wikitable&quot;
! 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 &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== 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 &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; 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}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{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:

&lt;math&gt;\eta = \frac{\sigma_P}{\sigma}&lt;/math&gt;

where:

&lt;math&gt;\sigma_P&lt;/math&gt; is a hypothesized minimal probability-resolution scale,
&lt;math&gt;\sigma&lt;/math&gt; 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:

&lt;math&gt;d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}&lt;/math&gt;

where:

&lt;math&gt;v&lt;/math&gt; is velocity,
&lt;math&gt;c&lt;/math&gt; is the speed of light,
&lt;math&gt;\eta = \sigma_P / \sigma&lt;/math&gt; encodes a proposed probability-resolution scale.

This expression is constructed such that standard special relativity is recovered exactly in the limit &lt;math&gt;\eta \to 0&lt;/math&gt;.

=== 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 &lt;math&gt;\sigma_P&lt;/math&gt; 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]]

==Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:==
== 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.

== Copyright and licensing ==

Text and original figures © Howard Richardson.

Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
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{{Original research}}
{{To be peer reviewed}}


== 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:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

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:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

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 &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== PDT transformation (probability reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''PDT transform''' &lt;math&gt;\widetilde{P}=\mathrm{PDT}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; 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:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The PDT-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

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:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

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:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive PDT transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the PDT transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;. 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:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative PDT transformation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

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:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the PDT update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

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=&quot;wikitable&quot;
! 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:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under iterative PDT dynamics:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

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=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! 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=&quot;wikitable&quot;
|+ 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:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; 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:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

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=&quot;wikitable&quot;
! 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:

&lt;math&gt;
D(x)
=
\exp\!\left(
\sum_k
\lambda_k
\exp\!\left(
-\frac{(x-x_k)^2}{2\sigma_k^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;x_k&lt;/math&gt; are the locations of the dilation peaks;
* &lt;math&gt;\lambda_k&gt;0&lt;/math&gt; control the amplification strength of each peak;
* &lt;math&gt;\sigma_k&lt;/math&gt; 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:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

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:

&lt;pre&gt;
Broad initial distribution
        ↓
Multiple localized amplifications
        ↓
Competing concentration regions
        ↓
Emergent multimodal probability structure
&lt;/pre&gt;

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:

&lt;math&gt;
D_n(x)
=
\exp\!\left(
\sigma \eta_n(x)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\eta_n(x)&lt;/math&gt; is a random field or stochastic fluctuation at iteration &lt;math&gt;n&lt;/math&gt;;
* &lt;math&gt;\sigma&gt;0&lt;/math&gt; 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:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D_n(x)P_n(x)
}{
\sum_y D_n(y)P_n(y)
}
&lt;/math&gt;

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:

&lt;pre&gt;
Broad initial distribution
        ↓
Random localized amplification
        ↓
Fluctuating concentration regions
        ↓
Dynamic probabilistic structure
&lt;/pre&gt;

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=&quot;wikitable&quot;
! 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 &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== 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 &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; 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}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{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:

&lt;math&gt;\eta = \frac{\sigma_P}{\sigma}&lt;/math&gt;

where:

&lt;math&gt;\sigma_P&lt;/math&gt; is a hypothesized minimal probability-resolution scale,
&lt;math&gt;\sigma&lt;/math&gt; 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:

&lt;math&gt;d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}&lt;/math&gt;

where:

&lt;math&gt;v&lt;/math&gt; is velocity,
&lt;math&gt;c&lt;/math&gt; is the speed of light,
&lt;math&gt;\eta = \sigma_P / \sigma&lt;/math&gt; encodes a proposed probability-resolution scale.

This expression is constructed such that standard special relativity is recovered exactly in the limit &lt;math&gt;\eta \to 0&lt;/math&gt;.

=== 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 &lt;math&gt;\sigma_P&lt;/math&gt; 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]]

==Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:==
== 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.

== Copyright and licensing ==

Text and original figures © Howard Richardson.

Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
      <sha1>nngsdwxxj9zlp4wnxh6n5h4kigt74bj</sha1>
    </revision>
  </page>
  <page>
    <title>Just sustainability transitions: a living review</title>
    <ns>0</ns>
    <id>326060</id>
    <revision>
      <id>2815490</id>
      <parentid>2815205</parentid>
      <timestamp>2026-06-13T13:20:35Z</timestamp>
      <contributor>
        <username>Jeanne Noiraud</username>
        <id>1366702</id>
      </contributor>
      <comment>/* Mapping a concept */</comment>
      <origin>2815490</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="64333" sha1="3cdk3mgdqelpsxixbe3qsyynfuj3wiv" xml:space="preserve">== Contributors ==
{| class=&quot;wikitable&quot;
|+
!Name
!Affiliation
!ORCID
!Contribution
|-
|Adélie Ranville
|IAE de Grenoble, CERAG lab (https://ror.org/0509qp208)
|https://orcid.org/0000-0002-3993-6135
|Research design, database search, article screening, knowledge modelling
|-
|Amélie Pereira
|
|
|Meta-data enrichement
|-
|
|
|
|
|}
Contribution statistics are visible here : https://xtools.wmcloud.org/pageinfo/en.wikiversity.org/Just_sustainability_transitions:_a_living_review

== Introduction ==

=== Definition of living review ===
The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked&lt;ref name=&quot;Why1&quot;&gt;{{Cite Q |Q40040379 }}&lt;/ref&gt;. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot;&gt;{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}&lt;/ref&gt;&lt;ref name=&quot;:6&quot;&gt;{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}&lt;/ref&gt;. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication&lt;ref name=&quot;Why1&quot;/&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;. Living systematic reviews exists to address this common problem&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/&lt;/ref&gt;. It is therefore particularly useful in rapidly evolving fields of research&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;, such as just transition. 

[[wikidata:Q33002955|Knowledge graphs]], a structured representation of knowledge in the form of a graph, linked together by relationships that encode explicit meanings between these entities, are very suitable for conducting living systematic reviews&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;Fotopoulou&quot;&gt;{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.&lt;/ref&gt;. Advances in AI could render certain older methodological types of living systematic reviews obsoletes&lt;ref&gt;{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}&lt;/ref&gt;, as IA are useful to extract, filter and classify datas&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;. [[Large language models]] (LLM) are &quot;on the rise&quot; (2025), but &quot;not yet ready for use&quot;&lt;ref&gt;{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}&lt;/ref&gt;.

The living review method relevant for just transition because it includes topic such as energy democracy which necessitate transdisciplinarity and consolidation of fragmented literature&lt;ref&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;.

===  Definitions of just transition : ===
* «a fair and equitable process of moving towards a post-carbon society’. »&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}&lt;/ref&gt;.


The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions&lt;ref name=&quot;:0&quot; /&gt;. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions&lt;ref name=&quot;:0&quot; /&gt;. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research &amp; Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}&lt;/ref&gt; and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 
 
=== Definition of Procedural justice ===
Procedural justice is about the fairness of decision-making processes related to transitions&lt;ref name=&quot;:4&quot; /&gt; such as the inclusion of those impacted by these decisions&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}&lt;/ref&gt;. Procedural justice can include issues of community and citizen participation in decision making, their political representation their consultation or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities&lt;ref&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. For example, the participation of affected communities in decisions related to the construction of new infrastructures&lt;ref name=&quot;:0&quot; /&gt;.     

== Methodology ==

=== Wikidata and the semantic web ===&lt;!-- Add introduction to what wikidata is and how the triplet works in a pedagogical manner

Example of good description here : https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k#fig1
 --&gt;
&quot;A knowledge graph is a structured representation of knowledge that captures information in a machine-readable format.&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}&lt;/ref&gt; A knowledge graph consists of a graph or network of interconnected data points, where each data point represents a piece of information or a concept, and the relationships between them are explicitly defined. Knowledge graphs organize and store data in a format that facilitates information retrieval, data analysis, and reasoning.&quot;&lt;ref&gt;{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}&lt;/ref&gt;

== Building a corpus and enriching bibliographic metadata ==

=== Database search ===
We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. 
 
{| class=&quot;wikitable&quot;
|+
!Keywords search
!Database
!Search date
!Filters
!Number of results
|-
|(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) 
|Web of Science (all databases, all dates)
|December 2025
|Document type: Review Article
|362
|}

=== Article screening ===
Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were 

* Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...)
* Not literature reviews (e.g. review of policies, initiatives, cases, review notes, book review...)
* Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors or included justice consideration only in “future research” suggestions
* Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy
* Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper

=== Importing selected articles into Wikidata ===
To import the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in the database. Next we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the &quot;export to Wikidata QuickStatements&quot; function of Zotero and use the QuickStatements tool to add them to Wikidata.

Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well  because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|&quot;Merge&quot; gadget]] on Wikidata. We checked manually for duplicated statments in those items.

=== Article classification through meta-data enrichement ===&lt;!-- Add : What is meta-data enrichement --&gt;

Existing review try to classify existing articles according to various criteria such as industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt).&lt;ref name=&quot;:5&quot; /&gt; We selected the most relevant properties in Wikidata to reflect these classifications : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe its geographical focus.

==== Main subjects ====
We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : 
{| class=&quot;wikitable&quot;
|+
!Qid
!Main topic
!Description
|-
|[[d:Q42377797|Q42377797]]
|acceptability
|characteristic of a thing being subject to acceptance for some purpose
|-
|[[d:Q2798912|Q2798912]]
|accountability
|concept of responsibility in ethics, governance and decision-making
|-
|[[d:Q421953|Q421953]]
|actor–network theory
|theory within social science
|-
|[[d:Q84459973|Q84459973]]
|affordability
|
|-
|[[d:Q185836|Q185836]]
|age of a person
|time elapsed since a person was born
|-
|[[d:Q4764988|Q4764988]]
|animal studies
|field in which animals are studied in a variety of cross-disciplinary ways
|-
|[[d:Q4338318|Q4338318]]
|awareness
|state or ability to perceive, to feel, or to be conscious of events, objects, or sensory patterns
|-
|[[d:Q4930066|Q4930066]]
|blue carbon
|carbon captured by the world's coastal ocean ecosystems
|-
|[[d:Q430460|Q430460]]
|capability approach
|economic theory
|-
|[[d:Q7569|Q7569]]
|child
|human between birth and puberty
|-
|[[d:Q4116870|Q4116870]]
|civic engagement
|individual or group activity addressing issues of public concern
|-
|[[d:Q125928|Q125928]]
|climate change
|human-caused changes to climate on Earth
|-
|[[d:Q260607|Q260607]]
|climate change 
adaptation
|process of adjustment to actual or expected climate change and its effects, seeking to moderate or avoid harm or exploit beneficial opportunities
|-
|[[d:Q1291678|Q1291678]]
|climate justice
|term linking the climate crisis with environmental and social justice
|-
|[[d:Q2270945|Q2270945]]
|co-creation
|product or service design process in which input from consumers plays a central role
|-
|[[d:Q16972712|Q16972712]]
|co-design
|approach to design attempting to actively involve all stakeholders
|-
|[[d:Q16324410|Q16324410]]
|coproduction
|product or service design process in which input from consumers plays a central role
|-
|[[d:Q11024|Q11024]]
|communication
|act of conveying intended meaning
|-
|[[d:Q177634|Q177634]]
|community
|social unit of human organisms who share common values
|-
|[[d:Q5154673|Q5154673]]
|community choice aggregation
|alternative energy supply system
|-
|[[d:Q113514984|Q113514984]]
|community energy
|delivery of community-led renewable energy, energy demand reduction and energy supply projects
|-
|[[d:Q65807646|Q65807646]]
|community participation
|The taking part by members of a community in decisionmaking processes related to the development of their community
|-
|[[d:Q188843|Q188843]]
|cosmopolitanism
|ideology that all human beings belong to a single community, based on a shared morality
|-
|[[d:Q11693783|Q11693783]]
|decarbonization
|change of economy, especially of energy industries, towards lower carbon dioxide emissions
|-
|[[d:Q284289|Q284289]]
|deliberative democracy
|form of democracy focusing on consensus
|-
|[[d:Q7174|Q7174]]
|democracy
|form of government
|-
|[[d:Q552284|Q552284]]
|distributive justice
|concept of the socially just allocation of goods
|-
|[[d:Q1230584|Q1230584]]
|diversity
|concept in sociology and political studies
|-
|[[d:Q1049066|Q1049066]]
|ecological economics
|research field on the interdependence of human economies and natural ecosystems
|-
|[[d:Q8134|Q8134]]
|economics
|social science that studies the production, distribution, and consumption of goods and services
|-
|[[d:Q868575|Q868575]]
|empowerment
|providing increased autonomy
|-
|[[d:Q295865|Q295865]]
|ecosystem service
|benefits created by nature, forests and environmental systems
|-
|[[d:Q138359220|Q138359220]]
|energy citizenship
|involvement of citizens in energy-related decisions
|-
|[https://www.wikidata.org/w/index.php?title=Q131444737&amp;redirect=no Q131444737]
|community energy
|[redirection]
|-
|[[d:Q16869822|Q16869822]]
|energy consumption
|amount of energy or power used
|-
|[[d:Q1358789|Q1358789]]
|senior
|elderly person
|-
|[[d:Q14944319|Q14944319]]
|energy democracy
|concept in environmental justice movement
|-
|[[d:Q192704|Q192704]]
|energy efficiency
|ratio between the useful energy output and the input of a machine
|-
|[[d:Q24965464|Q24965464]]
|energy modeling
|process of building computer models of energy systems in order to analyze them
|-
|[[d:Q1805337|Q1805337]]
|energy policy
|policy addressing energy issues
|-
|[[d:Q1341244|Q1341244]]
|energy poverty
|lack of access to modern energy services
|-
|[[d:Q3406659|Q3406659]]
|energy production
|conversion of energy from a primary source into a form useful to humans
|-
|[[d:Q117091181|Q117091181]]
|energy justice
|subconcept of economic equality
|-
|[[d:Q3456219|Q3456219]]
|energy renovation
|building works aimed at reducing energy consumption and decarbonising the energy sources used
|-
|[[d:Q2700433|Q2700433]]
|energy security
|national security considerations of energy availability
|-
|[[d:Q837718|Q837718]]
|energy storage
|capture of energy produced at one time for use at a later time
|-
|[[d:Q795757|Q795757]]
|energy transition
|long-term structural change towards sustainable energy systems
|-
|[[d:Q1479527|Q1479527]]
|environmental justice
|system of fairness
|-
|[[d:Q771773|Q771773]]
|fairness
|concept in sociology and generally the interaction of society
|-
|[[d:Q56395513|Q56395513]]
|farming system
|method of agricultural production defined by its physical practices and economic characteristics
|-
|[[d:Q5465532|Q5465532]]
|food system
|all processes and infrastructure involved in feeding a population
|-
|[[d:Q4421|Q4421]]
|forest
|dense collection of trees covering a relatively large area
|-
|[[d:Q48277|Q48277]]
|gender
|social concept which distinguish the different gender categories
|-
|[[d:Q1553864|Q1553864]]
|governance
|all of the processes of governing, whether undertaken by a government, market or network, whether over a family, tribe, formal or informal organization or territory and whether through the laws, norms, power or language of an organized society
|-
|[[d:Q8458|Q8458]]
|human rights
|inalienable fundamental rights to which a person is inherently entitled
|-
|[[d:Q11376059|Q11376059]]
|human rights violation
|act or omission which contravene the principles of human rights
|-
|[[d:Q103817|Q103817]]
|indigenous people
|first inhabitants of an area and their descendants
|-
|[[d:Q113561794|Q113561794]]
|indigenous science
|indigenous knowledge applied to the scientific method
|-
|[[d:Q770480|Q770480]]
|injustice
|quality relating to unfairness or undeserved outcomes
|-
|[[d:Q17142211|Q17142211]]
|interactional justice
|the perceived appropriateness of interpersonal treatment
|-
|[[d:Q1516555|Q1516555]]
|intersectionnality
|theoretical framework of multidimensional oppression
|-
|[[d:Q6316391|Q6316391]]
|just transition
|Framework developed by the trade union movement to encompass wide range of social interventions needed to secure decent work opportunities and a greener economy.
|-
|[[d:Q366139|Q366139]]
|legitimation
|the process of making something acceptable and normative to a group
|-
|[[d:Q3027857|Q3027857]]
|living lab
|user-centered, open innovation ecosystem integrating research and innovation in real life communities
|-
|[[d:Q59679511|Q59679511]]
|low income
|home with little money
|-
|[[d:Q43619|Q43619]]
|natural environment
|all living and non-living things occurring naturally on Earth or some region thereof
|-
|[[d:Q127514833|Q127514833]]
|nature-positive
|global goal to halt and reverse nature loss by 2030
|-
|[[d:Q13023682|Q13023682]]
|non-human
|organism not in the genus Homo
|-
|[[d:Q728646|Q728646]]
|partnership
|arrangement in which parties agree to cooperate to advance their mutual interests
|-
|[[d:Q3907287|Q3907287]]
|policy making
|the act of developing policy
|-
|[[d:Q9357091|Q9357091]]
|political theory
|class of theory
|-
|[[d:Q265425|Q265425]]
|postcolonialism
|academic discipline
|-
|[[d:Q25107|Q25107]]
|power
|ability to influence the behavior of others
|-
|[[d:Q442100|Q442100]]
|procedural justice
|fairness in the processes that resolve disputes and allocate resources
|-
|[[d:Q7249406|Q7249406]]
|project governance
|management framework
|-
|[[d:Q7257735|Q7257735]]
|public engagement
|Policy-making practice
|-
|[[d:Q541936|Q541936]]
|public participation
|participation of citizens in various policy decisions and planning processes
|-
|[[d:Q6142016|Q6142016]]
|recognition justice
|social philosophy theory
|-
|[[d:Q10509953|Q10509953]]
|renewable electricity
|electricity from renweable sources
|-
|[[d:Q12705|Q12705]]
|renewable energy
|energy collected from renewable resources
|-
|[[d:Q56510941|Q56510941]]
|renewable energy policy
|
|-
|[[d:Q1165392|Q1165392]]
|restorative justice
|approach to justice where victims and perpetrators mediate a restitution agreement
|-
|[[d:Q4414036|Q4414036]]
|rural population
|inhabitants of rural areas or of small towns classified as rural
|-
|[[d:Q17152351|Q17152351]]
|smart system
|adaptive intelligent systems
|-
|[[d:Q187588|Q187588]]
|social class
|group of people categorized in a hierarchy based on socioeconomic factors
|-
|[[d:Q264892|Q264892]]
|social justice
|concept that discrimination recognized in society should be remedied
|-
|[[d:Q34749|Q34749]]
|social science
|academic disciplines concerned with society and the relationships between individuals in society
|-
|[[d:Q2930198|Q2930198]]
|stakeholder participation
|involvement of groups or individuals affected by the actions of an entity
|-
|[[d:Q125359881|Q125359881]]
|sustainability transition
|
|-
|[[d:Q219416|Q219416]]
|sustainability
|ability of human civilization to coexist with the biosphere in a steady state
|-
|[[d:Q131201|Q131201]]
|sustainable development 
|mode of human development that meets current demands without compromising the needs of future generations
|-
|[[d:Q7649586|Q7649586]]
|Sustainable Development Goals
|set of United Nations-defined global development goals and climate change
|-
|[[d:Q69883|Q69883]]
|urban planning
|technical and political process concerned with the use of land and design of the urban environment
|-
|[[d:Q920600|Q920600]]
|urban renewal
|program of land redevelopment in cities, often where there is urban decay
|-
|[[d:Q3376054|Q3376054]]
|vulnerable population
|group of persons whose range of options is severely limited, are subjected to coercion, or who may be compromised in their ability to give informed consent
|-
|[[d:Q107389921|Q107389921]]
|water-management
|
|-
|[[d:Q7981051|Q7981051]]
|well-being
|measure of how well life is to someone or a group with factors such as health, happiness and satisfaction
|-
|[[d:Q467|Q467]]
|woman
|female adult human
|-
|[[d:Q188867|Q188867]]
|future studies
|study of possible, probable, and preferable social, technological and political futures
|-
|[[d:Q1038171|Q1038171]]
|participatory design
|active involvement of all stakeholders in the design process
|}
&lt;!-- include all below items using the wikidata link template
 --&gt;

Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords.

==== Study types ====
Our review included only litterature reviews. We first read abstracts to identify all the [https://angryloki.github.io/wikidata-graph-builder/?item=Q2412849&amp;property=P279&amp;mode=reverse different types of litterature reviews] present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved these method items using the methodological references cited in the reviewed papers. The types of reviews were : 
{| class=&quot;wikitable&quot;
|+
!Qid
!Study type
!Description
|-
|[[d:Q603441|Q603441]]
|bibliometrics
|statistical analysis of written publications, such as books or articles
|-
|[[d:Q472342|Q472342]]
|scientometrics
|study of measuring and analysing science, technology and innovation
|-
|[[d:Q815382|Q815382]]
|meta-analysis
|statistical method that summarizes data from multiple sources
|-
|[[d:Q1504425|Q1504425]]
|systematic review
|publication type, study that gathers, analyzes, and communicates the results of research and information on a topic
|-
|[[d:Q2412849|Q2412849]]
|literature review
|process of information search and text of a review article (Q7318358), which includes the current knowledge including substantive findings, as well as theoretical and methodological contributions to a particular topic
|-
|[[d:Q6822263|Q6822263]]
|meta-regression
|statistical tool used in meta-analyses
|-
|[[d:Q7301211|Q7301211]]
|realist evaluation
|[...]
|-
|[[d:Q17007303|Q17007303]]
|combinatorial meta-analysis 
|[...]
|-
|[[d:Q70470634|Q70470634]]
|network meta-analysis
|meta-analysis of randomized trials in which estimates of comparative treatment effects are visualized and interpreted from a network of interventions
|-
|[[d:Q101116078|Q101116078]]
|scoping review
|search for concepts by mapping the language and data which surrounds those concepts and adjusting the search method iteratively to synthesize evidence and assess the scope of an area of inquiry
|-
|[[d:Q110665014|Q110665014]]
|narrative review
|type of literature review, without structured method of retrieval and analysis
|-
|[[d:Q137174203|Q137174203]]
|conceptual review
|academic research aiming to review existing concepts and definitions in the litterature
|-
|[[d:Q137174450|Q137174450]]
|critical review
|type of literature review analysing strenghts, major contributions, mistakes and neglected issues in an academic field of research
|-
|[[d:Q137209848|Q137209848]]
|integrative literature review
|type of literature review
|-
|[[d:Q110665014|Q137211242]]
|narrative review
|type of literature review, without structured method of retrieval and analysis
|}Then, we added the {{Wikidata entity link|P8363}} of each articles based on the abstract and method sections. In case of doubt, we compared our interpretation.

==== Research site ====
When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article &quot;{{Wikidata entity link|Q137901202}}&quot; focused on {{Wikidata entity link|Q132959}}.

==== Results ====
The table listing all the papers in the sample can be visualized [https://tabernacle.toolforge.org/?#/tab/manual/Q137211155%0A%0A%0A%0A%0A%0A%0AQ114306483%0A%0A%0A%0A%0AQ137901181%0A%0A%0A%0AQ137901182%0A%0A%0A%0A%0A%0A%0A%0AQ137901183%0A%0A%0AQ114306476%0A%0A%0A%0A%0AQ137901184%0A%0A%0A%0A%0AQ137901185%0A%0A%0A%0A%0A%0AQ137901186%0A%0A%0A%0A%0A%0A%0AQ137901187%0A%0A%0A%0A%0A%0A%0AQ137901188%0A%0A%0A%0A%0AQ137210566%0A%0A%0A%0A%0AQ114306511%0A%0A%0A%0A%0A%0AQ137901191%0A%0A%0A%0A%0AQ137901192%0A%0A%0A%0A%0AQ137901193%0A%0A%0A%0A%0AQ135979013%0A%0A%0A%0A%0A%0A%0A%0AQ137901195%0A%0A%0A%0A%0A%0AQ137901196%0A%0A%0A%0A%0A%0A%0AQ137901197%0A%0A%0A%0A%0AQ136447761%0A%0A%0A%0AQ137901199%0A%0A%0A%0A%0A%0A%0AQ129652515%0A%0A%0A%0A%0A%0A%0AQ137901201%0A%0A%0A%0A%0A%0AQ137901202%0A%0A%0A%0A%0AQ137901203%0A%0A%0A%0AQ137901204%0A%0A%0A%0A%0A%0A%0A%0AQ137901205%0A%0A%0A%0A%0AQ137901206%0A%0A%0A%0A%0A%0A%0A%0A%0AQ137901207%0A%0A%0A%0A%0AQ129203992%0A%0A%0A%0A%0A%0A%0AQ114197507%0A%0A%0A%0AQ137901161%0A%0A%0A%0A%0A%0A%0A%0AQ137901209%0A%0A%0A%0A%0A%0AQ137901210%0A%0A%0A%0A%0A%0AQ137901211%0A%0A%0A%0A%0AQ11420462%0A%0AQ137901213%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0AQ104887325%0A%0A%0A%0A%0A%0AQ137901162%0A%0A%0AQ137901163%0A%0A%0A%0A%0AQ137901164%0A%0A%0A%0A%0A%0AQ137901215%0A%0A%0A%0A%0AQ137901216%0A%0A%0A%0A%0A%0A%0A%0A%0AQ137901217%0A%0A%0A%0A%0AQ115448818%0A%0A%0A%0A%0AQ137901218%0A%0A%0A%0AQ137901219%0A%0A%0A%0A%0AQ137901220%0A%0A%0A%0A%0A%0AQ137901221%0A%0A%0A%0A%0A%0AQ137901222%0A%0A%0A%0A%0AQ137901223%0A%0A%0AQ137901224%0A%0A%0A%0AQ137901225%0A%0A%0A%0A%0A%0A%0AQ137901226%0A%0A%0A%0AQ137901227%0A%0A%0AQ137901182/Len%3BP921%3BP6153%3BP8363%3BP50 here] (be careful if you are logged into Wikidata as the table is editable).

== Modelling knowledge ==
Concept maps can be a powerful literature review tool&lt;ref&gt;{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}&lt;/ref&gt; allowing to synthetize theoretical statements about relationship between concepts&lt;ref&gt;{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}&lt;/ref&gt;. In the present study, we explored how concept map can be used to model the knowledge present in the paper we selected. 

[define knowledge modelling]

==== Wikidata ontology ====
Wikidata &quot;supports multiple coexisting classification&quot; and allow multiple ontological frameworks to coexist.&lt;ref name=&quot;:8&quot;&gt;{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}&lt;/ref&gt;

It also supports epistemic pluralism : different worldviews can be represented in wikidata, even though scientific knowledge is preferred.&lt;ref name=&quot;:8&quot; /&gt;

See more on membership properties : https://www.wikidata.org/wiki/Help:Basic_membership_properties

See the discussion on cause modelling : https://www.wikidata.org/wiki/Help:Modeling_causes/en

==== Conceptual modelling ====
We first reflected on what kind of wikidata properties could be used to represent concepts and theories in wikidata. Capturing the content of a concept is not straightforward and there are various approaches coming from psychology and philosophy on the matter&lt;ref&gt;{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}&lt;/ref&gt; we summarize these approaches below and examine which wikidata properties exist to represent them.
* Definition: the content of a concept can be formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}...
* Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}}.
* Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}.
* Essence: the content of a concept is &quot;something&quot; deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can create it before really knowing what all its properties will be.
* Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}.

==== Categorization and conceptualisation practices in management sciences ====
In management sciences « systematic categorizing is the best and perhaps only method for clearing up semantic confusion, management scholars never take the classical approaches to categorizing that facilitated tremendous progress in the physical sciences, and seldomly build on extant categorial schemes. »&lt;ref&gt;{{Cite journal|last=Pierce|first=Jason R.|date=2025-01|title=Categorizing Concepts and Phenomena in Management Research: A Four-Phase Integrative Review and Recommendations|url=http://journals.aom.org/doi/full/10.5465/annals.2023.0052|journal=Academy of Management Annals|language=en|volume=19|issue=1|page=28|pages=9–37|doi=10.5465/annals.2023.0052|issn=1941-6520}}&lt;/ref&gt;.

Some scholars discussed how conceptualization should be done&lt;ref&gt;{{Cite journal|last=Podsakoff|first=Philip M.|last2=MacKenzie|first2=Scott B.|last3=Podsakoff|first3=Nathan P.|date=2016-04|title=Recommendations for Creating Better Concept Definitions in the Organizational, Behavioral, and Social Sciences|url=https://journals.sagepub.com/doi/10.1177/1094428115624965|journal=Organizational Research Methods|language=en|volume=19|issue=2|pages=159–203|doi=10.1177/1094428115624965|issn=1094-4281}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Makowski|first=Piotr Tomasz|date=2021-10|title=Optimizing Concepts: Conceptual Engineering in the Field of Management—The Case of Routines Research|url=http://journals.aom.org/doi/full/10.5465/amr.2019.0252|journal=Academy of Management Review|language=en|volume=46|issue=4|pages=702–724|doi=10.5465/amr.2019.0252|issn=0363-7425}}&lt;/ref&gt;.

==== Thematic networks ====
[[File:Thematic network example.jpg|thumb|547x547px|Structure of a thematic network (Source: Attride-Stirling 2001)]]
A thematic network is “simply a way of organizing a thematic analysis of qualitative data”&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}&lt;/ref&gt;. It is compatible with classical coding strategies such as [[grounded theory]]&lt;ref&gt;{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}&lt;/ref&gt;. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data&lt;ref name=&quot;:7&quot; /&gt;. The principle is to assemble basic themes into more general themes.   

Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding (e.g. grounded theory) to identify themes and sub-themes.   

However, the nature of the relationship between these various themes and sub-themes is often not specified.  

*

==== Causal networks ====
The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers sometime present models with boxes and arrows representing correlations and/or causations&lt;ref&gt;{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}&lt;/ref&gt;. In qualitative research, building grounded theory models is about &quot;[accounting] for not only all the major emergent concepts, themes, and dimensions, but also for their dynamic interrelationships. Speaking in classic boxes-and-arrows terms, this process amounts to assembling the constellation of boxes with a special focus on the arrows.&quot;&lt;ref&gt;{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}&lt;/ref&gt; Researchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can &quot;feedback&quot; (two variables can influence each other)&lt;ref&gt;{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}&lt;/ref&gt;.

Wikidata includes several properties to describe causal relationships:
* {{Wikidata entity link|P828}}
* {{Wikidata entity link|P1542}}
* {{Wikidata entity link|P1537}}
* {{Wikidata entity link|P1479}} : it is difficult to identify single causes for social phenomenons, many factors having an effect on the subject item will likely be contributing factors

== Testing concept modelling on {{Wikidata entity link|Q14944319}} ==
We started by experimenting the modelling of concept by focusing on the concept of {{Wikidata entity link|Q14944319}}. We selected a subset of papers which had energy democracy as main topic :

* {{Wikidata entity link|Q137901202}}
* {{Wikidata entity link|Q137901196}}
* {{Wikidata entity link|Q137901182}}
* {{Wikidata entity link|Q136447761}}
* {{Wikidata entity link|Q129652515}}
* {{Wikidata entity link|Q114306483}}

We read each paper and used them as source to enter statements in the item {{Wikidata entity link|Q14944319}}. For example, &quot;Energy democracy is both an ideal and a process&quot;&lt;ref&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;, we thus entered the wikidata statement {{Wikidata entity link|Q14944319}} is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q840396}}, using the paper as source. The result of this first step is visible in the archival version of the item (22 May 2026) here https://www.wikidata.org/w/index.php?title=Q14944319&amp;oldid=2495982191.

Ontology challenges:
*{{Wikidata entity link|P31}}: concepts may have a dual nature because they designate at the same time an idea and the entity that this idea represent. Energy democracy is a concept, an ideal, a process and an outcome. 
*'''Process versus outcome :''' For material processes, the distinction between process and outcome is rather simple. For example, in Wikidata, {{Wikidata entity link|Q11629}} (practice of applying paint) is different from {{Wikidata entity link|Q3305213}} (visual artwork), and this distinction is based on the criterion &quot;{{Wikidata entity link|Q127270577}}&quot;. However, this distinction is less straightforward for social processes that do not have an end. Such processes are ongoing and outcomes cannot be separated as clearly.
* '''Ideal versus reality :''' Concepts do not have goals in themselves, but the reality they represent can have goals. To distinguish goals from the process to reach it, we used {{Wikidata entity link|P3712}} to describe ideals and {{Wikidata entity link|P2670}} to describe processes.
* '''Phenomenon versus theory :''' Wikidata current items are not really suited to model &quot;meta-research&quot; statements. For example, modelling the idea tha the literature on energy democracy is fragmented would require creating an item representing the energy democracy literature, not just energy democracy in general. Similarly, it can be difficult to model the chronological evolution of the definition of an idea (although it could be technically possible). It is hard to represent in Wikidata affirmations related to missing knowlege, propositions of untested hypothesis, critique of existing research or research agenda recommandations
* '''Origin of discourses versus origin of practices :''' To distinguish the causes of the concepts/discourses and the causes of the phenomenon itself, we used {{Wikidata entity link|P3938}} to indicate the origins of the concept or the movements promoting it.


Some of the statements we added may seem contradictory. However, Wikidata supports &quot;because statements essentially point to referenceable sources of information and different sources may provide contradicting information, it's possible to represent a plurality of perspectives on Wikidata&quot;&lt;ref&gt;{{Cite web|url=https://www.wikidata.org/wiki/Help:Statements#Plurality_and_consensus|title=Help:Statements - Wikidata|website=www.wikidata.org|language=en|access-date=2026-06-08}}&lt;/ref&gt;. The {{Wikidata entity link|Q14944319}} concept could be split into more precise concepts to distinguish the social movement advocating for it, the political concept theorizing it and the concrete initiatives implementing it. However, the current sources do not make this distinction for now. 

Other challenges
* Wikidata does not seem to be the best tool to model quantitative statements, for example, the paper {{Wikidata entity link|Q137901196}} states that &quot;9.8% of the final energy consumed in developing countries comes from modern renewable energy sources&quot;. Including energy data in Wikidata require using or creating specific properties (e.g. {{Wikidata entity link|P6826}})
* When concepts are not precisely defined, statements cannot be modelled correctly. For example, in the sentence &quot;management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served&quot;&lt;ref&gt;{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research &amp; Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}&lt;/ref&gt;, &quot;choice&quot; could refer to {{Wikidata entity link|Q111986453}}, {{Wikidata entity link|Q1331926}}, or {{Wikidata entity link|Q12888920}} as &quot;choice&quot; can refer to the availability of different options, or the decision process to chose among them.


Advantages : 

* Link toward unique identifiers for concepts, but also laws (e.g. {{Wikidata entity link|Q139764294}})

== Interactions with the Wikidata community ==

* Some Wikidata contributors added labels for {{Wikidata entity link|Q14944319}} in other languages such as Armenian or Slovenian.

== Data visualisation ==

=== Filter statements ===

* Visualize only statements using a specitic source. Example : https://w.wiki/PFqH
* Visualize only items which are part to the present project (require that all items of the project include the statement {{Wikidata entity link|P6104}} {{Wikidata entity link|Q134545539}}).

=== Mapping a concept ===
Scholia request &quot;topic in context&quot; : [https://query.wikidata.org/#%23%20tool%3A%20scholia%0A%20%20%20%20%20%20%20%20PREFIX%20target%3A%20%3Chttp%3A%2F%2Fwww.wikidata.org%2Fentity%2FQ14944319%3E%0A%23defaultView%3AGraph%0APREFIX%20wd%3A%20%3Chttp%3A%2F%2Fwww.wikidata.org%2Fentity%2F%3E%0APREFIX%20wdt%3A%20%3Chttp%3A%2F%2Fwww.wikidata.org%2Fprop%2Fdirect%2F%3E%0APREFIX%20wikibase%3A%20%3Chttp%3A%2F%2Fwikiba.se%2Fontology%23%3E%0APREFIX%20rdf%3A%20%3Chttp%3A%2F%2Fwww.w3.org%2F1999%2F02%2F22-rdf-syntax-ns%23%3E%0A%0ASELECT%20%3Fnode%20%3FnodeLabel%20%3FnodeImage%20%3FchildNode%20%3FchildNodeLabel%20%3FchildNodeImage%20%3Frgb%20WHERE%20%7B%0A%20%20%7B%0A%20%20%20%20%7B%0A%20%20%20%20%20%20SELECT%20DISTINCT%20%3Fnode%20%3FchildNode%20WHERE%20%7B%0A%20%20%20%20%20%20%20%20BIND%20%28target%3A%20AS%20%3Fnode%29%0A%20%20%20%20%20%20%20%20%3Fnode%20%3Fp%20%3Fi%20.%0A%20%20%20%20%20%20%20%20%3FchildNode%20%3Fx%20%3Fp%20.%0A%20%20%20%20%20%20%20%20%3FchildNode%20rdf%3Atype%20wikibase%3AProperty.%0A%20%20%20%20%20%20%20%20FILTER%20%28STRSTARTS%28STR%28%3Fi%29%2C%22http%3A%2F%2Fwww.wikidata.org%2Fentity%2FQ%22%29%29%0A%20%20%20%20%20%20%20%20FILTER%20%28STRSTARTS%28STR%28%3FchildNode%29%2C%22http%3A%2F%2Fwww.wikidata.org%2Fentity%2FP%22%29%29%0A%20%20%20%20%20%20%7D%0A%20%20%20%20%20%20LIMIT%205000%0A%20%20%20%20%7D%0A%20%20%7D%0A%20%20UNION%20%7B%0A%20%20%20%20%7B%0A%20%20%20%20%20%20SELECT%20DISTINCT%20%3FchildNode%20%3Fnode%20%3Frgb%20WHERE%20%7B%0A%20%20%20%20%20%20%20%20BIND%20%28%22EFFBD8%22%20AS%20%3Frgb%29%0A%20%20%20%20%20%20%20%20target%3A%20%3Fp%20%3FchildNode%20.%0A%20%20%20%20%20%20%20%20%3Fnode%20%3Fx%20%3Fp%20.%0A%20%20%20%20%20%20%20%20%3Fnode%20rdf%3Atype%20wikibase%3AProperty.%0A%20%20%20%20%20%20%20%20FILTER%20%28STRSTARTS%28STR%28%3FchildNode%29%2C%22http%3A%2F%2Fwww.wikidata.org%2Fentity%2FQ%22%29%29%0A%20%20%20%20%20%20%7D%0A%20%20%20%20%20%20LIMIT%205000%0A%20%20%20%20%7D%0A%20%20%7D%0A%20%20OPTIONAL%20%7B%0A%20%20%20%20%7B%0A%20%20%20%20%20%20SELECT%20DISTINCT%20%3Fproperty%20WHERE%20%7B%0A%20%20%20%20%20%20%20%20%3Fproperty%20a%20wikibase%3AProperty%20%3B%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20wdt%3AP31%20wd%3AQ18610173%20%3B%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20wdt%3AP31%20wd%3AQ26940804%20.%0A%20%20%20%20%20%20%7D%0A%20%20%20%20%7D%0A%20%20%20%20%3Fproperty%20wikibase%3AdirectClaim%20%3Fnodeclaim%20.%0A%20%20%20%20%3Fnode%20%3Fnodeclaim%20%3FnodeImage%20.%0A%20%20%7D%0A%20%20OPTIONAL%20%7B%0A%20%20%20%20%7B%0A%20%20%20%20%20%20SELECT%20DISTINCT%20%3Fproperty%20WHERE%20%7B%0A%20%20%20%20%20%20%20%20%3Fproperty%20a%20wikibase%3AProperty%20%3B%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20wdt%3AP31%20wd%3AQ18610173%20%3B%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20wdt%3AP31%20wd%3AQ26940804%20.%0A%20%20%20%20%20%20%7D%0A%20%20%20%20%7D%0A%20%20%20%20%3Fproperty%20wikibase%3AdirectClaim%20%3FchildNodeclaim%20.%0A%20%20%20%20%3FchildNode%20%3FchildNodeclaim%20%3FchildNodeImage%20.%0A%20%20%7D%0A%0A%20%20OPTIONAL%20%7B%20%3Fnode%20%3Chttp%3A%2F%2Fwww.w3.org%2F2000%2F01%2Frdf-schema%23label%3E%20%3FnodeLabel.%20FILTER%28LANG%28%3FnodeLabel%29%20%3D%20%22fr%22%29%20%7D%0A%20%20%20%20OPTIONAL%20%7B%20%3Fnode%20%3Chttp%3A%2F%2Fwww.w3.org%2F2000%2F01%2Frdf-schema%23label%3E%20%3FnodeLabel.%20FILTER%28LANG%28%3FnodeLabel%29%20%3D%20%22fr-FR%22%29%20%7D%0A%20%20%20%20OPTIONAL%20%7B%20%3Fnode%20%3Chttp%3A%2F%2Fwww.w3.org%2F2000%2F01%2Frdf-schema%23label%3E%20%3FnodeLabel.%20FILTER%28LANG%28%3FnodeLabel%29%20%3D%20%22en-US%22%29%20%7D%0A%20%20%20%20OPTIONAL%20%7B%20%3Fnode%20%3Chttp%3A%2F%2Fwww.w3.org%2F2000%2F01%2Frdf-schema%23label%3E%20%3FnodeLabel.%20FILTER%28LANG%28%3FnodeLabel%29%20%3D%20%22en%22%29%20%7D%0A%20%20%20%20OPTIONAL%20%7B%20%3Fnode%20%3Chttp%3A%2F%2Fwww.w3.org%2F2000%2F01%2Frdf-schema%23label%3E%20%3FnodeLabel.%20FILTER%28LANG%28%3FnodeLabel%29%20%3D%20%22mul%22%29%20%7D%0A%20%20%20%20%0A%20%20%20%20OPTIONAL%20%7B%20%3FchildNode%20%3Chttp%3A%2F%2Fwww.w3.org%2F2000%2F01%2Frdf-schema%23label%3E%20%3FchildNodeLabel.%20FILTER%28LANG%28%3FchildNodeLabel%29%20%3D%20%22fr%22%29%20%7D%0A%20%20%20%20OPTIONAL%20%7B%20%3FchildNode%20%3Chttp%3A%2F%2Fwww.w3.org%2F2000%2F01%2Frdf-schema%23label%3E%20%3FchildNodeLabel.%20FILTER%28LANG%28%3FchildNodeLabel%29%20%3D%20%22fr-FR%22%29%20%7D%0A%20%20%20%20OPTIONAL%20%7B%20%3FchildNode%20%3Chttp%3A%2F%2Fwww.w3.org%2F2000%2F01%2Frdf-schema%23label%3E%20%3FchildNodeLabel.%20FILTER%28LANG%28%3FchildNodeLabel%29%20%3D%20%22en-US%22%29%20%7D%0A%20%20%20%20OPTIONAL%20%7B%20%3FchildNode%20%3Chttp%3A%2F%2Fwww.w3.org%2F2000%2F01%2Frdf-schema%23label%3E%20%3FchildNodeLabel.%20FILTER%28LANG%28%3FchildNodeLabel%29%20%3D%20%22en%22%29%20%7D%0A%20%20%20%20OPTIONAL%20%7B%20%3FchildNode%20%3Chttp%3A%2F%2Fwww.w3.org%2F2000%2F01%2Frdf-schema%23label%3E%20%3FchildNodeLabel.%20FILTER%28LANG%28%3FchildNodeLabel%29%20%3D%20%22mul%22%29%20%7D%0A%20%20%20%20%0A%7D Example with Energy democracy]

=== Mapping sources consensus ===
Visualise graphs and use the number of references to determine edge thickness/weight.

== Writing ==
To cite articles we used the [[Template:Cite Q|Cite Q template.]] Each reference is an item in Wikidata and the template retrieve the necessary data to generate the citation references below.


== Future research ==
The analysis of knowledge graph could in theory allow to make logical deduction to generate new data&lt;ref name=&quot;:9&quot; /&gt;.

Reflect on the future of scholarly communication : https://hal.science/hal-03277615/file/OPERAS_Future_of_Scholarly_Communication_06.2021.pdf

== Data == 
{| class=&quot;wikitable sortable&quot;
! QID !! Year !! DOI !! Title
|-
| [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review
|-
| [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review
|-
| [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review
|-
| [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter?
|-
| [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset.
|-
| [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies?
|-
| [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection
|-
| [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development
|-
| [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research
|-
| [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition
|-
| [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning
|-
| [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review
|-
| [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view
|-
| [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory
|-
| [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries
|-
| [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review
|-
| [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions
|-
| [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies
|-
| [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes
|-
| [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation
|-
| [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives
|-
| [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies
|-
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== References ==
{{References}}</text>
      <sha1>3cdk3mgdqelpsxixbe3qsyynfuj3wiv</sha1>
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  <page>
    <title>User:Dc.samizdat/Golden chords of the 120-cell</title>
    <ns>2</ns>
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        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 600-cell */</comment>
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      <text bytes="75118" sha1="8fhk94owqqyv9i62t9gn4hu49azg0pc" xml:space="preserve">= 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}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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), 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 5&lt;sup&gt;2&lt;/sup&gt; 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 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). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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 pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], 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 of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. 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. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
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:

:&lt;math&gt;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&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; 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]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;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&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of 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 &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic 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 &lt;math&gt;r_3&lt;/math&gt; 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° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° 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 &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation 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.

Because this is the rotation of the 16-cell taking its edge planes to each other, we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
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 &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; 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]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. 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 tetrahedra]]. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

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 &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. 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:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; 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}, which has chords:

:&lt;math&gt;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}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, 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.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; 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's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[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]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

In the 600-cell an isoclinic rotation 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.   

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. 

...

== Finally 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 &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== 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 isocclinic rotation characteristic 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 ''n'' dimensions. Application of the Fontaine and Hurley procedure in 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 &amp; 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 &amp; 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}}</text>
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      <text bytes="75118" sha1="6jd6r4gpjqlo721ogozdhbbyzvv8je4" xml:space="preserve">= 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}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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), 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 5&lt;sup&gt;2&lt;/sup&gt; 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 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). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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 pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], 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 of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. 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. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
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:

:&lt;math&gt;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&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; 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]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;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&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of 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 &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic 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 &lt;math&gt;r_3&lt;/math&gt; 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° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° 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 &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation 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.

Because this is the rotation of the 16-cell taking its edge planes to each other, we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
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 &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; 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]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. 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 tetrahedra]]. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

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 &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. 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:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; 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}, which has chords:

:&lt;math&gt;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}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, 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.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; 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's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

[[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]]
Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

In the 600-cell an isoclinic rotation 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.   

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. 

...

== Finally 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 &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== 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 isocclinic rotation characteristic 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 ''n'' dimensions. Application of the Fontaine and Hurley procedure in 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 &amp; 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 &amp; 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}}</text>
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      <text bytes="75118" sha1="rsy8sth5qggw22jp0od1olo93oos0i6" xml:space="preserve">= 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}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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), 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 5&lt;sup&gt;2&lt;/sup&gt; 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 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). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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 pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], 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 of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. 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. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
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|1.640~
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|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
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|1.618~
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|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
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|{{radic|2.5}}
|
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|- style=&quot;background: palegreen;&quot; |
|1.224~
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|1.581~
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|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
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|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
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| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
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|{{radic|2.309~}}
|
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|- style=&quot;background: gainsboro;&quot; |
|1.300~
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|1.520~
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|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
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|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
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|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
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|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
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|- style=&quot;background: seashell;&quot; |
|1.414~
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|1.414~
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|
|}

== The 8-point regular polytopes ==
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:

:&lt;math&gt;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&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; 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]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;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&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of 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 &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic 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 &lt;math&gt;r_3&lt;/math&gt; 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° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° 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 &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation 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.

Because this is the rotation of the 16-cell taking its edge planes to each other, we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
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 &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; 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]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. 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 tetrahedra]]. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

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 &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. 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:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; 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}, which has chords:

:&lt;math&gt;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}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; 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 &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, 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 these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

[[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]]
Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

In the 600-cell an isoclinic rotation 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.   

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. 

...

== Finally 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 &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== 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 isocclinic rotation characteristic 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 ''n'' dimensions. Application of the Fontaine and Hurley procedure in 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 &amp; 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 &amp; 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}}</text>
      <sha1>rsy8sth5qggw22jp0od1olo93oos0i6</sha1>
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      <text bytes="75118" sha1="nbc5e951mojxd8eafgy2beglws2rxdc" xml:space="preserve">= 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}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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), 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 5&lt;sup&gt;2&lt;/sup&gt; 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 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). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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 pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], 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 of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. 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. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
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:

:&lt;math&gt;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&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; 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]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;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&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of 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 &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic 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 &lt;math&gt;r_3&lt;/math&gt; 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° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° 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 &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation 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.

Because this is the rotation of the 16-cell taking its edge planes to each other, we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
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 &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; 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]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. 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 tetrahedra]]. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

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 &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. 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:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; 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}, which has chords:

:&lt;math&gt;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}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; 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 &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, 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 these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

[[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]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

In the 600-cell an isoclinic rotation 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.   

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. 

...

== Finally 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 &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== 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 isocclinic rotation characteristic 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 ''n'' dimensions. Application of the Fontaine and Hurley procedure in 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 &amp; 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 &amp; 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}}</text>
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      <text bytes="75133" sha1="s7s9teo1ioyp2to9gk77cib3de6iqab" xml:space="preserve">= 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}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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), 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 5&lt;sup&gt;2&lt;/sup&gt; 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 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). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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 pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], 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 of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. 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. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
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:

:&lt;math&gt;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&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; 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]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;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&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of 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 &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic 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 &lt;math&gt;r_3&lt;/math&gt; 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° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° 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 &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation 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.

Because this is the rotation of the 16-cell taking its edge planes to each other, we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
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 &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; 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]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. 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 tetrahedra]]. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

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 &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. 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:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; 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}, which has chords:

:&lt;math&gt;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}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; 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 &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, 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 these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

[[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]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

In the 600-cell an isoclinic rotation 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.   

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. 

...

== Finally 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 &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== 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 isocclinic rotation characteristic 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 ''n'' dimensions. Application of the Fontaine and Hurley procedure in 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 &amp; 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 &amp; 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}}</text>
      <sha1>s7s9teo1ioyp2to9gk77cib3de6iqab</sha1>
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    <revision>
      <id>2815505</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="75133" sha1="92m6jp0qpmj3dtsztz682apelfwrvch" xml:space="preserve">= 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}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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), 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 5&lt;sup&gt;2&lt;/sup&gt; 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 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). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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 pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], 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 of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. 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. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
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:

:&lt;math&gt;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&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; 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]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;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&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of 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 &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic 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 &lt;math&gt;r_3&lt;/math&gt; 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° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° 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 &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation 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.

Because this is the rotation of the 16-cell taking its edge planes to each other, we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
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 &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; 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]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. 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 tetrahedra]]. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

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 &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. 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:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

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 &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; 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}, which has chords:

:&lt;math&gt;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}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; 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 &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, 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 these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

[[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]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

In the 600-cell an isoclinic rotation 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.   

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. 

...

== Finally 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 &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== 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 isocclinic rotation characteristic 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 ''n'' dimensions. Application of the Fontaine and Hurley procedure in 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 &amp; 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 &amp; 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}}</text>
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      <text bytes="75133" sha1="s7s9teo1ioyp2to9gk77cib3de6iqab" xml:space="preserve">= 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}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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), 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 5&lt;sup&gt;2&lt;/sup&gt; 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 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). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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 pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], 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 of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. 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. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
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:

:&lt;math&gt;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&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; 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]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;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&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of 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 &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic 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 &lt;math&gt;r_3&lt;/math&gt; 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° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° 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 &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation 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.

Because this is the rotation of the 16-cell taking its edge planes to each other, we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
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 &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; 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]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. 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 tetrahedra]]. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

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 &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. 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:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; 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}, which has chords:

:&lt;math&gt;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}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; 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 &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, 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 these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

[[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]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

In the 600-cell an isoclinic rotation 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.   

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. 

...

== Finally 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 &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== 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 isocclinic rotation characteristic 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 ''n'' dimensions. Application of the Fontaine and Hurley procedure in 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 &amp; 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 &amp; 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}}</text>
      <sha1>s7s9teo1ioyp2to9gk77cib3de6iqab</sha1>
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    <revision>
      <id>2815507</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="75137" sha1="nokel2zrq24bbvo0lbke2ki58gpjin9" xml:space="preserve">= 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}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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), 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 5&lt;sup&gt;2&lt;/sup&gt; 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 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). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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 pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], 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 of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. 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. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
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:

:&lt;math&gt;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&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; 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]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;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&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of 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 &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic 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 &lt;math&gt;r_3&lt;/math&gt; 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° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° 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 &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation 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.

Because this is the rotation of the 16-cell taking its edge planes to each other, we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
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 &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; 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]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. 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 tetrahedra]]. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

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 &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. 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:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; 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}, which has chords:

:&lt;math&gt;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}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, and also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; 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 &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, 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 these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

[[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]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

In the 600-cell an isoclinic rotation 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.   

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. 

...

== Finally 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 &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== 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 isocclinic rotation characteristic 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 ''n'' dimensions. Application of the Fontaine and Hurley procedure in 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 &amp; 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 &amp; 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}}</text>
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      <text bytes="75135" sha1="lwew2qvmbgis0eouj78nu14u2llk4m0" xml:space="preserve">= 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}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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), 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 5&lt;sup&gt;2&lt;/sup&gt; 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 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). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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 pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], 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 of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. 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. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
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:

:&lt;math&gt;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&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; 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]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;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&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of 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 &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;2\pi&lt;/math&gt;, 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 &lt;math&gt;r_3&lt;/math&gt; 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° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° 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 &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation 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.

Because this is the rotation of the 16-cell taking its edge planes to each other, we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
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 &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; 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]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. 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 tetrahedra]]. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

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 &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. 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:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; 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}, which has chords:

:&lt;math&gt;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}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, and also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; 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 &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, 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 these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

[[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]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

In the 600-cell an isoclinic rotation 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.   

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. 

...

== Finally 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 &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== 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 isocclinic rotation characteristic 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 ''n'' dimensions. Application of the Fontaine and Hurley procedure in 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 &amp; 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 &amp; 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}}</text>
      <sha1>lwew2qvmbgis0eouj78nu14u2llk4m0</sha1>
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    <revision>
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        <username>Dc.samizdat</username>
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      <text bytes="75163" sha1="da9fl0o8t3cxrtibz9mv7glpyep8ann" xml:space="preserve">= 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}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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), 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 5&lt;sup&gt;2&lt;/sup&gt; 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 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). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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 pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], 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 of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. 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. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
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:

:&lt;math&gt;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&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; 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]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;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&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of 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 &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;2\pi&lt;/math&gt;, 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 &lt;math&gt;r_3&lt;/math&gt; 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° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° 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 &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation 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.

Because this is the rotation of the 16-cell taking its edge planes to each other, we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
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 &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; 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]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. 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 tetrahedra]]. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

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 &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. 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:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; 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}, which has chords:

:&lt;math&gt;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}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, and also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; 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 &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, 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 these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

[[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]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

In the 600-cell an isoclinic rotation 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.   

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. 

...

== Finally 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 &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== 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 isocclinic rotation characteristic 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 ''n'' dimensions. Application of the Fontaine and Hurley procedure in 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 &amp; 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 &amp; 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}}</text>
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      <text bytes="75163" sha1="hn4ct61fmoss4cuzt2ottifylswlqae" xml:space="preserve">= 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}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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), 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 5&lt;sup&gt;2&lt;/sup&gt; 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 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). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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 pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], 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 of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. 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. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
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:

:&lt;math&gt;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&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; 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]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;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&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of 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 &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;2\pi&lt;/math&gt;, 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 &lt;math&gt;r_3&lt;/math&gt; 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° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° 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 &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation 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.

Because this is the rotation of the 16-cell taking its edge planes to each other, we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
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 &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; 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]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. 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 tetrahedra]]. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

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 &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. 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:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; 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}, which has chords:

:&lt;math&gt;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}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, and also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; 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 &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, 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 these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

[[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]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

In the 600-cell an isoclinic rotation 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.   

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. 

...

== Finally 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 &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== 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 isocclinic rotation characteristic 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 ''n'' dimensions. Application of the Fontaine and Hurley procedure in 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 &amp; 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 &amp; 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}}</text>
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      <id>2815520</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* The 5-point (5-cell) 4-simplex */</comment>
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      <text bytes="75173" sha1="2crjpu2xi95804ufyk0pdy27a215b8q" xml:space="preserve">= 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}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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), 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 5&lt;sup&gt;2&lt;/sup&gt; 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 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). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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 pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], 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 of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. 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. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
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:

:&lt;math&gt;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&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; 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]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;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&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of 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 &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;2\pi&lt;/math&gt;, 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 &lt;math&gt;r_3&lt;/math&gt; 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° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° 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 &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation 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.

Because this is the rotation of the 16-cell taking its edge planes to each other, we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
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 &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; 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]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. 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 tetrahedra]]. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

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 &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. 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:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; 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}, which has chords:

:&lt;math&gt;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}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, and also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; 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 &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, 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 these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

[[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]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

In the 600-cell an isoclinic rotation 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.   

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. 

...

== Finally 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 &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== 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 isocclinic rotation characteristic 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 ''n'' dimensions. Application of the Fontaine and Hurley procedure in 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 &amp; 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 &amp; 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}}</text>
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      <text bytes="75155" sha1="dvypl3esiudmw53wa25k76z6anzptdt" xml:space="preserve">= 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}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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), 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 5&lt;sup&gt;2&lt;/sup&gt; 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 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). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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 pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], 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 of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. 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. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 16-cell ==
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:

:&lt;math&gt;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&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; 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]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;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&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of 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 &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;2\pi&lt;/math&gt;, 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 &lt;math&gt;r_3&lt;/math&gt; 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° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° 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 &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation 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.

Because this is the rotation of the 16-cell taking its edge planes to each other, we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
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 &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; 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]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. 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 tetrahedra]]. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

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 &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. 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:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; 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}, which has chords:

:&lt;math&gt;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}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, and also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; 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 &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, 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 these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

[[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]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

In the 600-cell an isoclinic rotation 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.   

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. 

...

== Finally 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 &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== 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 isocclinic rotation characteristic 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 ''n'' dimensions. Application of the Fontaine and Hurley procedure in 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 &amp; 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 &amp; 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}}</text>
      <sha1>dvypl3esiudmw53wa25k76z6anzptdt</sha1>
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      <id>2815522</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="75167" sha1="4x0omgxp6elwln1c217il9qfkkb5814" xml:space="preserve">= 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}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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), 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 5&lt;sup&gt;2&lt;/sup&gt; 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 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). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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 pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], 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 of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. 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. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== 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:

:&lt;math&gt;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&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; 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]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;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&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of 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 &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;2\pi&lt;/math&gt;, 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 &lt;math&gt;r_3&lt;/math&gt; 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° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° 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 &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation 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.

Because this is the rotation of the 16-cell taking its edge planes to each other, we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
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 &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; 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]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. 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 tetrahedra]]. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

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 &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. 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:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; 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}, which has chords:

:&lt;math&gt;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}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, and also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; 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 &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, 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 these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

[[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]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

In the 600-cell an isoclinic rotation 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.   

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. 

...

== Finally 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 &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== 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 isocclinic rotation characteristic 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 ''n'' dimensions. Application of the Fontaine and Hurley procedure in 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 &amp; 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 &amp; 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}}</text>
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      <text bytes="75148" sha1="1xn4nkyu9k4y3148ld08pld9vn56v3o" xml:space="preserve">= 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}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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), 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 5&lt;sup&gt;2&lt;/sup&gt; 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 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). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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 pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], 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 of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. 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. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== 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:

:&lt;math&gt;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&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; 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]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;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&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of 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 &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;2\pi&lt;/math&gt;, 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 &lt;math&gt;r_3&lt;/math&gt; 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° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° 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 &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation 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.

Because this is the rotation of the 16-cell taking its edge planes to each other, we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
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 &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; 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]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. 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 tetrahedra]]. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

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 &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. 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:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; 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}, which has chords:

:&lt;math&gt;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}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, and also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; 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 &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, 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 these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

[[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]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

In the 600-cell an isoclinic rotation 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.   

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. 

...

== Finally 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 &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== 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 isocclinic rotation characteristic 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 ''n'' dimensions. Application of the Fontaine and Hurley procedure in 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 &amp; 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 &amp; 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}}</text>
      <sha1>1xn4nkyu9k4y3148ld08pld9vn56v3o</sha1>
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    <revision>
      <id>2815559</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="75308" sha1="dyawv499dww11tihfze0y498n7ofp9u" xml:space="preserve">= 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}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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), 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 5&lt;sup&gt;2&lt;/sup&gt; 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 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). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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 pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], 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 of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. 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. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== 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:

:&lt;math&gt;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&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; 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]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;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&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of 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 &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;2\pi&lt;/math&gt;, 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 &lt;math&gt;r_3&lt;/math&gt; 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° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° 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 &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation 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.

Because this is the rotation of the 16-cell taking its edge planes to each other, we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
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 &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; 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]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. 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 tetrahedra]]. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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 isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

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 &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. 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:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; 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}, which has chords:

:&lt;math&gt;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}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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 &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_3&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, and also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; 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 &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, 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 these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

[[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]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

In the 600-cell an isoclinic rotation 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.   

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. 

...

== Finally 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 &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== 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 isocclinic rotation characteristic 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 ''n'' dimensions. Application of the Fontaine and Hurley procedure in 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 &amp; 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 &amp; 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}}</text>
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      <text bytes="75124" sha1="34csiz7c3wb1rj8dovx516c62ef1435" xml:space="preserve">= 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}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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), 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 5&lt;sup&gt;2&lt;/sup&gt; 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 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). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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 pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], 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 of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. 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. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== 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:

:&lt;math&gt;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&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; 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]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;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&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of 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 &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;2\pi&lt;/math&gt;, 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 &lt;math&gt;r_3&lt;/math&gt; 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° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° 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 &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation 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.

Because this is the rotation of the 16-cell taking its edge planes to each other, we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
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 &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; 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]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. 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 tetrahedra]]. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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 &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which visits each vertex of the tesseract once.

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 &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. 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:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; 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}, which has chords:

:&lt;math&gt;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}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_3&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; 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 &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, 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 these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

[[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]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

In the 600-cell an isoclinic rotation 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.   

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. 

...

== Finally 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 &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== 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 isocclinic rotation characteristic 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 ''n'' dimensions. Application of the Fontaine and Hurley procedure in 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 &amp; 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 &amp; 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}}</text>
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      <text bytes="75114" sha1="ly2gdesjk6ad7jjrb5bh1p3qyjm0yua" xml:space="preserve">= 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}}

&lt;blockquote&gt;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.&lt;/blockquote&gt;
== 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=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;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]].&quot;}} 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=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{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), 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 5&lt;sup&gt;2&lt;/sup&gt; 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 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). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (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 &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; 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 &lt;math&gt;\phi^6&lt;/math&gt; 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=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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 pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], 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 of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. 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. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== 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:

:&lt;math&gt;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&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; 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:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

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:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

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 &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; 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]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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:

:&lt;math&gt;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&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; 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.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of 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 &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; 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 &lt;math&gt;2\pi&lt;/math&gt;, 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 &lt;math&gt;r_3&lt;/math&gt; 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° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° 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 &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation 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.

Because this is the rotation of the 16-cell taking its edge planes to each other, we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
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 &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; 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]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. 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 tetrahedra]]. 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes 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 &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which visits each vertex of the tesseract once.

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 &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. 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:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; 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 &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; 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}, which has chords:

:&lt;math&gt;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}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; 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 &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_3&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; 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 &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. 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 between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, 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 these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

[[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]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation 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 &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

In the 600-cell an isoclinic rotation 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.   

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
== The 5-cell 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. 

...

== Finally 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 &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== 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 isocclinic rotation characteristic 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 &lt;math&gt;\sqrt{2}&lt;/math&gt; 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 ''n'' dimensions. Application of the Fontaine and Hurley procedure in 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 &amp; 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 &amp; 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}}</text>
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  <page>
    <title>Quantum Computing Algorithms in the NISQ Era/Quiz</title>
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      <id>2815616</id>
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      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
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      <minor />
      <comment>added [[Category:Algorithms]] using [[Help:Gadget-HotCat|HotCat]]</comment>
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      <text bytes="14943" sha1="hu70ib8ijf8htz48o88hf0g33uzx83f" xml:space="preserve">{{Author|Harold Foppele}}
{{Physics}}
{{Quiz}}
{{Learning project}}
=Learning Page: Quantum Computing Algorithms in the NISQ Era=
This sub-learning page provides a structured overview of quantum computing algorithms during the Noisy Intermediate-Scale Quantum (NISQ) era, based on key concepts from the Wikiversity resource. It includes summaries of main topics, explanations, examples, and visual aids to facilitate learning. At the end, there's a quiz to test your understanding, along with a conclusion summarizing key takeaways.

==Introduction to Quantum Computing in the NISQ Era==

Quantum computing represents a shift from classical computing by using '''qubits''' instead of bits. Qubits can exist in '''superposition''' (multiple states at once), '''entanglement''' (linked states), and leverage '''interference''' to process information in parallel. This enables potential speedups in areas like molecular simulation, optimization, machine learning, and search problems.

The NISQ era refers to current quantum devices with 50–1000 qubits that are noisy (error-prone) and lack full error correction. Despite limitations, hybrid quantum-classical approaches are advancing fields such as chemistry, materials science, logistics, finance, and AI. Key milestones include demonstrations of &quot;quantum supremacy&quot; and practical applications like modeling hydrogen chains or optimizing financial portfolios.

'''Key Terms:'''

* '''Quantum Advantage''': When quantum computers outperform classical ones for specific tasks.
* '''NISQ''': Noisy Intermediate-Scale Quantum – the current phase of quantum tech.

Example: In 2026, advancements like JPMorgan's quantum streaming for large datasets highlight real-world potential (source:[https://www.jpmorganchase.com/technology/quantum-computing JPMorgan Chase Research]

To visualize superposition, consider this simple image [[File:Riemann_Spin2States.jpg|thumb]]
'''Discussion Questions:'''

* How might the principles of superposition and entanglement change the way we approach problem-solving in fields like drug discovery or climate modeling?
* Discuss the ethical implications of achieving quantum advantage in areas such as cryptography or financial optimization. Who benefits, and who might be at risk?
* Compare the NISQ era to the early days of classical computing. What lessons from history could guide the development of quantum technologies?

==NISQ Algorithms Overview==

NISQ algorithms are hybrid methods that combine quantum circuits with classical optimization to handle noise without full error correction. They rely on the '''variational principle''', where the expectation value of a Hamiltonian (energy operator) is minimized to find approximate solutions: ⟨ψ|H|ψ⟩ ≥ E₀, where E₀ is the ground state energy.

'''Workflow:'''

* 1. Prepare a parameterized quantum state |ψ(θ)⟩ using a circuit (ansatz) from an initial state like |0⟩.
* 2. Measure the cost function C(θ) = ⟨ψ(θ)|H|ψ(θ)⟩, often decomposed into 3. Pauli operators (e.g., &lt;math&gt;H = \sum_k c_k P_k&lt;/math&gt;).&lt;br&gt; Use classical optimizers (e.g., gradient descent) to adjust parameters θ.

Challenges include '''barren plateaus''' (flat optimization landscapes where gradients vanish) and noise, often mitigated with AI techniques.

Example: Variational Quantum Algorithms (VQAs) for chemistry (e.g., molecular energy calculations) or finance (portfolio optimization). For more, explore IBM's Qiskit tutorials [https://qiskit.org/documentation/ Qiskit Documentation]

'''Discussion Questions:'''

* In what ways do hybrid quantum-classical approaches in NISQ algorithms bridge the gap between current hardware limitations and future fault-tolerant quantum computing?
* How could barren plateaus impact the scalability of NISQ algorithms, and what role might AI play in overcoming this challenge?
* Discuss potential real-world applications of VQAs beyond chemistry and finance. How might they transform industries like logistics or environmental science?

==Specific Algorithms==

===Variational Quantum Eigensolver (VQE)===

VQE approximates the ground or excited states of molecular Hamiltonians, useful in quantum chemistry. It uses an ansatz like the Unitary Coupled Cluster (UCC) to create trial states and minimizes the energy expectation value.

'''Mechanism:'''

* Generate |ψ(θ)⟩ with a parameterized circuit.
* Compute and minimize ⟨ψ|H|ψ⟩ via measurements.
* Innovations: Adaptive VQE dynamically adds operators to reduce qubit needs; subspace methods for excited states.

Challenges: Barren plateaus addressed with AI (e.g., reinforcement learning for parameter tuning).

Example: Google's use for hydrogen chain modeling; applications in drug discovery by 2026 (source:[https://quantumai.google/ Google Quantum AI])

Try a simple VQE simulation in Python with Qiskit:

```python

from qiskit import QuantumCircuit
# Basic example code snippet
qc = QuantumCircuit(1)

qc.h(0)  # Superposition
# Add more for full VQE
```

'''Discussion Questions for VQE:'''

* How does the choice of ansatz in VQE affect its accuracy and efficiency? What trade-offs might researchers face when selecting one?
* Discuss the potential of adaptive VQE in reducing resource requirements. Could this lead to broader accessibility of quantum computing for smaller organizations?
* In the context of drug discovery, how might VQE simulations accelerate the development process, and what limitations remain due to noise?

===Quantum Approximate Optimization Algorithm (QAOA)===

QAOA tackles combinatorial optimization problems (e.g., MaxCut) by mapping them to Ising Hamiltonians.

'''Mechanism:'''

* Start with a superposition state |+⟩^⊗n.
* Apply alternating layers: e^{-i γ H_C} (problem Hamiltonian, e.g., ∑ Z_i Z_j for edges) and e^{-i β H_B} (mixer, ∑ X_i). - Optimize parameters γ and β to minimize ⟨ψ|H_C|ψ⟩.

Variants: Recursive QAOA (RQAOA) iteratively reduces problem size; warm-starting with AI.

Challenges: Local minima in parameter space; hardware-aware versions reduce errors.

Example: Logistics and finance applications, like supply chain optimization on 30-qubit systems by 2026 (source: [https://arxiv.org/abs/1411.4028 arXiv Paper on QAOA])

'''Discussion Questions for QAOA:'''

* How does QAOA's approach to combinatorial problems differ from classical optimization methods, and in what scenarios might it provide a clear advantage? Note that advantages are still debated in literature.
* Explore the role of variants like RQAOA. How could iterative problem reduction improve performance on NISQ devices?
* Discuss the integration of AI in warm-starting QAOA. What synergies between quantum and classical AI could emerge in optimization tasks?

===Other Algorithms===

* '''Shor's Algorithm''': Factors large integers in polynomial time using Quantum Fourier Transform (QFT) for periodicity finding. It threatens RSA encryption; NISQ versions factor small numbers (e.g., 15=3*5). Mechanism: Prepare superposition, apply modular exponentiation, use QFT to find period, then classical post-processing.

* '''Grover's Algorithm''': Provides quadratic speedup (O(√N)) for unstructured search via amplitude amplification. Useful in machine learning for feature selection. Mechanism: Initialize superposition, apply oracle to mark solutions, amplify amplitudes iteratively.

* '''Amplitude Amplification''': Generalizes Grover to boost probabilities; applied in anomaly detection. It iteratively reflects states to increase desired amplitudes.

For diagrams, see a QFT circuit (placeholder for image).

'''Discussion Questions for Other Algorithms:'''

* What are the security implications of Shor's Algorithm in the NISQ era, even with its current limitations on small-scale factoring?
* How might Grover's Algorithm enhance machine learning tasks like feature selection, and what challenges arise from implementing it on noisy hardware?
* Discuss amplitude amplification as a generalization of Grover's. In what innovative ways could it be applied beyond search problems, such as in signal processing?

==Quantum Machine Learning (QML)==

QML leverages quantum mechanics for machine learning tasks, offering potential exponential speedups.

'''Key Methods:'''
- '''Harrow-Hassidim-Lloyd (HHL)''': 

* Solves linear systems Ax = b using phase estimation. - Quantum kernels and neural networks for classification, generative models.
* NISQ adaptations: Amplitude amplification for kernels; hybrid classical-quantum setups.
* Challenges: Noise and decoherence; mitigated with AI like shadow tomography.

Example: Google's Quantum Echo for NMR spectra in biomedicine by 2026 (source: [https://research.google/ Google Research])
{|class=&quot;wikitable&quot;
|+ '''Comparison Table'''
! Algorithm
! Key Use Case
! NISQ Adaptation
! Classical Counterpart
|-
| HHL
| Linear systems
| Hybrid setups
| Gaussian elimination
|-
| Quantum Kernels
| Classification
| Amplitude amplification
| SVM kernels
|}

'''Discussion Questions:'''

* How could quantum kernels in QML provide advantages over classical kernels in tasks like classification, and what datasets might benefit most?
* Discuss the HHL algorithm's potential for solving linear systems. In fields like finance or engineering, where could it outperform classical methods?
* Explore the challenges of noise in QML. How might techniques like shadow tomography evolve to make QML more practical in the NISQ era?

==Challenges in NISQ Computing==

* '''Error Correction''': Needs thousands of physical qubits per logical one (e.g., surface codes); partial progress by 2026.
* '''Scalability''': Current limits (e.g., 127 qubits on IBM Eagle); hybrids via cloud services like AWS Braket.
* '''Barren Plateaus''': Mitigated by specialized ansatzes and AI.
* '''Noise Mitigation''': Techniques like zero-noise extrapolation and AI decoders.
* Other: Decoherence, data loading issues, and the risk of &quot;dequantization&quot; (classical simulations outperforming quantum).

Future: Co-design of algorithms and hardware; transition to fault-tolerant quantum computing (FTQC) by 2030s. Ethical considerations include environmental impact from energy use and access inequality.

'''Discussion Questions:'''

* What strategies could accelerate the transition from NISQ to fault-tolerant quantum computing, and what role should governments or industries play?
* Discuss &quot;dequantization&quot; risks. How might classical simulations challenge the perceived advantages of quantum algorithms?
* In addressing scalability and error correction, how could cloud-based hybrids democratize access to quantum computing?

==Conclusion==

In summary, the NISQ era bridges classical and fault-tolerant quantum computing through hybrid algorithms like VQE and QAOA, despite challenges like noise and scalability. These tools are already impacting fields from chemistry to AI, with AI integrations promising further advances. As we move toward FTQC in the 2030s, continued innovation and ethical oversight will be key. 
==Quiz: Test Your Knowledge==

Here’s a 10-question quiz based on the content above. Questions are multiple choice or true/false. Each question has its answer hidden—click the toggle to reveal it.

'''1. What does NISQ stand for?'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
a) Noisy Intermediate-Scale Quantum
&lt;/div&gt;
&lt;/div&gt;
:a) Noisy Intermediate-Scale Quantum  
:b) New Integrated Superconducting Quantum  
:c) Non-Interfering Scalable Quantum  
:d) Noisy Infinite-Scale Quantum  

'''2. True or False: Superposition allows qubits to be in multiple states simultaneously, enabling parallel computation.'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
True
&lt;/div&gt;
&lt;/div&gt;
&lt;br&gt;&lt;br&gt;

'''3. In VQE, what is the primary goal?'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
b) To approximate ground states of Hamiltonians
&lt;/div&gt;
&lt;/div&gt;
:a) To factor large integers  
:b) To approximate ground states of Hamiltonians  
:c) To perform unstructured searches  
:d) To optimize neural networks  

'''4. What is a barren plateau in the context of NISQ algorithms?'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
a) A region where gradients vanish, making optimization difficult
&lt;/div&gt;
&lt;/div&gt;
:a) A region where gradients vanish, making optimization difficult  
:b) A type of quantum error correction code  
:c) A hardware limitation on qubit count  
:d) A method for amplitude amplification  

'''5. QAOA is primarily used for:'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
b) Combinatorial optimization problems
&lt;/div&gt;
&lt;/div&gt;
:a) Simulating molecular energies  
:b) Combinatorial optimization problems  
:c) Solving linear equations  
:d) Factoring primes  

'''6. True or False: Shor's Algorithm provides an exponential speedup over classical factoring methods.'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
True (polynomial time vs. exponential classical)
&lt;/div&gt;
&lt;/div&gt;&lt;br&gt;.&lt;br&gt;'''7. Grover's Algorithm offers what kind of speedup for unstructured search?'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
b) Quadratic
&lt;/div&gt;
&lt;/div&gt;
:a) Linear  
:b) Quadratic  
:c) Exponential  
:d) None  

'''8. In Quantum Machine Learning, what does HHL solve?'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
b) Linear systems of equations
&lt;/div&gt;
&lt;/div&gt;
:a) Combinatorial problems  
:b) Linear systems of equations  
:c) Optimization landscapes  
:d) Eigenvalue approximations  

'''9. Which technique is NOT commonly used for noise mitigation in NISQ?'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
c) Full fault-tolerant error correction (not yet feasible in NISQ)
&lt;/div&gt;
&lt;/div&gt;
:a) Zero-noise extrapolation  
:b) Probabilistic error cancellation  
:c) Full fault-tolerant error correction  
:d) AI decoders  

'''10. True or False: The variational principle states that the expectation value of the Hamiltonian is always greater than or equal to the ground state energy.'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
True
&lt;/div&gt;
&lt;/div&gt;

[[Category:Algorithms]]</text>
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        <id>2888076</id>
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      <comment>added [[Category:Quantum computing]] using [[Help:Gadget-HotCat|HotCat]]</comment>
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      <text bytes="14974" sha1="edtj4zdxonri3py3mebwdfok2s4nnjn" xml:space="preserve">{{Author|Harold Foppele}}
{{Physics}}
{{Quiz}}
{{Learning project}}
=Learning Page: Quantum Computing Algorithms in the NISQ Era=
This sub-learning page provides a structured overview of quantum computing algorithms during the Noisy Intermediate-Scale Quantum (NISQ) era, based on key concepts from the Wikiversity resource. It includes summaries of main topics, explanations, examples, and visual aids to facilitate learning. At the end, there's a quiz to test your understanding, along with a conclusion summarizing key takeaways.

==Introduction to Quantum Computing in the NISQ Era==

Quantum computing represents a shift from classical computing by using '''qubits''' instead of bits. Qubits can exist in '''superposition''' (multiple states at once), '''entanglement''' (linked states), and leverage '''interference''' to process information in parallel. This enables potential speedups in areas like molecular simulation, optimization, machine learning, and search problems.

The NISQ era refers to current quantum devices with 50–1000 qubits that are noisy (error-prone) and lack full error correction. Despite limitations, hybrid quantum-classical approaches are advancing fields such as chemistry, materials science, logistics, finance, and AI. Key milestones include demonstrations of &quot;quantum supremacy&quot; and practical applications like modeling hydrogen chains or optimizing financial portfolios.

'''Key Terms:'''

* '''Quantum Advantage''': When quantum computers outperform classical ones for specific tasks.
* '''NISQ''': Noisy Intermediate-Scale Quantum – the current phase of quantum tech.

Example: In 2026, advancements like JPMorgan's quantum streaming for large datasets highlight real-world potential (source:[https://www.jpmorganchase.com/technology/quantum-computing JPMorgan Chase Research]

To visualize superposition, consider this simple image [[File:Riemann_Spin2States.jpg|thumb]]
'''Discussion Questions:'''

* How might the principles of superposition and entanglement change the way we approach problem-solving in fields like drug discovery or climate modeling?
* Discuss the ethical implications of achieving quantum advantage in areas such as cryptography or financial optimization. Who benefits, and who might be at risk?
* Compare the NISQ era to the early days of classical computing. What lessons from history could guide the development of quantum technologies?

==NISQ Algorithms Overview==

NISQ algorithms are hybrid methods that combine quantum circuits with classical optimization to handle noise without full error correction. They rely on the '''variational principle''', where the expectation value of a Hamiltonian (energy operator) is minimized to find approximate solutions: ⟨ψ|H|ψ⟩ ≥ E₀, where E₀ is the ground state energy.

'''Workflow:'''

* 1. Prepare a parameterized quantum state |ψ(θ)⟩ using a circuit (ansatz) from an initial state like |0⟩.
* 2. Measure the cost function C(θ) = ⟨ψ(θ)|H|ψ(θ)⟩, often decomposed into 3. Pauli operators (e.g., &lt;math&gt;H = \sum_k c_k P_k&lt;/math&gt;).&lt;br&gt; Use classical optimizers (e.g., gradient descent) to adjust parameters θ.

Challenges include '''barren plateaus''' (flat optimization landscapes where gradients vanish) and noise, often mitigated with AI techniques.

Example: Variational Quantum Algorithms (VQAs) for chemistry (e.g., molecular energy calculations) or finance (portfolio optimization). For more, explore IBM's Qiskit tutorials [https://qiskit.org/documentation/ Qiskit Documentation]

'''Discussion Questions:'''

* In what ways do hybrid quantum-classical approaches in NISQ algorithms bridge the gap between current hardware limitations and future fault-tolerant quantum computing?
* How could barren plateaus impact the scalability of NISQ algorithms, and what role might AI play in overcoming this challenge?
* Discuss potential real-world applications of VQAs beyond chemistry and finance. How might they transform industries like logistics or environmental science?

==Specific Algorithms==

===Variational Quantum Eigensolver (VQE)===

VQE approximates the ground or excited states of molecular Hamiltonians, useful in quantum chemistry. It uses an ansatz like the Unitary Coupled Cluster (UCC) to create trial states and minimizes the energy expectation value.

'''Mechanism:'''

* Generate |ψ(θ)⟩ with a parameterized circuit.
* Compute and minimize ⟨ψ|H|ψ⟩ via measurements.
* Innovations: Adaptive VQE dynamically adds operators to reduce qubit needs; subspace methods for excited states.

Challenges: Barren plateaus addressed with AI (e.g., reinforcement learning for parameter tuning).

Example: Google's use for hydrogen chain modeling; applications in drug discovery by 2026 (source:[https://quantumai.google/ Google Quantum AI])

Try a simple VQE simulation in Python with Qiskit:

```python

from qiskit import QuantumCircuit
# Basic example code snippet
qc = QuantumCircuit(1)

qc.h(0)  # Superposition
# Add more for full VQE
```

'''Discussion Questions for VQE:'''

* How does the choice of ansatz in VQE affect its accuracy and efficiency? What trade-offs might researchers face when selecting one?
* Discuss the potential of adaptive VQE in reducing resource requirements. Could this lead to broader accessibility of quantum computing for smaller organizations?
* In the context of drug discovery, how might VQE simulations accelerate the development process, and what limitations remain due to noise?

===Quantum Approximate Optimization Algorithm (QAOA)===

QAOA tackles combinatorial optimization problems (e.g., MaxCut) by mapping them to Ising Hamiltonians.

'''Mechanism:'''

* Start with a superposition state |+⟩^⊗n.
* Apply alternating layers: e^{-i γ H_C} (problem Hamiltonian, e.g., ∑ Z_i Z_j for edges) and e^{-i β H_B} (mixer, ∑ X_i). - Optimize parameters γ and β to minimize ⟨ψ|H_C|ψ⟩.

Variants: Recursive QAOA (RQAOA) iteratively reduces problem size; warm-starting with AI.

Challenges: Local minima in parameter space; hardware-aware versions reduce errors.

Example: Logistics and finance applications, like supply chain optimization on 30-qubit systems by 2026 (source: [https://arxiv.org/abs/1411.4028 arXiv Paper on QAOA])

'''Discussion Questions for QAOA:'''

* How does QAOA's approach to combinatorial problems differ from classical optimization methods, and in what scenarios might it provide a clear advantage? Note that advantages are still debated in literature.
* Explore the role of variants like RQAOA. How could iterative problem reduction improve performance on NISQ devices?
* Discuss the integration of AI in warm-starting QAOA. What synergies between quantum and classical AI could emerge in optimization tasks?

===Other Algorithms===

* '''Shor's Algorithm''': Factors large integers in polynomial time using Quantum Fourier Transform (QFT) for periodicity finding. It threatens RSA encryption; NISQ versions factor small numbers (e.g., 15=3*5). Mechanism: Prepare superposition, apply modular exponentiation, use QFT to find period, then classical post-processing.

* '''Grover's Algorithm''': Provides quadratic speedup (O(√N)) for unstructured search via amplitude amplification. Useful in machine learning for feature selection. Mechanism: Initialize superposition, apply oracle to mark solutions, amplify amplitudes iteratively.

* '''Amplitude Amplification''': Generalizes Grover to boost probabilities; applied in anomaly detection. It iteratively reflects states to increase desired amplitudes.

For diagrams, see a QFT circuit (placeholder for image).

'''Discussion Questions for Other Algorithms:'''

* What are the security implications of Shor's Algorithm in the NISQ era, even with its current limitations on small-scale factoring?
* How might Grover's Algorithm enhance machine learning tasks like feature selection, and what challenges arise from implementing it on noisy hardware?
* Discuss amplitude amplification as a generalization of Grover's. In what innovative ways could it be applied beyond search problems, such as in signal processing?

==Quantum Machine Learning (QML)==

QML leverages quantum mechanics for machine learning tasks, offering potential exponential speedups.

'''Key Methods:'''
- '''Harrow-Hassidim-Lloyd (HHL)''': 

* Solves linear systems Ax = b using phase estimation. - Quantum kernels and neural networks for classification, generative models.
* NISQ adaptations: Amplitude amplification for kernels; hybrid classical-quantum setups.
* Challenges: Noise and decoherence; mitigated with AI like shadow tomography.

Example: Google's Quantum Echo for NMR spectra in biomedicine by 2026 (source: [https://research.google/ Google Research])
{|class=&quot;wikitable&quot;
|+ '''Comparison Table'''
! Algorithm
! Key Use Case
! NISQ Adaptation
! Classical Counterpart
|-
| HHL
| Linear systems
| Hybrid setups
| Gaussian elimination
|-
| Quantum Kernels
| Classification
| Amplitude amplification
| SVM kernels
|}

'''Discussion Questions:'''

* How could quantum kernels in QML provide advantages over classical kernels in tasks like classification, and what datasets might benefit most?
* Discuss the HHL algorithm's potential for solving linear systems. In fields like finance or engineering, where could it outperform classical methods?
* Explore the challenges of noise in QML. How might techniques like shadow tomography evolve to make QML more practical in the NISQ era?

==Challenges in NISQ Computing==

* '''Error Correction''': Needs thousands of physical qubits per logical one (e.g., surface codes); partial progress by 2026.
* '''Scalability''': Current limits (e.g., 127 qubits on IBM Eagle); hybrids via cloud services like AWS Braket.
* '''Barren Plateaus''': Mitigated by specialized ansatzes and AI.
* '''Noise Mitigation''': Techniques like zero-noise extrapolation and AI decoders.
* Other: Decoherence, data loading issues, and the risk of &quot;dequantization&quot; (classical simulations outperforming quantum).

Future: Co-design of algorithms and hardware; transition to fault-tolerant quantum computing (FTQC) by 2030s. Ethical considerations include environmental impact from energy use and access inequality.

'''Discussion Questions:'''

* What strategies could accelerate the transition from NISQ to fault-tolerant quantum computing, and what role should governments or industries play?
* Discuss &quot;dequantization&quot; risks. How might classical simulations challenge the perceived advantages of quantum algorithms?
* In addressing scalability and error correction, how could cloud-based hybrids democratize access to quantum computing?

==Conclusion==

In summary, the NISQ era bridges classical and fault-tolerant quantum computing through hybrid algorithms like VQE and QAOA, despite challenges like noise and scalability. These tools are already impacting fields from chemistry to AI, with AI integrations promising further advances. As we move toward FTQC in the 2030s, continued innovation and ethical oversight will be key. 
==Quiz: Test Your Knowledge==

Here’s a 10-question quiz based on the content above. Questions are multiple choice or true/false. Each question has its answer hidden—click the toggle to reveal it.

'''1. What does NISQ stand for?'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
a) Noisy Intermediate-Scale Quantum
&lt;/div&gt;
&lt;/div&gt;
:a) Noisy Intermediate-Scale Quantum  
:b) New Integrated Superconducting Quantum  
:c) Non-Interfering Scalable Quantum  
:d) Noisy Infinite-Scale Quantum  

'''2. True or False: Superposition allows qubits to be in multiple states simultaneously, enabling parallel computation.'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
True
&lt;/div&gt;
&lt;/div&gt;
&lt;br&gt;&lt;br&gt;

'''3. In VQE, what is the primary goal?'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
b) To approximate ground states of Hamiltonians
&lt;/div&gt;
&lt;/div&gt;
:a) To factor large integers  
:b) To approximate ground states of Hamiltonians  
:c) To perform unstructured searches  
:d) To optimize neural networks  

'''4. What is a barren plateau in the context of NISQ algorithms?'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
a) A region where gradients vanish, making optimization difficult
&lt;/div&gt;
&lt;/div&gt;
:a) A region where gradients vanish, making optimization difficult  
:b) A type of quantum error correction code  
:c) A hardware limitation on qubit count  
:d) A method for amplitude amplification  

'''5. QAOA is primarily used for:'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
b) Combinatorial optimization problems
&lt;/div&gt;
&lt;/div&gt;
:a) Simulating molecular energies  
:b) Combinatorial optimization problems  
:c) Solving linear equations  
:d) Factoring primes  

'''6. True or False: Shor's Algorithm provides an exponential speedup over classical factoring methods.'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
True (polynomial time vs. exponential classical)
&lt;/div&gt;
&lt;/div&gt;&lt;br&gt;.&lt;br&gt;'''7. Grover's Algorithm offers what kind of speedup for unstructured search?'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
b) Quadratic
&lt;/div&gt;
&lt;/div&gt;
:a) Linear  
:b) Quadratic  
:c) Exponential  
:d) None  

'''8. In Quantum Machine Learning, what does HHL solve?'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
b) Linear systems of equations
&lt;/div&gt;
&lt;/div&gt;
:a) Combinatorial problems  
:b) Linear systems of equations  
:c) Optimization landscapes  
:d) Eigenvalue approximations  

'''9. Which technique is NOT commonly used for noise mitigation in NISQ?'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
c) Full fault-tolerant error correction (not yet feasible in NISQ)
&lt;/div&gt;
&lt;/div&gt;
:a) Zero-noise extrapolation  
:b) Probabilistic error cancellation  
:c) Full fault-tolerant error correction  
:d) AI decoders  

'''10. True or False: The variational principle states that the expectation value of the Hamiltonian is always greater than or equal to the ground state energy.'''  &lt;div class=&quot;mw-collapsible mw-collapsed&quot; data-expandtext=&quot;Show Answer&quot; data-collapsetext=&quot;Hide Answer&quot;&gt;
&lt;div class=&quot;mw-collapsible-content&quot;&gt;
True
&lt;/div&gt;
&lt;/div&gt;

[[Category:Algorithms]]
[[Category:Quantum computing]]</text>
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  </page>
  <page>
    <title>User:Juandev/R/Compression stocking</title>
    <ns>2</ns>
    <id>329166</id>
    <revision>
      <id>2815499</id>
      <parentid>2811829</parentid>
      <timestamp>2026-06-13T16:03:52Z</timestamp>
      <contributor>
        <username>Juandev</username>
        <id>2651</id>
      </contributor>
      <comment>/* Generic questions */ update</comment>
      <origin>2815499</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11589" sha1="lmz0qdzgdckacqws8qiored9nqw9ejt" xml:space="preserve">{{contrib-creator}}
{{User:Juandev/T/QA AI contribution}}
{{medicine}}
{{non-formal education}}
{{research}}
== How does this course work? ==
This course is built on a question-and-answer format. Anyone can ask a question, and anyone can answer any question. It is for those interested in [[w:en:Compression stockings|Compression stocking]], for those who enjoy researching and solving problems. Answering the questions is up to you. Ask a question and then write an answer to it. You can find it in the literature, on YouTube, via LLM, or through your research (experiment). You can also answer other people's questions as part of the exercise. We would greatly appreciate it if you could attach free images and videos and upload them to Wikimedia Commons. This will help others better understand the problem.

== Questions ==

=== Generic questions ===
''These are questions when you can adequately name things and structure your answer.'' 
{| class=&quot;wikitable&quot;
!No.
!Question
!Answer
!Visual explanation
!Notes
|-
|GQ.1
|What is a function of compression stocking?
|They create pressure on the veins under the skin, helping blood flow upwards. This works both by narrowing the vein diameter and by pressing the vein valves together, as the vein valves prevent blood from falling downwards.
|
|
|-
|GQ.2
|What are the degrees of compression of stockings?
|
# CCL 1 – common prevention for people who sit or stand for long periods of time.
# CCL 2 – for varicose veins, after surgeries.
# CCL 3 – for example, for extensive swelling or treatment of a leg ulcer
# CCL 4 – for extreme lymphedema.
|
|
|-
|GQ.3
|Why there are different levels of compression?
|
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|
|-
|GQ.4
|What are the types of socks in terms of height and how are they marked?
|These classes are distinguished according to the RAL GZ-387 standard&lt;ref&gt;http://www.tagungsmanagement.org/icc/images/stories/PDF/ral_gz_387_englisch.pdf p. 13&lt;/ref&gt;:

* AD – calf stocking, ends below the knee
* AF – mid-thigh stocking, ends mid-thigh. These stockings may ride down because the thigh is tapered, worked.
* AG – thigh-high stocking, ends below the crotch.
* AT – tights, reaching to the navel and covering both legs
* AG-G{{Citation needed}} – one-leg stocking with waist strap
|
|
|-
|GQ.5
|How does the AG-G stocking look like?
|
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|-
|GQ.6
|Which brands produce compression stockings for Europe?
|
* Medi&lt;ref&gt;{{Cite web|url=https://www.medi.de/en/products/compression-stockings/|title=Compression stockings by medi – modern and individual {{!}} medi|website=www.medi.de|language=en-DE|access-date=2026-04-19}}&lt;/ref&gt; (Germany)
* Bauerfeind&lt;ref&gt;{{Cite web|url=https://www.bauerfeind-group.com/en/products/compression-therapy/compression-stockings-vein-treatment-compression-therapy|title=One moment, please...|website=www.bauerfeind-group.com|language=en|access-date=2026-04-19}}&lt;/ref&gt; (Germany)
* Sigvaris&lt;ref&gt;{{Cite web|url=https://origin-www.sigvaris.com/en-us/catalog/medical/varicose-veins|title=Varicose veins|website=origin-www.sigvaris.com|language=en|access-date=2026-04-19}}&lt;/ref&gt; (Switzerland)
* Jobst&lt;ref&gt;{{Cite web|url=https://www.jobst.cz/produkty/zdravotni-komprese.html|title=Zdravotní komprese|website=Jobst|language=cs|access-date=2026-04-19}}&lt;/ref&gt; (Germany)
* Maxis&lt;ref&gt;{{Cite web|url=http://www.maxis-medica.cz/?sl=CZ|title=MAXIS a.s. - Zdravotní kompresivní punčochy, pažní návleky|website=www.maxis-medica.cz|access-date=2026-04-19}}&lt;/ref&gt; (Medi, Czech Republic)
* Aries&lt;ref&gt;https://cz.aries.eu/avicenum_phlebo_cz.pdf&lt;/ref&gt; (Czech Republic)
*
|
|
|-
|GQ.7
|After what time, or distance traveled, does a foot swell to the point where it is no longer good to measure it?
|Either immediately after waking up, or within one hour of regular exercise, but preferably within 30 minutes.
|
|
|-
|GQ.8
|And is it possible to let the night go by, for example, putting my legs above my head for 20 minutes?
|It can help, but it is not 100 % same as after waking up.
|
|
|-
|GQ.9
|Is it necessary to put on compression socks in the morning?
|Its the best, they could be put on later during the day, but even after few minutes with feet up, feet are still bigger so the stocking doesnt work so well as after waking up in the morning.
|
|
|-
|GQ.10
|Is it possible to swim with stockings?
|Yes, but their material is demaged especially in pools by chemical composition of the water.
|
|
|-
|GQ.11
|Does a sock that constricts more than a compression stocking affect leg constriction?
|This can be a problem for patients with varicose veins because blood pools under the constriction, putting more pressure on other blood vessels, which can then dilate.
|
|
|-
|GQ.12
|What circumferences are measured for AG stockings? Is it measured the same for all manufacturers?
|Each manufacturer requires a combination of different anatomical points, but they are generally standardized. It is therefore better to measure more than one and then make a selection. Ideally, measure the points:

* b – '''above ankle''', the most important measurement
* c – the widest point on the '''calf'''
* d – lust '''below the knee''' joint
* g – '''thigh''', specifically 5 cm below the crotch
|
|
|-
|GQ.13
|Is the leg measured for stockings lying down or standing up?
|Standing, without pressure and without straining the leg.
|
|
|-
|GQ.14
|Which circuits are most important for an AT stocking?
|
|
|
|-
|GQ.15
|Why are stockings shorter the day after they are put on?
|There may be several reasons:

* different way of putting them on, for example, you pulled them more on the first day when putting them on,
* unwashed material and thus limited elasticity - it is common to wash stockings every day to remove grease and skin residues
* the reason may also be putting them on after partial swelling of the leg, i.e. they were not put on immediately after waking up
* stockings are also not good to put on on oily, sweaty and wet legs. For example, after washing your feet, you need to wait at least 30 minutes for the skin to dry Its better to shower the body in the evening,morning sweat is better to remove by a wet cloth as morning shower may introduce swelling too.
|
|
|-
|GQ.16
|Why aren't socks made up to the crotch?
|Gemini assisted: Because the skin in the crotch is softer, there is friction, heating and the skin could be damaged by the hem, or the stocking could slide down from there. That is why the longest size is AG, where the G point is usually 5 cm below the crotch. If someone needs longer, AT tights are made, for example.
|
|
|-
|GQ.17
|Why are stockings the same length when they are at rest, but one of them stretches more when put on?
|Gemini assisted: Because it is a knitting system, but also about how well the given foot was measured. Some manufacturers simply knit in a way that the fibers can stretch more. But it also depends on the correct measurement, if the stocking is well adjusted to all measured circumferences, it can also be pulled out correctly.
|
|
|-
|GQ.18
|What causes the stocking hem to bend? Is it subcutaneous fat or swelling of the foot?
|
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|-
|GQ.19
|Is it correct that a stocking without a toe covers part of the little toe?
|Partially yes, the stockings should reach the knuckles of the fingers. The problem is that the knuckles of the fingers are not level and the little finger is thus moved closer to the heel. So it can happen that part of the little finger covers the hem of the sock. But it should not be a large part, the hem should end somewhere above the knuckle of the little finger.
|
|The reason may be incorrect fitting of the stockings, or incorrect measurements and ordering stockings of inappropriate sizes.
|-
|GQ.20
|What is the mechanism behind the accumulation of fabric below the knee when worn?
|
|
|
|-
|GQ.21
|What is the mechanism behind the accumulation of substance in the popliteal fossa after stretching?
|This could be because:

* the stocking is not tight enough on the calf, but because it is tight on the thigh, the thigh pulls it up, but because it is wide, this buildup gets into the knee socket
* or it could be because the person is turning the stocking when putting it on. In other words, if the heel is down and the dark crease line is visible on one side, then the dark crease line should always be on one side in the same place on each part of the limb
|
|The rotation of the sock does not have to occur throughout the entire putting on. It is more likely due to a bad fit at the beginning on the heel. This creates a deflection angle that continues. With AG socks, it can happen that there is a 120-degree deflection at the end of the stocking. This can be prevented, for example, by using a wire frame tensioner, because here we can center the heel in the middle.&lt;ref&gt;See: https://www.youtube.com/watch?v=i3QmY9BZQSw&lt;/ref&gt;
Perhaps a person could draw some marks on the stocking with a body marker first, but since they can't see the exact position of the knee, etc., this can be difficult. In addition, there is a risk of losing the warranty or damaging the stocking.
|-
|GQ.22
|What to do with the accumulation of material in the knee pit when wearing?
|Some of the accumulated fabric is natural. However, it should not be too much and the user should not feel that it is digging into their leg. It is natural to pull the stockings as high as possible, even above the usual level (e.g. for AG stockings below 5 cm from the crotch) and possibly pull them up during the day. The fabric accumulated in the bend of the pit can be grasped with the fingers and distributed upwards.
|
|
|-
|GQ.23
|How to take off a stocking?
|Roll it up to the ankle, then use your fingers to spread it out and pull it over the heel. Definitely don't pull with force or tug on the hem.
|
|
|-
|GQ.24
|Does the donner serve to evenly distribute the stocking on the limb?
|No, its main purpose is to put the stocking on the leg.
|
|
|-
|GQ.24
|How to distribute AG stocking evenly on the limb?
|
|
|
|}

=== Personal problems ===
''Here are questions when you cannot correctly name things and describe them. Thus, it is necessary to include photographs, videos, or drawings to describe your problem visually.''
{| class=&quot;wikitable&quot;
!No.
!Question
!Visual documentation
!Answer
!Visual explanation
!Notes
!Discussion
|-
|PP.1
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|PP.2
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|PP.3
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|PP.4
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|PP.5
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|PP.6
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|PP.7
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|PP.8
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|PP.9
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|PP.10
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|PP.11
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|PP.12
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|PP.13
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|PP.14
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|PP.15
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|}

=== Related questions ===
''This includes questions that are not related to compression stockings, but related things.''
{| class=&quot;wikitable&quot;
!No.
!Question
!Visual documentation
!Answer
!Notes
!Discussion
|-
|RQ.1
|Under what license is Gemini AI output?
|
|In the case of Gemini, services must be used in the European Union after careful consideration&lt;ref&gt;{{Cite web|url=https://policies.google.com/terms/generative-ai?hl=cs|title=Dodatečné smluvní podmínky generativní umělé inteligence|website=policies.google.com|access-date=2026-04-19}}&lt;/ref&gt; and their originator must not be hidden.&lt;ref&gt;{{Cite web|url=https://policies.google.com/terms/generative-ai/use-policy?hl=cs|title=Zásady zakázaného používání generativní umělé inteligence|website=policies.google.com|access-date=2026-04-19}}&lt;/ref&gt;
|
|
|-
|RQ.2
|Where can I get a sock that doesn't constrict my foot but doesn't fall down?
|
|Try stretch socks for runners and cyclists.
|
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|-
|RQ.3
|
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|}

== References ==
&lt;references /&gt;</text>
      <sha1>lmz0qdzgdckacqws8qiored9nqw9ejt</sha1>
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  </page>
  <page>
    <title>User:Fortuna imperatrix mundi/IaR2</title>
    <ns>2</ns>
    <id>329439</id>
    <revision>
      <id>2815488</id>
      <parentid>2812665</parentid>
      <timestamp>2026-06-13T12:11:00Z</timestamp>
      <contributor>
        <username>ShakespeareFan00</username>
        <id>6645</id>
      </contributor>
      <minor />
      <origin>2815488</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="49455" sha1="79vypwir4quzxcqtoypob3t89dbqvrj" xml:space="preserve">==Background==

===Richard II===

===Elizabeth I===

====Essex's rebellion====

==IaR2==

===Political anxiety and tension===

===Censorship===

==Consequences and aftermath==

== Notes ==
{{reflist|group=note}}

==References==
{{Reflist|20em}}

==Bibliography==
{{refbegin|30em|indent=yes}}
* [https://link.springer.com/chapter/10.1057/9780230307261_2 Orgel, S. (2011). Prologue: I am Richard II. In: Petrina, A., Tosi, L. (eds) Representations of Elizabeth I in Early Modern Culture. Palgrave Macmillan, London. https://doi.org/10.1057/9780230307261_2]
* [https://broadlytextual.com/2017/12/15/i-am-richard-ii-know-ye-not-that-drama-and-political-anxiety-in-shakespeares-london/ Hixon, E., Syracuse Univ]
* Bate, Jonathan (2008). Soul of the Age. London: Penguin. pp. 256–286. ISBN 978-0-670-91482-1.
* [https://theconversation.com/richard-ii-by-william-shakespeare-why-the-divine-right-of-kings-still-matters-186648 McFarlane, K., Univ South Australia]
* [https://muse.jhu.edu/article/31090/summary Lemon, Rebecca. &quot;The Faulty Verdict in &quot;The Crown v. John Hayward&quot;.&quot; SEL Studies in English Literature 1500-1900, vol. 41 no. 1, 2001, p. 109-132. Project MUSE, https://dx.doi.org/10.1353/sel.2001.0009]
* [https://brill.com/display/book/9789401211666/B9789401211666-s009.xml?language=en&amp;srsltid=AfmBOoqTUaTQr0msNWRdtEZZ9qQ3rNcVkSoBvDPXuGv7nRvHrC8t5OzH Kizelbach, U. (2014), In The Pragmatics of Early Modern Politics: Power and Kingship in Shakespeare’s History Plays. Leiden, The Netherlands: Brill. https://doi.org/10.1163/9789401211666_009]
* [https://www.google.co.uk/books/edition/Richard_II/z-kOAQAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Lopez, R2]
* [https://www.manchesterhive.com/display/9781526130532/9781526130532.00008.xml?print rgel, S. (2017). &quot;I am Richard II&quot;. In Spectacular Performances. Manchester, England: Manchester University Press. Retrieved Apr 22, 2026, from https://doi.org/10.7765/9781526130532.00008]
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* [https://www.google.co.uk/books/edition/Elizabethan_Literature_and_the_Law_of_Fr/vBFdAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Elizabethan Literature and the Law of Fraudulent Conveyance]
* [https://www.google.co.uk/books/edition/Hamlet_History_and_commentary/yxgrAQAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Hamlet History etc]
* [https://www.google.co.uk/books/edition/William_Shakespeare_A_Popular_Life/1BuaAAAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover William Shakespeare: a popular life]
* [https://www.google.co.uk/books/edition/A_State_of_Mind/pCiIAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover A State of Mind?]
* [https://www.google.co.uk/books/edition/Shakespeare_s_Dramatic_Genres/J5FlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespeare's Dramatic Genres]
* [https://www.google.co.uk/books/edition/Literature_Criticism_from_1400_to_1800/O6VkAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Literature Criticism from 1400 to 1800 Volume 89]
* [https://www.google.co.uk/books/edition/Shakespeare_the_Papist/LPwNAQAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespeare the Papist]
* [https://www.google.co.uk/books/edition/Goslicius_Ideal_Senator_and_His_Cultural/1jc7AQAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Goslicius' Ideal Senator and His Cultural Impact Over the Centuries]
* [https://www.google.co.uk/books/edition/Shakespeare_by_Another_Name/FqllAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover &quot;Shakespeare&quot; by Another Name]
* [https://www.google.co.uk/books/edition/Shakespeare_s_Friends/AlZlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespeare's Friends]
* [https://www.google.co.uk/books/edition/Dr_Simon_Forman/qHceAQAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Dr Simon Forman]
* [https://www.google.co.uk/books/edition/William_Shakespeare_the_Wars_of_the_Rose/dZFlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover William Shakespeare, the Wars of the Roses and the historians]
* [https://www.google.co.uk/books/edition/Shakespearean_Criticism/2TdlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespearean Criticism]
* [https://www.google.co.uk/books/edition/Poetry_History_and_Antiquarianism_in_Ref/jwIhAQAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Poetry, History, and Antiquarianism in Reformation England]
* [https://www.google.co.uk/books/edition/Shakespeare_and_the_Prince_of_Love/D0QgAQAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespeare and the Prince of Love]
* [https://www.google.co.uk/books/edition/Paper_Bullets_of_the_Brain/o0YgAQAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Paper Bullets of the Brain]
* [https://www.google.co.uk/books/edition/As_You_Like_It/GiBaAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover As You Like It: Third Series]
* [https://www.google.co.uk/books/edition/William_Shakespeare/WCqaAAAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover William Shakespeare, Dominic Shellard]
* [https://www.google.co.uk/books/edition/Poets_and_God/0XZlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Poets and God]
* [https://www.google.co.uk/books/edition/Law_and_Literature/Ax5MAQAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Law and Literature Volume 16]
* [https://www.google.co.uk/books/edition/The_Embodied_Word/FV8sAQAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover The Embodied Word]
* [https://www.google.co.uk/books/edition/The_Case_for_Shakespeare/WaRlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover The Case for Shakespeare: The End of the Authorship Question]
* [https://www.google.co.uk/books/edition/Explorations_in_Renaissance_Culture/_SYrAQAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Explorations in Renaissance Culture Volumes 33-34]
* [https://www.google.co.uk/books/edition/The_Touch_of_the_Real/ewdaAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover The Touch of the Real: Essays in Early Modern Culture in Honour of Stephen Greenblatt]
* [https://www.google.co.uk/books/edition/Wotton_and_His_Worlds/ZfgNAQAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Wotton and his Worlds]
* [https://www.google.co.uk/books/edition/Theatre_and_Religion/wo1lAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Theatre and Religion Lancastrian Shakespeare]
* [https://www.google.co.uk/books/edition/Trying_Treason/TOKxAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Trying Treason]
* [https://www.google.co.uk/books/edition/Willing_Subjects/IEX0sGwT1QQC?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Willing Subjects]
* [https://www.google.co.uk/books/edition/Symbolism/Bt0ZAQAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Symbolism]
* [https://www.google.co.uk/books/edition/Performing_Shakespeare/35pQAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Performing Shakespeare]
* [https://www.google.co.uk/books/edition/Soul_of_the_Age/e0UgAQAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Soul of the Age]
* [https://www.google.co.uk/books/edition/England/aD9nAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover England]
* [https://www.google.co.uk/books/edition/Elizabeth_I/-GtnAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Elizabeth I]
* [https://www.google.co.uk/books/edition/King_Richard_II/oGUMX4RntjgC?hl=en&amp;gbpv=1&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;pg=PA25&amp;printsec=frontcover King Richard II]
* [https://www.google.co.uk/books/edition/Shakespeare_and_the_Legal_Imagination/OXPvBqQLw-4C?hl=en&amp;gbpv=1&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;pg=PA38&amp;printsec=frontcover Shakespeare and the legal imagination]
* [https://www.google.co.uk/books/edition/Shakespeare_s_Theatre/GxN3ue9_r3oC?hl=en&amp;gbpv=1&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;pg=PA69&amp;printsec=frontcover Shakespeare's Theatre]
* [https://www.google.co.uk/books/edition/Critical_Essays_on_Shakespeare_s_Richard/AaYoAQAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Critical Essays on Shakespeare's Richard II]
* [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II_Essays_in_Honour/y3xnAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover The Reign of Richard II: Essays in Honour of May McKisack]
* [https://www.google.co.uk/books/edition/Poetry_and_the_Realm_of_Politics/oQFaAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Poetry and the Realm of Politics]
* [https://www.google.co.uk/books/edition/Shakespeare/BM0mAQAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespeare]
* [https://www.google.co.uk/books/edition/Shakespearean_Politics/oTdlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespearean Politics]
* [https://www.google.co.uk/books/edition/Shakespeare_the_Theatrical_Dimension/wl4gAQAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespeare, the Theatrical Dimension]
* [https://www.google.co.uk/books/edition/Who_was_Kit_Marlowe/zQ1aAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Kit Marlowe etc]
* [https://www.google.co.uk/books/edition/From_Page_to_Performance/beQKAQAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover From Page to Performance]
* [https://www.google.co.uk/books/edition/Exploring_Tudor_England/ax56AAAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Exploring Tudor England]
* [https://www.google.co.uk/books/edition/The_Movement_Towards_Subversion/vyJaAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover The Movement Towards Subversion]
* [https://www.google.co.uk/books/edition/Shakespeare_s_Typological_Satire/G5BlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespeare's Typological Satire]
* [https://www.google.co.uk/books/edition/Shakespeare_Recycled/zzNlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespeare Recycled]
* [https://www.google.co.uk/books/edition/Reinventing_the_Middle_Ages_the_Renaissa/fXFnAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Reinventing the Middle Ages &amp; the Renaissance]
* [https://www.google.co.uk/books/edition/The_Mysterious_William_Shakespeare/WnllAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover The Mysterious William Shakespeare]
* [https://www.google.co.uk/books/edition/Richard_II/GHhlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Richard II Critical Essays]
* [https://www.google.co.uk/books/edition/William_Shakespeare/WJvC6gu_I0gC?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover William Shakespeare: Records and Images]
* [https://www.google.co.uk/books/edition/Shakespeare_and_the_Actors/HYtlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespeare and the Actors]
* [https://www.google.co.uk/books/edition/Shakespeare_the_Man/BVdlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespeare the man]
* [https://www.google.co.uk/books/edition/Henry_V/zXllAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Henry V: A Guide to the Play]
* [https://www.google.co.uk/books/edition/Shakespearean_Contingencies/Cw1NAQAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespearean Contingencies]
* [https://www.google.co.uk/books/edition/Renaissance_Drama/E60kAQAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Renaissance Drama 1990]
* [https://www.google.co.uk/books/edition/Language_Discourse_Sign/uH4oAQAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Language, Discourse, Sign]
* [https://www.google.co.uk/books/edition/Mock_Kings_in_Medieval_Society_and_Renai/T98KAQAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Mock kings in medieval society and Renaissance drama]
* [https://www.google.co.uk/books/edition/Shakespeare_Invention_of_the_Human/ojHirImrtYoC?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespeare: Invention of the Human]
* [https://www.google.co.uk/books/edition/Shakespeare/wn5lAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespeare]
* [https://www.google.co.uk/books/edition/Persons_in_Groups/rQ24AAAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Persons in Groups]
* [https://www.google.co.uk/books/edition/All_Semblative_a_Woman_s_Part/0DlaAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover All Semblative a Woman's Part?]
* [https://www.google.co.uk/books/edition/Crossing_the_Mirror/qRZNAQAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Crossing the Mirror]
* [https://www.google.co.uk/books/edition/De_Vere_is_Shakespeare/dKJlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover De Vere is Shakespeare]
* [https://www.google.co.uk/books/edition/William_Lambarde_Elizabethan_Antiquary_1/x1RnAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover William Lambarde, Elizabethan Antiquary, 1536-1601]
* [https://www.google.co.uk/books/edition/The_Power_of_Forms_in_the_English_Renais/cPtZAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover The Power of Forms in the English Renaissance]
* [https://www.google.co.uk/books/edition/Ravishment_and_Rememberance/G31LAQAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Ravishment and Rememberance]
* [https://www.google.co.uk/books/edition/Shakespeare_and_His_Theatre/8A5ZQq3uOVQC?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespeare and His Theatre]
* [https://www.google.co.uk/books/edition/Critical_Hermeneutics_and_Shakespeare_s/O10gAQAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Critical Hermeneutics and Shakespeare's History Plays]
* [https://www.google.co.uk/books/edition/Christian_England/K-WfAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Christian England]
* [https://www.google.co.uk/books/edition/Shakespeare_s_Religious_Background/xDSaAAAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespeare's Religious Background]
* [https://www.google.co.uk/books/edition/Shylock/N4RlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shylock]
* [https://www.google.co.uk/books/edition/The_Shakespeare_Legacy/MM5XAAAAYAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover The Shakespeare Legacy]
* [https://www.google.co.uk/books/edition/Renaissance_Genres/0uFZAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Renaissance Genres]
* [https://www.google.co.uk/books/edition/Cannibals_Witches_and_Divorce/qZRpAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Cannibals, Witches, and Divorce]
* [https://www.google.co.uk/books/edition/The_Problem_of_Religious_Knowledge/C29LAAAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover The Problem of Religious Knowledge]
* [https://www.google.co.uk/books/edition/Essex_and_the_Great_Revolt_of_1381/J8RzAAAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Essex and the Great Revolt of 1381]
* [https://www.google.co.uk/books/edition/Transactions_of_the_London_and_Middlesex/4dtJAQAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover TLMAS]
* [https://www.google.co.uk/books/edition/Shakespeare_Politics_and_the_State/Mn9lAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespeare, Politics and the State]
* [https://www.google.co.uk/books/edition/Allegories_of_Power_in_the_England_of_El/LIYgAQAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Allegories of Power in the England of Elizabeth]
* [https://www.google.co.uk/books/edition/William_Shakespeare/rIVlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover William Shakespeare]
* [https://www.google.co.uk/books/edition/Women_s_Matters/PDRlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Women's Matters]
* [https://www.google.co.uk/books/edition/The_Weak_King_Dilemma_in_the_Shakespeare/0bJlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover The Weak King Dilemma in the Shakespearean History Play]
* [https://www.google.co.uk/books/edition/The_Book_Known_as_Q/S2tlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover The Book Known as Q]
* [https://www.google.co.uk/books/edition/Fields_of_Vision/OD0eAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Fields of Vision]
* [https://www.google.co.uk/books/edition/Ungodly_Delights/RKgcAQAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Ungodly Delights]
* [https://www.google.co.uk/books/edition/The_Shakespeare_Handbook/rLRlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover The Shakespeare HandboOK]
* [https://www.google.co.uk/books/edition/Humanities/y5FZAAAAYAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Humanities]
* [https://www.google.co.uk/books/edition/Richard_II_by_William_Shakespeare/Bb3yAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Richard II by William Shakespeare]
* [https://www.google.co.uk/books/edition/King_Richard_II/50NnAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover King Richard II]
* [https://www.google.co.uk/books/edition/Murder_Under_Trust_Or_The_Topical_Macbet/0oNlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Murder under trust]
* [https://www.google.co.uk/books/edition/The_Shakespearean_Kings/tHBlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover The Shakespearean Kings]
* [https://www.google.co.uk/books/edition/America_the_Mabr_e_y_Experience/mRQ3AAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover America, the Mabr(e)y Experience: Resistance, Revolution &amp; Civil War]
* [https://www.google.co.uk/books/edition/Richard_II/ZDEkAQAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Richard II: An Annotated Bibliography, Volume 2]
* [https://www.google.co.uk/books/edition/The_Batsford_Companion_to_Medieval_Engla/ev78b9EJQy0C?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover The Batsford Companion to Medieval England]
* [https://www.google.co.uk/books/edition/Shakespeare_s_Unruly_Women/FKFlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespeare's Unruly Women]
* [https://www.google.co.uk/books/edition/Shakespeare_and_Others/iFEgAQAAIAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Shakespeare and Others]
* [https://www.google.co.uk/books/edition/Kings_and_Chroniclers/L1wpAAAAYAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Kings and Chroniclers]
* [https://www.google.co.uk/books/edition/A_Kingdom_for_a_Stage/UzxlAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover A Kingdom for a Stage]
* [https://www.google.co.uk/books/edition/The_House_of_Commons/Ezz4OZuYVFYC?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover The History of Parliament: The House of Commons 1558-1603 (3 v.)]
* [https://www.google.co.uk/books/edition/Shakespeare_Soul_of_the_Age/nMYCAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover shakespeare, Soul of the Age]
* [https://www.google.co.uk/books/edition/After_Poststructuralism/TOaEAAAAIAAJ?hl=en&amp;gbpv=0&amp;bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 After Poststructuralism: Interdisciplinarity and Literary Theory]
* [https://www.google.co.uk/books/edition/The_Unschooled_Mind/C7WnYtt219IC?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover The Unschooled Mind]
* [https://www.google.co.uk/books/edition/Elizabeth_I/hHZnAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Eliz I]
* [https://www.google.co.uk/books/edition/Dramas_of_Christian_Time/mnIqAQAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Dramas of Christian Time]
* [https://www.google.co.uk/books/edition/Elizabeth_I/XjQmAQAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover Elizabeth I: The Shrewdness of Virtue]
* [https://www.google.co.uk/books/edition/John_Dryden/9Q1aAAAAMAAJ?hl=en&amp;gbpv=1&amp;bsq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;printsec=frontcover John Dryden]
* [https://www.google.co.uk/books/edition/Shakespeare_and_Early_Modern_Political_T/DUwhAwAAQBAJ?hl=en&amp;gbpv=1&amp;dq=%22shakespeare%22+%2B+%22political+propaganda%22&amp;pg=PA259&amp;printsec=frontcover Shakespeare and Early Modern Political Thought]
* [https://www.google.co.uk/books/edition/The_English_History_Play_in_the_age_of_S/5TT-AQAAQBAJ?hl=en&amp;gbpv=1&amp;dq=%22shakespeare%22+%2B+%22political+propaganda%22&amp;pg=PA158&amp;printsec=frontcover The English History Play in the Age of Shakespeare]
* [https://www.google.co.uk/books/edition/Shakespeare_and_the_Political/rEcREQAAQBAJ?hl=en&amp;gbpv=1&amp;dq=%22shakespeare%22+%2B+%22political+propaganda%22&amp;pg=PA215&amp;printsec=frontcover Shakespeare and the Political]
* [https://www.google.co.uk/books/edition/William_Shakespeare_Subject_of_the_Crown/a7G6DAAAQBAJ?hl=en&amp;gbpv=1&amp;dq=%22shakespeare%22+%2B+%22political+propaganda%22&amp;pg=PT18&amp;printsec=frontcover William Shakespeare - Subject of the Crown?]
* [https://www.google.com/search?q=%22shakespeare%22+%2B+%22political+propaganda%22&amp;client=firefox-b-d&amp;hs=4AQ&amp;sca_esv=6d4ade7bd26771c9&amp;udm=36&amp;biw=2510&amp;bih=1307&amp;tbs=cdr%3A1%2Ccd_min%3A2000%2Ccd_max%3A2099&amp;sxsrf=ANbL-n6I6Pkwl7mmdHK6N1xPQXLbGBIOSg%3A1776853062010&amp;ei=RqDoaZUvztiFsg_I5bToDw&amp;ved=0ahUKEwiV6tC8nYGUAxVObEEAHcgyDf0Q4dUDCBM&amp;uact=5&amp;oq=%22shakespeare%22+%2B+%22political+propaganda%22&amp;gs_lp=EhBnd3Mtd2l6LW1vZGVsZXNzIiYic2hha2VzcGVhcmUiICsgInBvbGl0aWNhbCBwcm9wYWdhbmRhIjIIECEYoAEYwwRInQlQxgZYuwdwAXgAkAEAmAF_oAHPAaoBAzEuMbgBA8gBAPgBAZgCAqACVsICCxAAGIAEGKIEGLADmAMAiAYBkAYCkgcBMqAHowOyBwExuAdTwgcDMC4yyAcEgAgB&amp;sclient=gws-wiz-modeless The Nazi Appropriation of Shakespeare: Cultural Politics in]
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'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Chet cunningham!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

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You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Quinlan83|Quinlan83]] ([[User talk:Quinlan83|discussione]] • [[Special:Contributions/Quinlan83|contributi]]) 17:40, 5 June 2026 (UTC)
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&lt;!-- Please add categories and interwikis to the bottom of this page, and also add interwikis to Wikidata. --&gt;
== About this template ==
This is the '''2FA-required''' template. It is used on pages about user rights on [[Wikibooks]] in which [[m:H:2FA|two-factor authentication]] (2FA) is required for both legal and security reasons. For example, 2FA is required for the [[Wikibooks:Interface administrators|interface administrator]] user right on Wikibooks.
=== Usage ===
Add {{tlx|2FA-required}} to a page about a user right on Wikibooks, which will produce the following message:
{{2FA-required}}
Please note that this template should be used '''only''' if the [[m:Wikimedia Foundation|Wikimedia Foundation]] requires 2FA for the role.

== TemplateData ==
{{TemplateData header}}
&lt;templatedata&gt;
{
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}
&lt;/templatedata&gt;

== General information ==
=== What is two-factor authentication? ===
{{see also|m:Help:Two-factor authentication}}
'''Two-factor authentication''' is the addition of security to your Wikimedia global account to prevent your account being '''[[m:Help:Compromised accounts|compromised]]'''. Two-factor authentication contains two factors:
* '''First factor:''' Your password for your account.
* '''Second factor:''' Verification code retrieved from an app on a mobile device or computer.
Before enabling 2FA, please ensure that you have a strong password that is exclusively used for Wikibooks. The Wikimedia Foundation recommends that the password for your global account is not the same as the password for your account on other websites.
=== Why is two-factor authentication required? ===
Occasionally, the Wikimedia Foundation requires two-factor authentication for roles for both legal and security reasons.

== See also ==
* [[m:Template:2FA-required|The same template on Meta-Wiki]]
* [[species:Template:2FA-required|The same template on Wikispecies]]

&lt;includeonly&gt;
&lt;!-- Categories and interwikis go here, and interwikis also go on Wikidata. --&gt;

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This is the '''2FA-required''' template. It is used on pages about user rights on [[Wikibooks]] in which [[m:H:2FA|two-factor authentication]] (2FA) is required for both legal and security reasons. For example, 2FA is required for the [[Wikibooks:Interface administrators|interface administrator]] user right on Wikibooks.
=== Usage ===
Add {{tlx|2FA-required}} to a page about a user right on Wikibooks, which will produce the following message:
{{2FA-required}}
Please note that this template should be used '''only''' if the [[m:Wikimedia Foundation|Wikimedia Foundation]] requires 2FA for the role.

== TemplateData ==
{{TemplateData header}}
&lt;templatedata&gt;
{
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	&quot;format&quot;: &quot;inline&quot;
}
&lt;/templatedata&gt;

== General information ==
=== What is two-factor authentication? ===
{{see also|m:Help:Two-factor authentication}}
'''Two-factor authentication''' is the addition of security to your Wikimedia global account to prevent your account being '''[[m:Help:Compromised accounts|compromised]]'''. Two-factor authentication contains two factors:
* '''First factor:''' Your password for your account.
* '''Second factor:''' Verification code retrieved from an app on a mobile device or computer.
Before enabling 2FA, please ensure that you have a strong password that is exclusively used for Wikibooks. The Wikimedia Foundation recommends that the password for your global account is not the same as the password for your account on other websites.
=== Why is two-factor authentication required? ===
Occasionally, the Wikimedia Foundation requires two-factor authentication for roles for both legal and security reasons.

== See also ==
* [[m:Template:2FA-required|The same template on Meta-Wiki]]
* [[species:Template:2FA-required|The same template on Wikispecies]]

&lt;includeonly&gt;
&lt;!-- Categories and interwikis go here, and interwikis also go on Wikidata. --&gt;

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== About this template ==
This is the '''2FA-required''' template. It is used on pages about user rights on Wikiversity in which [[m:H:2FA|two-factor authentication]] (2FA) is required for both legal and security reasons. For example, 2FA is required for the [[Project:Bureaucratship|bureaucrats]] and [[Project:Interface administrators|interface administrator]] user groups on Wikiversity.

=== Usage ===
Add {{tlx|2FA-required}} to a page about a user right on Wikiversity, which will produce the following message:
{{2FA-required}}
Please note that this template should be used '''only''' if the [[m:Wikimedia Foundation|Wikimedia Foundation]] requires 2FA for the role.

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== General information ==
=== What is two-factor authentication? ===
{{see also|m:Help:Two-factor authentication}}
'''Two-factor authentication''' is the addition of security to your Wikimedia global account to prevent your account being '''[[m:Help:Compromised accounts|compromised]]'''. Two-factor authentication contains two factors:
* '''First factor:''' Your password for your account.
* '''Second factor:''' Verification code retrieved from an app on a mobile device or computer.
Before enabling 2FA, please ensure that you have a strong password that is exclusively used for Wikibooks. The Wikimedia Foundation recommends that the password for your global account is not the same as the password for your account on other websites.
=== Why is two-factor authentication required? ===
Occasionally, the Wikimedia Foundation requires two-factor authentication for roles for both legal and security reasons.

== See also ==
* [[m:Template:2FA-required|The same template on Meta-Wiki]]
* [[species:Template:2FA-required|The same template on Wikispecies]]

&lt;includeonly&gt;
&lt;!-- Categories and interwikis go here, and interwikis also go on Wikidata. --&gt;

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== About this template ==
This is the '''2FA-required''' template. It is used on pages about user rights on Wikiversity in which [[m:H:2FA|two-factor authentication]] (2FA) is required for both legal and security reasons. For example, 2FA is required for the [[Project:Bureaucratship|bureaucrat]] and [[Project:Interface administrators|interface administrator]] user groups on Wikiversity.

=== Usage ===
Add {{tlx|2FA-required}} to a page about a user right on Wikiversity, which will produce the following message:
{{2FA-required}}
Please note that this template should be used '''only''' if the [[m:Wikimedia Foundation|Wikimedia Foundation]] requires 2FA for the role.

== TemplateData ==
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== General information ==
=== What is two-factor authentication? ===
{{see also|m:Help:Two-factor authentication}}
'''Two-factor authentication''' is the addition of security to your Wikimedia global account to prevent your account being '''[[m:Help:Compromised accounts|compromised]]'''. Two-factor authentication contains two factors:
* '''First factor:''' Your password for your account.
* '''Second factor:''' Verification code retrieved from an app on a mobile device or computer.
Before enabling 2FA, please ensure that you have a strong password that is exclusively used for Wikibooks. The Wikimedia Foundation recommends that the password for your global account is not the same as the password for your account on other websites.
=== Why is two-factor authentication required? ===
Occasionally, the Wikimedia Foundation requires two-factor authentication for roles for both legal and security reasons.

== See also ==
* [[m:Template:2FA-required|The same template on Meta-Wiki]]
* [[species:Template:2FA-required|The same template on Wikispecies]]

&lt;includeonly&gt;
&lt;!-- Categories and interwikis go here, and interwikis also go on Wikidata. --&gt;

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== Licensing ==
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    <title>Probability Dilation Theory/Quantum Computing in Dilation Fields</title>
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      <text bytes="2512" sha1="p7yi9e29yb4ndocia1vv1a4r3uc5gi3" xml:space="preserve">&lt;nowiki&gt;= Quantum Computing in Dilation Fields =&lt;/nowiki&gt;

This subpage explores speculative applications of Probability Dilation Theory (PDT) to quantum information processing under nonuniform probabilistic weighting environments.

The discussion is exploratory and does not represent established quantum computing theory. No experimental evidence currently supports the existence of physical PDT dilation fields.

The purpose of this page is to investigate possible mathematical consequences of differential probabilistic weighting on quantum systems and information processing.

== Thought experiment: differential dilation fields ==
Consider two hypothetical quantum computers, A and B, operating in different probabilistic environments represented by PDT dilation fields:

and

The PDT framework asks whether differing probabilistic weighting environments could influence statistical properties of quantum information processing, such as decoherence rates, error probabilities, entropy production, or state distinguishability.

At present no experimental evidence supports the existence of physical PDT dilation fields. The discussion below is therefore exploratory and conceptual.

Standard quantum mechanics is recovered in the limiting case:

In this limit, no probabilistic reweighting occurs and ordinary quantum evolution is obtained.

=== Illustrative probabilistic weighting ===
PDT transforms a baseline probability measure according to

where

ensures normalization of the transformed measure.

Within this framework, different dilation environments may produce distinct effective probability distributions over measurement outcomes or computational trajectories.

=== Illustrative decoherence hypothesis ===
As a purely heuristic example, one may define an effective decoherence rate

where

\Gamma_0 denotes a baseline decoherence rate;

\Gamma_D denotes a hypothetical dilation-modified rate.

This relation is illustrative only and is not derived from established quantum theory.

=== Open research questions ===
Can PDT transformations be formulated on Hilbert spaces?

Are PDT transformations compatible with quantum channels?

Can Fisher information geometry describe dilation evolution?

Do dilation flows correspond to geodesics in information geometry?

Could differential dilation environments influence entropy production in quantum systems?

The present discussion is exploratory and intended to generate mathematically testable questions rather than established physical predictions.</text>
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      <comment>/* Thought experiment: differential dilation fields */ quantum computing</comment>
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      <text bytes="2755" sha1="ohdsvplhgsu3sk19j70i2xdpaqqta3y" xml:space="preserve">&lt;nowiki&gt;= Quantum Computing in Dilation Fields =&lt;/nowiki&gt;

This subpage explores speculative applications of Probability Dilation Theory (PDT) to quantum information processing under nonuniform probabilistic weighting environments.

The discussion is exploratory and does not represent established quantum computing theory. No experimental evidence currently supports the existence of physical PDT dilation fields.

The purpose of this page is to investigate possible mathematical consequences of differential probabilistic weighting on quantum systems and information processing.

== Thought experiment: differential dilation fields ==

Consider two hypothetical quantum computers, A and B, operating in different probabilistic environments represented by PDT dilation fields:

&lt;math&gt;D_A(x)&lt;/math&gt;

and

&lt;math&gt;D_B(x)&lt;/math&gt;.

The PDT framework asks whether differing probabilistic weighting environments could influence statistical properties of quantum information processing, such as decoherence rates, error probabilities, entropy production, or state distinguishability.

At present no experimental evidence supports the existence of physical PDT dilation fields. The discussion below is therefore exploratory and conceptual.

Standard quantum mechanics is recovered in the limiting case:

&lt;math&gt;D_A(x)=D_B(x)=1&lt;/math&gt;.

In this limit, no probabilistic reweighting occurs and ordinary quantum evolution is obtained.

=== Illustrative probabilistic weighting ===

PDT transforms a baseline probability measure according to

&lt;math&gt;
dP_D(x)=\frac{D(x)\,dP(x)}{Z},
&lt;/math&gt;

where

&lt;math&gt;
Z=\int_{\Omega} D(x)\,dP(x)
&lt;/math&gt;

ensures normalization of the transformed measure.

Within this framework, different dilation environments may produce distinct effective probability distributions over measurement outcomes or computational trajectories.

=== Illustrative decoherence hypothesis ===

As a purely heuristic example, one may define an effective decoherence rate

&lt;math&gt;
\Gamma_D=D\,\Gamma_0,
&lt;/math&gt;

where:

* &lt;math&gt;\Gamma_0&lt;/math&gt; denotes a baseline decoherence rate;
* &lt;math&gt;\Gamma_D&lt;/math&gt; denotes a hypothetical dilation-modified rate.

This relation is illustrative only and is not derived from established quantum theory.

=== Open research questions ===

* Can PDT transformations be formulated on Hilbert spaces?
* Are PDT transformations compatible with quantum channels?
* Can Fisher information geometry describe dilation evolution?
* Do dilation flows correspond to geodesics in information geometry?
* Could differential dilation environments influence entropy production in quantum systems?

The present discussion is exploratory and intended to generate mathematically testable questions rather than established physical predictions.</text>
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      <text bytes="2757" sha1="qtx1yziabksyyb4ki1qkzazor2j774p" xml:space="preserve">&lt;nowiki&gt;== Quantum Computing in Dilation Fields ==&lt;/nowiki&gt;

This subpage explores speculative applications of Probability Dilation Theory (PDT) to quantum information processing under nonuniform probabilistic weighting environments.

The discussion is exploratory and does not represent established quantum computing theory. No experimental evidence currently supports the existence of physical PDT dilation fields.

The purpose of this page is to investigate possible mathematical consequences of differential probabilistic weighting on quantum systems and information processing.

== Thought experiment: differential dilation fields ==

Consider two hypothetical quantum computers, A and B, operating in different probabilistic environments represented by PDT dilation fields:

&lt;math&gt;D_A(x)&lt;/math&gt;

and

&lt;math&gt;D_B(x)&lt;/math&gt;.

The PDT framework asks whether differing probabilistic weighting environments could influence statistical properties of quantum information processing, such as decoherence rates, error probabilities, entropy production, or state distinguishability.

At present no experimental evidence supports the existence of physical PDT dilation fields. The discussion below is therefore exploratory and conceptual.

Standard quantum mechanics is recovered in the limiting case:

&lt;math&gt;D_A(x)=D_B(x)=1&lt;/math&gt;.

In this limit, no probabilistic reweighting occurs and ordinary quantum evolution is obtained.

=== Illustrative probabilistic weighting ===

PDT transforms a baseline probability measure according to

&lt;math&gt;
dP_D(x)=\frac{D(x)\,dP(x)}{Z},
&lt;/math&gt;

where

&lt;math&gt;
Z=\int_{\Omega} D(x)\,dP(x)
&lt;/math&gt;

ensures normalization of the transformed measure.

Within this framework, different dilation environments may produce distinct effective probability distributions over measurement outcomes or computational trajectories.

=== Illustrative decoherence hypothesis ===

As a purely heuristic example, one may define an effective decoherence rate

&lt;math&gt;
\Gamma_D=D\,\Gamma_0,
&lt;/math&gt;

where:

* &lt;math&gt;\Gamma_0&lt;/math&gt; denotes a baseline decoherence rate;
* &lt;math&gt;\Gamma_D&lt;/math&gt; denotes a hypothetical dilation-modified rate.

This relation is illustrative only and is not derived from established quantum theory.

=== Open research questions ===

* Can PDT transformations be formulated on Hilbert spaces?
* Are PDT transformations compatible with quantum channels?
* Can Fisher information geometry describe dilation evolution?
* Do dilation flows correspond to geodesics in information geometry?
* Could differential dilation environments influence entropy production in quantum systems?

The present discussion is exploratory and intended to generate mathematically testable questions rather than established physical predictions.</text>
      <sha1>qtx1yziabksyyb4ki1qkzazor2j774p</sha1>
    </revision>
    <revision>
      <id>2815584</id>
      <parentid>2815583</parentid>
      <timestamp>2026-06-13T20:44:21Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <minor />
      <comment>Changing intro.</comment>
      <origin>2815584</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2716" sha1="1chhnxe0pb4mbfcdvdv1etzr2dth9kn" xml:space="preserve">== Introduction ==

This subpage explores speculative applications of Probability Dilation Theory (PDT) to quantum information processing under nonuniform probabilistic weighting environments.

The discussion is exploratory and does not represent established quantum computing theory. No experimental evidence currently supports the existence of physical PDT dilation fields.

The purpose of this page is to investigate possible mathematical consequences of differential probabilistic weighting on quantum systems and information processing.

== Thought experiment: differential dilation fields ==

Consider two hypothetical quantum computers, A and B, operating in different probabilistic environments represented by PDT dilation fields:

&lt;math&gt;D_A(x)&lt;/math&gt;

and

&lt;math&gt;D_B(x)&lt;/math&gt;.

The PDT framework asks whether differing probabilistic weighting environments could influence statistical properties of quantum information processing, such as decoherence rates, error probabilities, entropy production, or state distinguishability.

At present no experimental evidence supports the existence of physical PDT dilation fields. The discussion below is therefore exploratory and conceptual.

Standard quantum mechanics is recovered in the limiting case:

&lt;math&gt;D_A(x)=D_B(x)=1&lt;/math&gt;.

In this limit, no probabilistic reweighting occurs and ordinary quantum evolution is obtained.

=== Illustrative probabilistic weighting ===

PDT transforms a baseline probability measure according to

&lt;math&gt;
dP_D(x)=\frac{D(x)\,dP(x)}{Z},
&lt;/math&gt;

where

&lt;math&gt;
Z=\int_{\Omega} D(x)\,dP(x)
&lt;/math&gt;

ensures normalization of the transformed measure.

Within this framework, different dilation environments may produce distinct effective probability distributions over measurement outcomes or computational trajectories.

=== Illustrative decoherence hypothesis ===

As a purely heuristic example, one may define an effective decoherence rate

&lt;math&gt;
\Gamma_D=D\,\Gamma_0,
&lt;/math&gt;

where:

* &lt;math&gt;\Gamma_0&lt;/math&gt; denotes a baseline decoherence rate;
* &lt;math&gt;\Gamma_D&lt;/math&gt; denotes a hypothetical dilation-modified rate.

This relation is illustrative only and is not derived from established quantum theory.

=== Open research questions ===

* Can PDT transformations be formulated on Hilbert spaces?
* Are PDT transformations compatible with quantum channels?
* Can Fisher information geometry describe dilation evolution?
* Do dilation flows correspond to geodesics in information geometry?
* Could differential dilation environments influence entropy production in quantum systems?

The present discussion is exploratory and intended to generate mathematically testable questions rather than established physical predictions.</text>
      <sha1>1chhnxe0pb4mbfcdvdv1etzr2dth9kn</sha1>
    </revision>
    <revision>
      <id>2815617</id>
      <parentid>2815584</parentid>
      <timestamp>2026-06-14T02:59:08Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <minor />
      <comment>added [[Category:Quantum computing]] using [[Help:Gadget-HotCat|HotCat]]</comment>
      <origin>2815617</origin>
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      <format>text/x-wiki</format>
      <text bytes="2748" sha1="rz8hqli5xjr8rnkvbqt9w91i69gmd35" xml:space="preserve">== Introduction ==

This subpage explores speculative applications of Probability Dilation Theory (PDT) to quantum information processing under nonuniform probabilistic weighting environments.

The discussion is exploratory and does not represent established quantum computing theory. No experimental evidence currently supports the existence of physical PDT dilation fields.

The purpose of this page is to investigate possible mathematical consequences of differential probabilistic weighting on quantum systems and information processing.

== Thought experiment: differential dilation fields ==

Consider two hypothetical quantum computers, A and B, operating in different probabilistic environments represented by PDT dilation fields:

&lt;math&gt;D_A(x)&lt;/math&gt;

and

&lt;math&gt;D_B(x)&lt;/math&gt;.

The PDT framework asks whether differing probabilistic weighting environments could influence statistical properties of quantum information processing, such as decoherence rates, error probabilities, entropy production, or state distinguishability.

At present no experimental evidence supports the existence of physical PDT dilation fields. The discussion below is therefore exploratory and conceptual.

Standard quantum mechanics is recovered in the limiting case:

&lt;math&gt;D_A(x)=D_B(x)=1&lt;/math&gt;.

In this limit, no probabilistic reweighting occurs and ordinary quantum evolution is obtained.

=== Illustrative probabilistic weighting ===

PDT transforms a baseline probability measure according to

&lt;math&gt;
dP_D(x)=\frac{D(x)\,dP(x)}{Z},
&lt;/math&gt;

where

&lt;math&gt;
Z=\int_{\Omega} D(x)\,dP(x)
&lt;/math&gt;

ensures normalization of the transformed measure.

Within this framework, different dilation environments may produce distinct effective probability distributions over measurement outcomes or computational trajectories.

=== Illustrative decoherence hypothesis ===

As a purely heuristic example, one may define an effective decoherence rate

&lt;math&gt;
\Gamma_D=D\,\Gamma_0,
&lt;/math&gt;

where:

* &lt;math&gt;\Gamma_0&lt;/math&gt; denotes a baseline decoherence rate;
* &lt;math&gt;\Gamma_D&lt;/math&gt; denotes a hypothetical dilation-modified rate.

This relation is illustrative only and is not derived from established quantum theory.

=== Open research questions ===

* Can PDT transformations be formulated on Hilbert spaces?
* Are PDT transformations compatible with quantum channels?
* Can Fisher information geometry describe dilation evolution?
* Do dilation flows correspond to geodesics in information geometry?
* Could differential dilation environments influence entropy production in quantum systems?

The present discussion is exploratory and intended to generate mathematically testable questions rather than established physical predictions.

[[Category:Quantum computing]]</text>
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  </page>
  <page>
    <title>Probability Dilation Theory/Fisher Geometry and Dilation Flows</title>
    <ns>0</ns>
    <id>330193</id>
    <revision>
      <id>2815587</id>
      <timestamp>2026-06-13T20:59:37Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>Create subpage for Probability distributions as geometric objects.</comment>
      <origin>2815587</origin>
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      <format>text/x-wiki</format>
      <text bytes="3045" sha1="2ohi25oq1xd1i8vrmoc0k6w9qaio1o3" xml:space="preserve">
== Introduction ==
This subpage explores possible relationships between Probability Dilation Theory (PDT), information geometry, and Fisher metrics on spaces of probability distributions.

The discussion is exploratory and does not represent established physical theory. The purpose of this page is to investigate whether iterative PDT transformations may be interpreted geometrically as flows on statistical manifolds.

== Probability distributions as geometric objects ==
Information geometry studies spaces of probability distributions as geometric manifolds equipped with a metric structure. Within this framework, probability distributions may be viewed as points on a statistical manifold.

PDT transformations modify probability measures through positive dilation fields. This suggests the possibility that iterative PDT transformations may generate trajectories through spaces of probability distributions.

At present this interpretation is exploratory and no formal geometric structure for PDT has been established.

=== Fisher information metric ===
One of the central metrics in information geometry is the Fisher information metric. For a parametric family of probability distributions p(x;\theta), the Fisher information matrix is given by

The Fisher metric provides a natural notion of distance on statistical manifolds and has applications in statistics, information theory, and physics.

=== Dilation flows on probability space ===
Given a probability measure P and dilation field D(x), PDT defines the transformed measure

where

ensures normalization.

Repeated application of PDT transformations generates a sequence of probability measures

which may be interpreted as a trajectory through probability space.

=== Geodesic hypothesis ===
A natural question is whether certain classes of PDT transformations approximate geodesics under the Fisher metric.

No such result is presently known. The possibility that some dilation flows may approximate geodesic motion on statistical manifolds remains an open mathematical question.

=== Fisher distance and entropy evolution ===
PDT studies often examine entropy evolution under repeated transformations. One possible direction for future work is to compare changes in Shannon entropy with geometric distances induced by Fisher information.

This raises questions such as:

Does entropy reduction correspond to shorter Fisher distances?

Do concentrating regimes converge toward special geometric structures?

Can fixed points of iterative PDT transformations be characterized geometrically?

=== Open research questions ===
Can PDT be formulated directly on statistical manifolds?

Are dilation flows compatible with information geometry?

Under what conditions do PDT transformations converge?

Do attractor-like probability structures possess geometric interpretations?

Can curvature emerge naturally from iterative probabilistic reweighting?

The present discussion is exploratory and intended to generate mathematically testable questions rather than established results.</text>
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  </page>
  <page>
    <title>User talk:Morsecodeworld</title>
    <ns>3</ns>
    <id>330194</id>
    <revision>
      <id>2815608</id>
      <timestamp>2026-06-14T02:46:37Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <comment>advert1 ([[m:User:ZbVl/VD|Vandoom]])</comment>
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== 2026-06-14 ==
&lt;div class=&quot;mw-content-ltr&quot; dir=&quot;ltr&quot; style=&quot;text-align: left&quot; lang=&quot;en&quot;&gt;[[File:Information.svg|25px|alt=Information icon]] Hello. Apologies for writing this in English, but I wanted to let you know that one or more of [[Special:Contributions/Morsecodeworld|your recent contributions]] have been undone because they appeared to be promotional. [[:m:en:WP:SOAPBOX|Advertising or using &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikiversity&lt;/span&gt; as a &quot;soapbox&quot;]] are not permitted. Take a look at the welcome pages to learn more about &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikiversity&lt;/span&gt;. Thanks. &lt;/div&gt;&lt;!-- Glow-advert1 @ 1781405200613.9s --&gt;&lt;nowiki&gt;&lt;/nowiki&gt; [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:46, 14 June 2026 (UTC)</text>
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  </page>
  <page>
    <title>User talk:Nancyduncan70</title>
    <ns>3</ns>
    <id>330195</id>
    <revision>
      <id>2815609</id>
      <timestamp>2026-06-14T02:47:20Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <comment>advert1 ([[m:User:ZbVl/VD|Vandoom]])</comment>
      <origin>2815609</origin>
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      <text bytes="809" sha1="n042mc8zy8umi6olpav91iequh0eppp" xml:space="preserve">

== 2026-06-14 ==
&lt;div class=&quot;mw-content-ltr&quot; dir=&quot;ltr&quot; style=&quot;text-align: left&quot; lang=&quot;en&quot;&gt;[[File:Information.svg|25px|alt=Information icon]] Hello. Apologies for writing this in English, but I wanted to let you know that one or more of [[Special:Contributions/Nancyduncan70|your recent contributions]] have been undone because they appeared to be promotional. [[:m:en:WP:SOAPBOX|Advertising or using &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikiversity&lt;/span&gt; as a &quot;soapbox&quot;]] are not permitted. Take a look at the welcome pages to learn more about &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikiversity&lt;/span&gt;. Thanks. &lt;/div&gt;&lt;!-- Glow-advert1 @ 1781405241196.4s --&gt;&lt;nowiki&gt;&lt;/nowiki&gt; [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:47, 14 June 2026 (UTC)</text>
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  <page>
    <title>User talk:~2026-34761-52</title>
    <ns>3</ns>
    <id>330196</id>
    <revision>
      <id>2815611</id>
      <timestamp>2026-06-14T02:49:22Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <comment>advert1 ([[m:User:ZbVl/VD|Vandoom]])</comment>
      <origin>2815611</origin>
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== 2026-06-14 ==
&lt;div class=&quot;mw-content-ltr&quot; dir=&quot;ltr&quot; style=&quot;text-align: left&quot; lang=&quot;en&quot;&gt;[[File:Information.svg|25px|alt=Information icon]] Hello. Apologies for writing this in English, but I wanted to let you know that one or more of [[Special:Contributions/&amp;#126;2026-34761-52|your recent contributions]] have been undone because they appeared to be promotional. [[:m:en:WP:SOAPBOX|Advertising or using &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikiversity&lt;/span&gt; as a &quot;soapbox&quot;]] are not permitted. Take a look at the welcome pages to learn more about &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikiversity&lt;/span&gt;. Thanks. &lt;/div&gt;&lt;!-- Glow-advert1 @ 1781405366206.5s --&gt;&lt;nowiki&gt;&lt;/nowiki&gt; [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:49, 14 June 2026 (UTC)</text>
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  </page>
  <page>
    <title>Category:Quantum computing</title>
    <ns>14</ns>
    <id>330197</id>
    <revision>
      <id>2815612</id>
      <timestamp>2026-06-14T02:56:29Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <comment>Created page with &quot; [[Category:Physics]] [[Category:Computer science]]&quot;</comment>
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[[Category:Physics]]
[[Category:Computer science]]</text>
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