User talk:Jtneill

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&lt;!-- {{Long wikibreak|image=Leaf_1_web.jpg|[[User:Jtneill|Jtneill]]|mid-Jan, 2012.}} --&gt;
{{{{TALKPAGENAME}}/Header}}  {{TOCright}}

== Your feedback is welcome at [[User talk:Username142857]] ==

Dear my mentor, I believe we have already seen [[User:Username142857]] making too many non-Wikiversity questions at [[Wikiversity:Candidates for Custodianship/MathXplore]] and [[Wikiversity talk:Custodianship/Archive 6]]. In the beginning, I answered them one by one as part of demonstrating my competency to answer questions as a custodian candidate (and they were somewhat related to my global contributions) and courtesy to discussion participants. However, by facing [[special:diff/2631774]] and [[special:diff/2618170]] (editing discussion archives, re-opening closed discussions), I started to believe that we should bring an end to their excessive non-Wikiversity usage of Wikiversity (talk) namespaces. According to [[:w:User talk:Username142857]] (especially [[:w:special:diff/1073391896]]), [[User:Username142857]] is evaluated as {{tq|the other editors are tired to waste their time to read and answer your non-useful edits.}} and I think they are doing the similar thing at Wikiversity. Our community may have limited tolerance for such behavior. If you had any experience of handling such issues in the past, your feedback may be helpful to allow [[User:Username142857]] to improve their behavior. Thank you for your attention and mentoring. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:21, 9 June 2024 (UTC)
: {{ping|MathXplore}} Thanks for the heads up. Sorry for slow response. I'm recovering from COVID, but on way back. Thankyou for your very patient, clear, and supportive feedback on Username142857's talk page which, along with Mikeu, seems to have communicated the concerns and hopefully lead to a change/improvement in behaviour. What a great example of handling challenging behaviour courteously. Fingers crossed. Keep well. Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:39, 22 June 2024 (UTC)

== [[:b:Motivation and emotion/Book/2024/Free will and neuroscience]] ==

Hello, can this be related to your project? Should this be imported here? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:10, 30 July 2024 (UTC)
: Sorry, the page has been deleted, should we request temporary restoration for import, or should we just ask the author to resubmit to Wikiversity? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:29, 30 July 2024 (UTC)
::Thank-you for pointing this out. Yes, it does look like one of my students' editing. It is a little puzzling how the user ended up on Wikibooks. It is OK that that the wikibooks page has been deleted because the user also appears to be underway here: [[Motivation and emotion/Book/2024/Free will and neuroscience]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:53, 30 July 2024 (UTC)

== [[Template:Subst:ME/BCS]] ==

Hello, should this template be kept for your project? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:42, 31 July 2024 (UTC)

:Yes, please - but it could be moved from Template into a subpage of [[Motivation and emotion]]. Note that we are actively using the template at the moment to help build out the [[Motivation and emotion/Book/2024]] pages. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:43, 1 August 2024 (UTC)

== [[:File:Rejection sensitivity chart.webp]] ==

One of your students uploaded this image to Commons as part of [[Motivation and emotion/Book/2024/Rejection sensitivity]]. Unfortunately, it's meaningless AI-generated sludge. Can this image be removed from the chapter to allow it to be deleted from Commons?

(You may want to have a word with your students about AI-generated content; I think some of the text in this chapter was generated by ChatGPT as well.) [[User:Omphalographer|Omphalographer]] ([[User talk:Omphalographer|discuss]] • [[Special:Contributions/Omphalographer|contribs]]) 02:52, 6 August 2024 (UTC)
: {{ping|Omphalographer}} Great, thanks for picking this up and letting me know. Yes please, delete. I've given the student a heads-up here: [[User talk:Yonis Yousufzai]]. We're covering genAI in classes this week {{smile}}. Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:25, 6 August 2024 (UTC)

== [[Wikiversity:Bots/Status#Leaderbot]] ==

Hi, is there a chance you can approve this bot request (or otherwise let me know if there are any issues)? Thanks in advance. [[User:Leaderboard|Leaderboard]] ([[User talk:Leaderboard|discuss]] • [[Special:Contributions/Leaderboard|contribs]]) 15:03, 15 September 2024 (UTC)

== VDT - U3126684 chapter ==

Hi James ! I saw you added the hanging indent which is amazing, thank you so much! However, I had a few references missing and I tried to add them in but they didn't keep the required APA formatting. I deleted the template and reused the hanging indent template but it won't keep any formatting. Can you please help me fix it? 
[[Motivation and emotion/Book/2024/Vulnerable dark triad, motivation, and emotion|Motivation and emotion/Book/2024/Vulnerable dark triad, motivation, and emotion - Wikiversity]] [[User:U3126684|U3126684]] ([[User talk:U3126684|discuss]] • [[Special:Contributions/U3126684|contribs]]) 11:16, 3 October 2024 (UTC)

:James, I figured it out! I was just missing the &quot;}}&quot; at the end of the text... all solved! [[User:U3126684|U3126684]] ([[User talk:U3126684|discuss]] • [[Special:Contributions/U3126684|contribs]]) 11:31, 3 October 2024 (UTC)

== Your feedback may be needed at [[User talk:Tule-hog]] ==

Hello, user:Dan Polansky is currently communicating with a participant on this talk page. As Dan's mentor, I thought you may want to provide feedback so I came here for a notice. ({{ping|Guy vandegrift}} Your feedback is also welcome). [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 06:20, 7 October 2024 (UTC)

:Thanks for bringing this to my attention.  I will keep up with further developments. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 00:07, 8 October 2024 (UTC)

== [[General health and well-being]] ==
This page was in the proposed-deletion state for over 3 months, with no opposition. Should I feel free to delete the page? I guess it seemed to be a good idea back in 2011 (at least as a stub to get things started), but no one expanded it into anything really useful during all these years. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:24, 11 October 2024 (UTC)

:Hi Dan - thanks for checking - yes, it can go - I've removed the one incoming link to this page. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:39, 11 October 2024 (UTC)

== Enquiry about Correct Setup of Wikiversity? ==
Hi James, 
I just had a few questions regarding my Setup on Wikiversity:

1. We are asked to enable the Visual Editor. Have I done this correctly? Or how do I do it if I have not?

2. Have I chosen a book chapter and inserted my name correctly?

3. There isn’t a discussion forum page on our UCLearn for me to comment on, for the assessment, so where should I comment?

Thank you, I look forward to hearing back from you.
[[User:Hcoad|Hcoad]] ([[User talk:Hcoad|discuss]] • [[Special:Contributions/Hcoad|contribs]]) 14:27, 2 August 2025 (UTC)

:@[[User:Hcoad|Hcoad]]:
:# To access the Visual Editor, use &quot;Create&quot; for the first edit on a page, or &quot;Edit&quot; thereafter
:# Sign-up looks good
:# You can create a new discussion thread on UCLearn about a topic of interest or respond to existing threads such as &quot;What do you really want to learn about?&quot;
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:34, 2 August 2025 (UTC)

== Problem with curator ==

Reading above, may i address you as James? If so, hello James, i have a problem with a curator and would ask if you are a contact to talk about it. If not, sorry to bother you. Kind regards, [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 21:19, 10 October 2025 (UTC)

:Hi Harold,
:Thanks for getting in touch.
:Sorry about the teething issues in getting underway with your contributions to Wikiversity.
:Let's hopefully have a constructive discussion here, which you've initiated: [[Wikiversity:Request custodian action#Contest removal of article]]
:Sincerely,
:James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:38, 11 October 2025 (UTC)
::@[[User:Jtneill|Jtneill]] Hi James, 
::Thank you very much for sending me the article text, I really appriciate that. If not to much to ask, could you also send me the template? Template:Condensed matter physics see: User:Harold Foppele/Quantum A Matter Of Size.
::Did you read the disucussion with Dan Polansky? I think its rather weird. I answered all his questions truthfully, since i have nothing to hide. (see my user page) And than he started some trivia about the double slit expiriment, went on without listening. Like the article was a sort of explosive that must be removed ASAP. That is not the way a curator should behave (my opinion). 
::I could acctually use a mentor physics to avoid mistakes in the future.
::I know both my articles have flaws but i can fix that in time.
::Do you maybe have suggestions?
::Last but not least, thanks again for the time you took to help me !!! Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 09:14, 12 October 2025 (UTC)
: @James: To reduce or eliminate further risk that I am abusing my curator priviledges in relation to suspected copyright violation (I don't think I am, but my point of view can be skewed), I can start tagging material for copyright violation using a template (does not require curator privileges). That should address concerns? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:01, 13 October 2025 (UTC)
::@[[User:Dan Polansky|Dan Polansky]] As long as you remove the insulting (in my opinion) remarks on both articles and remove the tag -since it does not violate '''[[creativecommons:by-sa/3.0/|CC-BY-SA 4.0]] license'''- i will be satisfied. As i explained, Wikipedia use a free-to-use policy. Also could you please clarify this code: &lt;nowiki&gt;{{subst:&lt;/nowiki&gt;[[Template:No thanks|no thanks]]|pg=User:Harold Foppele/Quantum A Matter Of Size|url=&lt;nowiki&gt;{{{url}}}&lt;/nowiki&gt;}} [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • . After this is resolved i'm willing to consider this complaint closed. Maybe we can start over with a new and different conversation, since I strongly believe in AGF.  You have a way much longer experience on Wikiversity than I do, so perhaps you could help me in a friendly and constructive way? It seems we have a lot in common and I shall gladly listen to any comments.
::CC @[[User:Jtneill|Jtneill]] Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 09:16, 13 October 2025 (UTC)
::: The page [[User:Harold Foppele/Quantum A Matter Of Size]] currently features multiple sentences from a CC-BY-SA source without using quotation marks. My determination is that the page shows copyright violation (failure to ''attribute'') of CC-BY-SA and should therefore be deleted.
::: If you, James, remove the copyright violation tagging, I will understand it as you taking responsibility for a possible copyright violation and I will probably disengage (or do I have a duty to take more pains and try to override your assessment?) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:31, 13 October 2025 (UTC)
::: As for &quot;As i explained, Wikipedia use a free-to-use policy&quot;: that seems to be a misunderstanding or too vague understanding; Wikipedia uses CC-BY-SA copyright license, which requires proper ''attribution'' of authorship, which could have been done in the edit summary that created the article, but was not done. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:35, 13 October 2025 (UTC)
::::@[[User:Dan Polansky|Dan Polansky]] It has already been added, as you would have seen upon checking. I would still appreciate a response to the other points I mentioned earlier, if you are willing to continue the discussion. If not, your choise. CC:@[[User:Jtneill|Jtneill]] Cheers[[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 10:08, 13 October 2025 (UTC)
: James, as my mentor in my role of a custodian, if you want me to do something, or if you have a recommendation for me, please let me know on my talk page. I am struggling to figure out how to navigate these waters. You can also use email if it seems better from some perspective. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:21, 13 October 2025 (UTC)
::@[[User:Dan Polansky|Dan Polansky]] Why not take a step back? I offered you a solution and a possibility to cooperate instead of continuing a conflict. I still believe that working together is more productive than arguing over small details. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 10:26, 13 October 2025 (UTC)
:::The discussion at this talk page ended not very fruitfully.
:::Pitty, i really tried to make piece.
:::Yet I am not the only one complainting about Dan’s behaviour.
::: 
:::Anything I can do (or you) ?
:::Am I free to remove remarks and/or tags?
:::I dont want to end up in an editwar.
::: 
:::Sorry to have asked so much of your time [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 15:54, 13 October 2025 (UTC)

Thanks, both. May I suggest:
* {{ping|Harold Foppele}}: Any text you don't write yourself needs appropriate attribution or removal, otherwise it runs the risk of copyright violation. For example, this message appears on each edit source screen underneath the edit summary box: &quot;Do not copy text from other websites without permission. It will be deleted.&quot; If text is copied from Wikipedia it needs to be acknowledged as such because it is licensed under CC-by-SA which allows re-use but requires acknowledgement. Such acknowledgement could be made in the edit summary when the contribution is first made. If not, then the next best could be to put quotation marks around copied text and a link to the source(s) of the text.
* {{ping|Dan Polansky}}: Appreciate your administrative work. Let's try to AGF and work constructively with new users who are learning how to contribute. Wikiversity is a learning environment.
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 20:42, 13 October 2025 (UTC)

:@[[User:Jtneill|Jtneill]] Thank you very much. I hope it will work out since Dan does not respond, to me that is. Could you find time to look at the revised [[User:Harold Foppele/Quantum A Matter Of Size]] i made additions to it, but since it is a mix of WP, other sources and OR, it is alomost impossible to keep quoting. So i made a general intro. Is that enough? Also 99% of the [[]] refer directly to WP since WV does not have most of the words/pages. I also recreated the template so that it shows all original text/items. The new section ==Tunneling== is not cited yet, but it wiil be when I have time. Can I remove the tags myself? Thanks again [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 21:21, 13 October 2025 (UTC)
::Looks like a solid chunk is copied from Wikipedia: https://www.copyscape.com/view.php?o=4829&amp;u=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FMesoscopic_physics&amp;t=1760433515&amp;s=https%3A%2F%2Fen.wikiversity.org%2Fwiki%2FUser%3AHarold_Foppele%2FQuantum_A_Matter_Of_Size&amp;w=66&amp;i=1&amp;r=10
::without appropriate acknowledgement.
::Some ways to deal with this appropriately include:
::# Acknowledge the source in the edit summary when content is added to the page
::# Using quotation marks and citations to indicate the source of any content which you haven't authored yourself
::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:02, 14 October 2025 (UTC)
:::The &quot;chunk&quot; is correct :) I took that since it fits perfect to the article. At the top of the page I quoted:
:::{Wikipedia [[wikipedia:Mesoscopic_physics|Mesoscopic physics]]&lt;nowiki&gt;}}&lt;/nowiki&gt;
:::[[creativecommons:by-sa/4.0/|License CC-BY-SA 4.0]]
:::In Edit summary: The first section of this article is copied from Wikipedia &quot;Mesoscopic physics&quot;
:::Is that sufficient ?
:::I did cite almost everything what is not so much requested in Wikiversity as far as i found out, but is a first requirement in Wikipedia.
:::Is it OK if I remove the tags ? Thanks [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 10:51, 14 October 2025 (UTC)
::::I think it would be more transparent and demonstrate greater academic integrity to use quotation marks for text which is copied from elsewhere, especially because there was no appropriate edit summary when the text was added to the page.
::::[https://en.wikiversity.org/w/index.php?title=User%3AHarold_Foppele%2FQuantum_A_Matter_Of_Size&amp;diff=2760582&amp;oldid=2760574 Example of how this might be done].
::::I don't suggest removing the copyright tag until copied text is more clearly quoted and cited and there is consensus that it [[wikt:pass muster|passes muster]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:52, 14 October 2025 (UTC)
:::::Thank you SO MUCH !!  I had no idea that a &lt;blockquote existed nor what it does. This is the first time i used a Wikipedia copy into Wikiversity. So a simple explanation, as you gave me now, would have prevented all this. :) I changed the layout a bit to make it view nicer. Is this required also for my own publications on Wikipedia? Thanks again!! and a goodnight to you [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 12:28, 14 October 2025 (UTC)
::::::I decided to re-write the copyrighted text in my own words. It feels better this way, what do you think? [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 13:07, 14 October 2025 (UTC)
:::::::Great, I think that makes a big difference to rewrite in your own words. I've removed the copyright tag.
:::::::Let me know if I can do anything else as you go along. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:03, 15 October 2025 (UTC)
:::::::: The page still contains copyright violation. I am starting to track problems at [[User:Dan Polansky/Problem reports (about Wikiversity problems)]]. I will disengage from Harold Foppele; this is not being productive and can lead to my harm and thereby harm to the English Wikiversity. I have seen this kind of people elsewhere: I explained a class/type of a problem to the person and pointed to an example for clarity and the person corrected just the single item I gave as an example. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:17, 15 October 2025 (UTC)
:::::::::@[[User:Dan Polansky|Dan Polansky]] Since you want to take this personally instead of having a civilized conversation, I will not engage in a mud-throwing contest or labeling people as “this kind of people&quot;. I saw your problem report and I seriously question your objectivity as a science debater. You took ONE paragraph from an article—a paragraph that had been modified (as your question mark even shows)—plus a scientific debate over a previously accepted article on Wikipedia. You completely ignored the accepted contributions I have made to Wikipedia. Yet this alone is enough for you to request that a contributor be blocked.
:::::::::What do I gain from spending hours and hours doing research for a new article? Hours and hours searching for proper references? Hours writing and rewriting the text? How much do I get paid? Nothing. How much honor or credit do I receive? None. So what &quot;kind of people&quot; am I? [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 08:21, 15 October 2025 (UTC)
:::::::::: DFX. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:26, 15 October 2025 (UTC)
:::::::::::Exactly my point. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 09:19, 15 October 2025 (UTC)
:Thanks [[User:Harold Foppele|Harold]] and [[User:Dan Polansky|Dan]] — I appreciate your considerations and communications. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:51, 15 October 2025 (UTC)

== Peer review ==

@[[User:Jtneill|Jtneill]] Hello James, I hope you are doing well. The 2 articles I wrote are now ready to be published. Is there some kind of peer review possible? I tried to find some help at [[Portal:Particle physics]] but all data there is very old. How can we move forward from this? Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 09:52, 16 October 2025 (UTC)

:Perhaps try [[Wikiversity:Colloquium]] - that's the general way to communicate with English Wikiversity users/editors. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:08, 17 October 2025 (UTC)

== Hello James, I need your help. ==

Could join the discussion with us in [[Wikiversity:Colloquium#Concern regarding curator conduct User:Dan Polansky]]

We would like to solicit your input on this matter.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 03:54, 17 October 2025 (UTC)

== Quantum ==

Hello James, If you have time could you lease look at [[Quantum]]. An essay like page with simple information, that might attract students. I Know its not your field, but maybe it appeals to you. Thanks, [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 23:39, 18 October 2025 (UTC)

== ShakespeareFan00 ==

Goodevening, please, if you have time, take a look at the edits made by this user. A few hundred in 2 days ! Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 20:35, 31 October 2025 (UTC)

== When is a quote or blockquote needed? ==

Hi James, I hope you are doing well. I did wrote some articles and parts off them at Wikipedia. If i want to use parts of it at Wikiversity do i still need to quote that parts? Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:19, 2 November 2025 (UTC)

:Basically, if you didn't author text which is being added, then the genesis of the text needs to be made clear (e.g, edit summary, quotation etc.) It is also possible to import pages (e.g., from Wikipedia) which brings in the full edit history. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:38, 3 November 2025 (UTC)

== Publishing transcripts ==

Hi James, Is it allowed to publish a transcript in Wikiversity as per my example at [[User:Harold Foppele/sandbox-2]]. If not, then I remove the page ofcourse. I think it could be nice if I edit it to make it easy accessible in various Wikipages.
But again, if its not allowed, i remove it. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:28, 6 November 2025 (UTC)

== User:Dan Polansky ==

@Jtneill , Hi James, You are a curator/bureaucrat, if i'm not mistaken. Please look at: [[User:Dan Polansky/Problem reports (about Wikiversity problems)]] I feel outright insulted and ask you (if you can) to put an end to it. Thanks [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:59, 6 November 2025 (UTC)
: I wrote: &quot;The user account created articles in the subject of quantum mechanics that use wiki-voice and do not state the author. Since it is very likely that he does not understand quantum mechanics as per evidence in the revision history of his user talk page, it is also likely that they contain countless errors. The articles are presented to the reader as valid referenced content, not as one person's exercise in who-knows-what. Preventing the user account from creating new pages and moving all his articles to user space would address the issue.&quot;
: I think it is accurate. By now, we have enough evidence I think that the user account is a troll account, an intentional disruptor. There are multiple behavioral signs, both in Wikipedia and in Wikiversity.
: I propose an indef block of the user account. An alternative is not to feed into this troll account. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:03, 6 November 2025 (UTC)
::Well well here we go again [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 18:18, 6 November 2025 (UTC)
::: I opened [[Wikiversity:Request custodian action#Indefinite block for Harold_Foppele]]. I fear it will be in vain. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:26, 6 November 2025 (UTC)
::::You are allowed to hope [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 18:42, 6 November 2025 (UTC)

== Moving to personal namespace ==

What are the policies or customs on Wikiversity for moving pages to personal userspace? Isn't there a risk that Wikiversity will turn into a blogging platform where many users will cultivate pages in their userspace and the outside world will not benefit from it?

I see moving to ns user as a frequent suggestion in Requests for deletion (RFD). I would understand moving to ns Draft, which is clearly defined and there is a chance that the resource will then get into the main ns, thus serving the community. I would understand the suggestion to move to another wikiproject, where the text will serve the community. But I don't really understand the frequent moves to personal ns. Since it's in the RFD, it should either be kept or deleted. If someone contributes to Wikiversity, they automatically agree to its policies and also to the fact that they don't own the pages and someone can put them up for deletion. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:36, 22 November 2025 (UTC)

I personally don't need a free website to host my pages. How would I get rid of the unfinished [[Pomology]] meta course if it was moved to my NS? ([https://en.wikiversity.org/wiki/Wikiversity:Requests_for_Deletion#c-Dan_Polansky-20251121091100-Juandev-20251120220900 Moving it to my own NS is suggested in RFD]). I'm putting it in the Request for deletion because, even though I started it, it looks like other editors had significant input there. Will I have the right to request speedy deletion if the pages are moved to my user ns?

I think this tactic of moving to personal space is poorly thought out, but it has become the norm.

Is there any guideline or discussion from before? If something appears in a deletion request, the majority decides that it should be moved to user ns, how can the person in question defend themselves that they don't want it in their own ns? It seems the community is pressuring the original author to agree to deletion. It seems that the user ns is an untouchable territory into which the community has the right to throw whatever it thinks from the main ns. So why aren't those pages deleted when the community decides that they don't belong in the main ns? --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:30, 22 November 2025 (UTC)

{{ping|Juandev}} I replied on your talk page. But here's another version: Personally, in general, I try to keep my notes etc. in user space. Then if I have something more developed to share and collaborate on, then main space. Draft could be helpful to keep main space tidy, but is very quiet/unused, so in reality most drafts are in main space. But if the content is dubious, underdeveloped, lacking citation/peer review etc. then delete, or user space if it could still be developed. That's roughly how I see it. But everyone has a slightly different view/preference, so discuss to develop consensus. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:48, 22 November 2025 (UTC)

== Ninefold Resonance Theory ==
Dear Jtneill, I noticed that when you deleted [[Ninefold Resonance Theory]], you accidentally deleted the article in my own user space as well. However, I got the impression that most users felt that it should be allowed to exist in my own user space. I thought long and hard about my theory and I'm disappointed that it's gone now... Could you move the article back to my own user space, so not in the main space? I look forward to hearing from you! Kind regards, [[User:S. Perquin|S. Perquin]] ([[User talk:S. Perquin|overleg]] • [[Special:Contributions/S. Perquin|bijdragen]]) 06:22, 28 November 2025 (UTC)

:Nevermind. I will move all my ideas to everybodywiki.com. 😄 Kind regards, [[User:S. Perquin|S. Perquin]] ([[User talk:S. Perquin|overleg]] • [[Special:Contributions/S. Perquin|bijdragen]]) 06:36, 28 November 2025 (UTC)
::Could you please e-mail me the source code of the deleted page? Kind regards, [[User:S. Perquin|S. Perquin]] ([[User talk:S. Perquin|overleg]] • [[Special:Contributions/S. Perquin|bijdragen]]) 06:42, 28 November 2025 (UTC)
:[[User:S. Perquin|S. Perquin]]: Apologies, the user page version was accidentally deleted. It has now been restored. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:25, 29 November 2025 (UTC)
::Thank you! ☺️ Kind regards, [[User:S. Perquin|S. Perquin]] ([[User talk:S. Perquin|overleg]] • [[Special:Contributions/S. Perquin|bijdragen]]) 06:58, 29 November 2025 (UTC)
:::All pages in my user space have been moved to EverybodyWiki. Could you perhaps delete all the pages with the {{tl|speedy}} template on it? Kind regards, [[User:S. Perquin|S. Perquin]] ([[User talk:S. Perquin|overleg]] • [[Special:Contributions/S. Perquin|bijdragen]]) 07:08, 29 November 2025 (UTC)
::::[[User:S. Perquin|S. Perquin]]: The main space redirects and all your user sub-pages have been deleted. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:25, 1 December 2025 (UTC)
:::::Thank you! Kind regards, [[User:S. Perquin|S. Perquin]] ([[User talk:S. Perquin|overleg]] • [[Special:Contributions/S. Perquin|bijdragen]]) 08:24, 1 December 2025 (UTC)

== Vandalism ==

{{ping|Jtneill}} May I draw your attantion to this!
==== 6 December 2025 ====

* cur[https://en.wikiversity.org/w/index.php?title=Chaos_Theory_Extended&amp;diff=prev&amp;oldid=2778412 prev] &lt;bdi&gt;[https://en.wikiversity.org/w/index.php?title=Chaos_Theory_Extended&amp;oldid=2778412 13:15, 6 December 2025]&lt;/bdi&gt; [[User:Revolving Doormat|&lt;bdi&gt;Revolving Doormat&lt;/bdi&gt;]] [[User talk:Revolving Doormat|discuss]] [[Special:Contributions/Revolving Doormat|contribs]]  75,351 bytes +279  request speedy delete under CSD1 [https://en.wikiversity.org/w/index.php?title=Chaos_Theory_Extended&amp;action=edit&amp;undoafter=2777042&amp;undo=2778412 undo][[Special:Thanks/2778412|thank]] [[Special:Tags|Tag]]: [[Wikiversity:VisualEditor|Visual edit: Switched]]

[[User:Revolving Doormat|&lt;bdi&gt;Revolving Doormat&lt;/bdi&gt;]] account created today 
at the same time as = &lt;bdi&gt;~2025-38873-79&lt;/bdi&gt; =

So I assume they are all the same.
Am I allowed to remove the delete template by myself?
Greetings [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 16:41, 6 December 2025 (UTC)

:We are not the same person. I came here from an AfD on Wikipedia and your page creation ban here: https://en.wikipedia.org/wiki/Wikipedia:Administrators%27_noticeboard/Incidents#c-Ldm1954-20251205133800-Requesting_page_creation_block_of_User:Harold_Foppele
:The temp user already identified that I notified WP about the same activity on WV, and that brought them here. [[User:Revolving Doormat|Revolving Doormat]] ([[User talk:Revolving Doormat|discuss]] • [[Special:Contributions/Revolving Doormat|contribs]]) 17:08, 6 December 2025 (UTC)
::Its so coincidental that you all share the same IP range isn't it? Using an empty account? [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:19, 6 December 2025 (UTC)
:::The user already identified their WP account and my WP user id is the same one I have here. I don't believe you have access to our IP addresses, but but based on their WP biography, that would also be impossible. I will not be engaging with you further. [[User:Revolving Doormat|Revolving Doormat]] ([[User talk:Revolving Doormat|discuss]] • [[Special:Contributions/Revolving Doormat|contribs]]) 17:25, 6 December 2025 (UTC)
::::What you believe or not is up to you [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:41, 6 December 2025 (UTC)

== User Dan Polansky ==

I want to draw your attention to the edits (mainly copy/paste) by [[user:Dan Polansky|Dan Polansky]] today. Still trying to act as curator? They continue their previous harassment. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:07, 12 December 2025 (UTC)

== Happy New Year, Jtneill! ==
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{{Center|{{resize|179%|'''''[[New Year|Happy New Year]]!'''''}}}}
'''Jtneill''',&lt;br /&gt;Have a prosperous, productive and enjoyable [[New Year]], and thanks for your contributions to Wikiversity.
&lt;br /&gt;[[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:10, 2 January 2026 (UTC)&lt;br /&gt;&lt;br /&gt;
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&amp;nbsp;&amp;nbsp;&amp;nbsp;''{{resize|88%|Send New Year cheer by adding {{tls|Happy New Year fireworks}} to user talk pages.}}''
{{clear}}&lt;!-- From template:Happy New Year fireworks --&gt;

== Please delete [[MediaWiki:Gadget-WikiSign.js]] ==

Reason: This is a request by the author (major contributor). Custodians don't have interface admin rights, so custodians cannot delete this page. Bureaucrats can delete this page by temporarily adding themselves to the interface admin user group ([[User_talk:Jtneill/Archive/2024#Please_delete_MediaWiki:Wikidebate.js]]). Thank you for your attention. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 09:11, 11 February 2026 (UTC)

== DELETE request ==

Please DELETE [[Creating Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future]] to [[Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future]]. I created the article with an erroneous name. I will recreate it with the name I want. Thanks, [[User:DavidMCEddy|DavidMCEddy]] ([[User talk:DavidMCEddy|discuss]] • [[Special:Contributions/DavidMCEddy|contribs]]) 20:15, 11 February 2026 (UTC)
: {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:12, 13 February 2026 (UTC)

== Archiving ==

Hi and hello @[[User:Jtneill|Jtneill]] I did some archiving from Colloquium and RCA. If you have time that I'm on the right track? It where only a few, so if I did wrong, its easily undone, otherwise I continue as per request. Thanks [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 19:21, 12 February 2026 (UTC)

:@[[User:Harold Foppele|Harold Foppele]] Please remember to user &lt;nowiki&gt;{{archive|Wikiversity:Colloquium}}&lt;/nowiki&gt; instead of &lt;nowiki&gt;{{archive}}&lt;/nowiki&gt; so that people who find themselves in the archives know where to go if they are unsure of anything. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 07:12, 13 February 2026 (UTC)
::@[[User:PieWriter|PieWriter]] I have literally no idea what you are talking about. So elaborate please. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 08:53, 13 February 2026 (UTC)
:::Ahhh I see what you mean. Strange that you comment on MY edits only. NONE of the archive templates at WC archive have that. Did you overlook that?[[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 09:13, 13 February 2026 (UTC)
::::That’s why the discussion parameter is red linked, I am working on that. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:22, 13 February 2026 (UTC)
:::::Well, you could have said that instead. I think it's a bit overdone, since the page title is reads already Archive. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 09:26, 13 February 2026 (UTC)
::::::New users will click on the red linked, which brings them to create the talk page, which is not watched so they won’t receive a response to their question. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:15, 13 February 2026 (UTC)
:::::::That is true [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 12:58, 13 February 2026 (UTC)

== Email ==

I sent you an email about a private abuse filter, feel free to take a look. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:39, 15 April 2026 (UTC)

== AI slop, ownership, and wikilawyering. ==

Using AI images is worse than no images. Your constant reverting of reasonable edits removing images you prompted on pages you wrote would be considered [[w:wp:OWN]]ership on Wikipedia; even if there is no general guideline on Wikiversity the spirit of not having the final say because just you made the page is applicable to all Wikimedia wikis. Reverting a reasonable edit because it lacks an image seems like [[w:wp:WIKILAWYER]]ing— I don’t know if edit summaries are ''required'' here, but I doubt it, and on most wikis they are simply recommended. Not having one doesn’t invalidate the edit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 05:27, 26 April 2026 (UTC)

:I understand that you don't like many AI images because you consider them slop. My view is that some of these AI images can be useful for educational purposes.
:I understand that you think an alternative or no image is better than some AI images. My view is that some AI images are better than no image and are either useful in addition to alternative images or more useful than some alternatives.
:May I suggest deciding first on Commons whether to keep an image, rather than removing from Wikiversity and then nominating for deletion on Commons because of no use.
:I have no interest in edit warring. I'll invite [[WV:RCA]] to review your recent edits. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:20, 26 April 2026 (UTC)

== You may be an eligible candidate for the U4C election ==

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

The [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee (U4C)]] seeks candidates for the 2026 election. The U4C is the global committee responsible for overseeing enforcement of the [[foundation:Special:MyLanguage/Policy:Universal Code of Conduct|Universal Code of Conduct]]. Elections are held annually, if elected a committee member serves for two years.

This year the U4C requires candidates to hold administrator rights on at least one wiki, which is why you are being contacted as you appear to hold this right. There are other requirements, such as candidates must be at least 18 years old and may not be employed by the Wikimedia Foundation or other related chapters and affiliates. You can find more information in the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026#Call_for_Candidates|call for candidates on Meta-wiki]]. Additionally, the committee's working language is English; some ability to communicate in English is required.

The election opens on 18 May, if you are eligible and interested you have until 10 May to submit your candidacy. There will week between for candidates to answer questions from the community. Voting takes place privately in [[m:Special:MyLanguage/SecurePoll|SecurePoll]], successful candidates must receive at least 60% support. More information is available on [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|the 2026 Elections page]], including timelines and other candidacy information. If you read over the material and consider yourself qualified, please consider submitting your name to run for the committee. If you think someone else in your community might be interested and qualified, please encourage them to run.

In partnership with the U4C -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User_talk:Keegan (WMF)|talk]]) 18:32, 28 April 2026 (UTC) &lt;/div&gt;
&lt;!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Keegan_(WMF)/test&amp;oldid=30471751 --&gt;

== Thoughts about Wikinews closure ==

I think Wikiversity could bring in Wikinews users possibly. Thoughts? @[[User:Jtneill|Jtneill]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 23:05, 13 May 2026 (UTC)

:Welcome. Sorry for the loss of Wikinews. I hope WN editors can find their way into contributing to WMF sister projects most aligned with their interests and skills, including Wikiversity. For me, the key here is alignment with [[Wikiversity:Mission]]. It may take some time to work out what's possible. As @[[User:Koavf|koavf]] suggests, a good place to start could be building on [[:Category:Journalism]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:22, 13 May 2026 (UTC)
::Thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 23:23, 13 May 2026 (UTC)

== Hi. Would it be ok to post on your talk page using &quot;AI&quot;/LLMs? ==

Hello! Would it be ok if I posted some future messages that were generated by an &quot;AI&quot;/AI/LLM? If yes, would you prefer the generated message to be ie. max 100 words, less words or the talk message to include both original and generated message? Any other preferences/requirements? So far, 1 user has responded to this type of inquiry. They prefer 100 words max of generated talk page message. Best wishes [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 21:04, 22 June 2026 (UTC)
: You are welcome to post directly to my talk page if you think that is a good place for a conversation. Personally, I don't much care whether or not content is AI-generated, but note the principles suggested by [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:09, 23 June 2026 (UTC)

== Resources suitable for the main namespace ==

I noticed that we have a Draft namespace on Wikiversity and that there were discussions about moving pages to Draft. Some colleagues also hold the opinion that pages should be moved to the user ns. However, I did not understand what the criteria are for such transfers, in other words, what page deserves to be on Wikiversity, but cannot be in the main ns.

Now that I have [[:cs:Wikiverzita:Diskusní prostor#Ukončení činnosti na projektuh Wikiversity|finally left the Czech Wikiversity]], I am wondering if it is worth cloning my resources to the English one, or continuing on a personal wiki. For example, due to the resistance against AI-generated files on Commons, which has also spilled over to en.wv, I decided that I would not continue with [[Audio-visual German language materials]], because I wanted to generate the missing recordings and files in AI. This means that I will finish this course on my PC, rather than falling into eternal conjectures about why the AI ​​illustration of cherries is bad or good.

And I have a similar concern with my other creations, where there was already pressure about a year ago to move them to a personal ns. What I have been creating in recent years has been education/learning through research. A person interested in a given topic asks a question and then researches the literature, or experiments and writes down the answer. Another person interested does the same, or as part of the training, looks for answers to other people's questions. The system may resemble Stack Overflow and the like, but the goal is not to create full texts together, but to go through the process of searching for information and learning from that. Of course, if the page is then too long, it can be turned into full-text study material and, for example, a new page of a similar nature can be founded.

An example of such a project is needed [[Sweet Home 3D|here]] or [[User:Juandev/R/Compression stocking|here]], but there was an arumentation, they are underdeveloped and they should be moved to user ns. So that's why I'm asking what the evaluation criteria are, so that it doesn't end up in a way that the pages are moved away from the main ns and I end up finding out that I have to move it to my own wiki anyway. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:37, 3 July 2026 (UTC)
: Oh, sorry to hear that you are finished with Czech Wikiversity, but maybe that is good for en.wv.
: I guess we'll never really have any guarantees about anything placed on a publicly editable wiki because practices and users can change.
: I share your concerns about actual, or threats of, rather blunt approaches to educational use of AI. Of course, AI can be educational useful, and of course we are capable of finding nuanced, reasonable ways to include and use it. But as we see e.g., on Commons, there is a strong, simplistic anti-AI sentiment within the Wikimedia community.
: I don't recall much discussion about, or use of the Draft ns on en.wv. I think it was probably created very early on, to replicate Wikipedia, where a draft article makes sense before being moved to main space. I think the Draft or User space is welcome to be used for almost anything within scope, without much tension or debate. Then there is the issue around what some users consider acceptable or not for the main space on en.wv. Personally, I'm quite open. en.wv is still in early days of experimentation and trying things is needed, so I'm included to be inclusive and accepting, rather than shunting projects into Draft ns.
: I don't use Draft:, but I do use User: subpages and of course main space. I haven't had any issues with others asking me to justify main space content or proposals to move content to Draft or User.
: I'm sorry this doesn't provide any guarantees, except I guess to say I feel good about using en.wv as a working environment.
: Sincerely,&lt;br&gt; James
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:25, 3 July 2026 (UTC)
::Well, yeah. I was just wondering if there was a debate around Draft, but if werent its about the opinion of the future community. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:51, 17 July 2026 (UTC)

== LLM-generated content: Talk post about my neurodiversity draft ==

Hello Jtneill — does the topic I’ve brought up interest you? If so, would you be willing to spend about 5 minutes skimming through my “idea” on Wikiversity (“[[Draft:The Neurodiversity-inspired Idea]]”)? No need to overthink it, and I’m not expecting a reply soon—tomorrow, in a month, or even later would all be welcome. Thank you.

Metadata: Since edit summaries are not present on talk pages like this one, I'll link to the LLM interaction history that I saved in my Wikiversity user space here: [[User:ThinkingScience/All_General_AI_Prompt_History_Archive#Goal:_Interact_with_User:Jtneill_at_July_5,_2026]] so that I follow [[Wikiversity:Artificial intelligence]]. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 04:38, 5 July 2026 (UTC)

== Nodal Current Splitter ==

Hello Jtneill  ( I have been E-Mailed by you a few times in the past )

I had placed a page called Nodal Current Splitter on Wikiversity &amp; it was a way to solve unbalanced bridge circuits using Ohms Law rather then having to use Kirchhoff's mesh equation method.  It greatly reduced the amount of work that was required to solve such problems.  It once showed up on your site when searching Wikiversity as Nodal Current Splitter.  Again it allowed you to use Ohms Law instead of the harder method laid out by Kirchhoff. Can you tell me why the page was removed please ?   I have another page that's still on Wikiversity called Trirectangular Tetrahedrons &amp; I use the user name Tet-Math, 1 through 5 because Wikiversity prevents me from simply using Tet-Math more than once.  Thank You.

lm@start.ca [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 05:21, 20 July 2026 (UTC)</text>
      <sha1>kqobs4cs363rc5qtuw4csleybb8xxo5</sha1>
    </revision>
    <revision>
      <id>2818594</id>
      <parentid>2818592</parentid>
      <timestamp>2026-07-20T05:37:01Z</timestamp>
      <contributor>
        <username>Tet-Math4</username>
        <id>3063842</id>
      </contributor>
      <comment>/* Please change the name for me. */ new section</comment>
      <origin>2818594</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="53268" sha1="0j2ix2dew3vbd6qkv9jsycjuqrihn7f" xml:space="preserve">&lt;!-- {{Out of town}} --&gt;
&lt;!-- {{Long wikibreak|image=Leaf_1_web.jpg|[[User:Jtneill|Jtneill]]|mid-Jan, 2012.}} --&gt;
{{{{TALKPAGENAME}}/Header}}  {{TOCright}}

== Your feedback is welcome at [[User talk:Username142857]] ==

Dear my mentor, I believe we have already seen [[User:Username142857]] making too many non-Wikiversity questions at [[Wikiversity:Candidates for Custodianship/MathXplore]] and [[Wikiversity talk:Custodianship/Archive 6]]. In the beginning, I answered them one by one as part of demonstrating my competency to answer questions as a custodian candidate (and they were somewhat related to my global contributions) and courtesy to discussion participants. However, by facing [[special:diff/2631774]] and [[special:diff/2618170]] (editing discussion archives, re-opening closed discussions), I started to believe that we should bring an end to their excessive non-Wikiversity usage of Wikiversity (talk) namespaces. According to [[:w:User talk:Username142857]] (especially [[:w:special:diff/1073391896]]), [[User:Username142857]] is evaluated as {{tq|the other editors are tired to waste their time to read and answer your non-useful edits.}} and I think they are doing the similar thing at Wikiversity. Our community may have limited tolerance for such behavior. If you had any experience of handling such issues in the past, your feedback may be helpful to allow [[User:Username142857]] to improve their behavior. Thank you for your attention and mentoring. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:21, 9 June 2024 (UTC)
: {{ping|MathXplore}} Thanks for the heads up. Sorry for slow response. I'm recovering from COVID, but on way back. Thankyou for your very patient, clear, and supportive feedback on Username142857's talk page which, along with Mikeu, seems to have communicated the concerns and hopefully lead to a change/improvement in behaviour. What a great example of handling challenging behaviour courteously. Fingers crossed. Keep well. Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:39, 22 June 2024 (UTC)

== [[:b:Motivation and emotion/Book/2024/Free will and neuroscience]] ==

Hello, can this be related to your project? Should this be imported here? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:10, 30 July 2024 (UTC)
: Sorry, the page has been deleted, should we request temporary restoration for import, or should we just ask the author to resubmit to Wikiversity? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:29, 30 July 2024 (UTC)
::Thank-you for pointing this out. Yes, it does look like one of my students' editing. It is a little puzzling how the user ended up on Wikibooks. It is OK that that the wikibooks page has been deleted because the user also appears to be underway here: [[Motivation and emotion/Book/2024/Free will and neuroscience]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:53, 30 July 2024 (UTC)

== [[Template:Subst:ME/BCS]] ==

Hello, should this template be kept for your project? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:42, 31 July 2024 (UTC)

:Yes, please - but it could be moved from Template into a subpage of [[Motivation and emotion]]. Note that we are actively using the template at the moment to help build out the [[Motivation and emotion/Book/2024]] pages. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:43, 1 August 2024 (UTC)

== [[:File:Rejection sensitivity chart.webp]] ==

One of your students uploaded this image to Commons as part of [[Motivation and emotion/Book/2024/Rejection sensitivity]]. Unfortunately, it's meaningless AI-generated sludge. Can this image be removed from the chapter to allow it to be deleted from Commons?

(You may want to have a word with your students about AI-generated content; I think some of the text in this chapter was generated by ChatGPT as well.) [[User:Omphalographer|Omphalographer]] ([[User talk:Omphalographer|discuss]] • [[Special:Contributions/Omphalographer|contribs]]) 02:52, 6 August 2024 (UTC)
: {{ping|Omphalographer}} Great, thanks for picking this up and letting me know. Yes please, delete. I've given the student a heads-up here: [[User talk:Yonis Yousufzai]]. We're covering genAI in classes this week {{smile}}. Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:25, 6 August 2024 (UTC)

== [[Wikiversity:Bots/Status#Leaderbot]] ==

Hi, is there a chance you can approve this bot request (or otherwise let me know if there are any issues)? Thanks in advance. [[User:Leaderboard|Leaderboard]] ([[User talk:Leaderboard|discuss]] • [[Special:Contributions/Leaderboard|contribs]]) 15:03, 15 September 2024 (UTC)

== VDT - U3126684 chapter ==

Hi James ! I saw you added the hanging indent which is amazing, thank you so much! However, I had a few references missing and I tried to add them in but they didn't keep the required APA formatting. I deleted the template and reused the hanging indent template but it won't keep any formatting. Can you please help me fix it? 
[[Motivation and emotion/Book/2024/Vulnerable dark triad, motivation, and emotion|Motivation and emotion/Book/2024/Vulnerable dark triad, motivation, and emotion - Wikiversity]] [[User:U3126684|U3126684]] ([[User talk:U3126684|discuss]] • [[Special:Contributions/U3126684|contribs]]) 11:16, 3 October 2024 (UTC)

:James, I figured it out! I was just missing the &quot;}}&quot; at the end of the text... all solved! [[User:U3126684|U3126684]] ([[User talk:U3126684|discuss]] • [[Special:Contributions/U3126684|contribs]]) 11:31, 3 October 2024 (UTC)

== Your feedback may be needed at [[User talk:Tule-hog]] ==

Hello, user:Dan Polansky is currently communicating with a participant on this talk page. As Dan's mentor, I thought you may want to provide feedback so I came here for a notice. ({{ping|Guy vandegrift}} Your feedback is also welcome). [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 06:20, 7 October 2024 (UTC)

:Thanks for bringing this to my attention.  I will keep up with further developments. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 00:07, 8 October 2024 (UTC)

== [[General health and well-being]] ==
This page was in the proposed-deletion state for over 3 months, with no opposition. Should I feel free to delete the page? I guess it seemed to be a good idea back in 2011 (at least as a stub to get things started), but no one expanded it into anything really useful during all these years. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:24, 11 October 2024 (UTC)

:Hi Dan - thanks for checking - yes, it can go - I've removed the one incoming link to this page. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:39, 11 October 2024 (UTC)

== Enquiry about Correct Setup of Wikiversity? ==
Hi James, 
I just had a few questions regarding my Setup on Wikiversity:

1. We are asked to enable the Visual Editor. Have I done this correctly? Or how do I do it if I have not?

2. Have I chosen a book chapter and inserted my name correctly?

3. There isn’t a discussion forum page on our UCLearn for me to comment on, for the assessment, so where should I comment?

Thank you, I look forward to hearing back from you.
[[User:Hcoad|Hcoad]] ([[User talk:Hcoad|discuss]] • [[Special:Contributions/Hcoad|contribs]]) 14:27, 2 August 2025 (UTC)

:@[[User:Hcoad|Hcoad]]:
:# To access the Visual Editor, use &quot;Create&quot; for the first edit on a page, or &quot;Edit&quot; thereafter
:# Sign-up looks good
:# You can create a new discussion thread on UCLearn about a topic of interest or respond to existing threads such as &quot;What do you really want to learn about?&quot;
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:34, 2 August 2025 (UTC)

== Problem with curator ==

Reading above, may i address you as James? If so, hello James, i have a problem with a curator and would ask if you are a contact to talk about it. If not, sorry to bother you. Kind regards, [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 21:19, 10 October 2025 (UTC)

:Hi Harold,
:Thanks for getting in touch.
:Sorry about the teething issues in getting underway with your contributions to Wikiversity.
:Let's hopefully have a constructive discussion here, which you've initiated: [[Wikiversity:Request custodian action#Contest removal of article]]
:Sincerely,
:James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:38, 11 October 2025 (UTC)
::@[[User:Jtneill|Jtneill]] Hi James, 
::Thank you very much for sending me the article text, I really appriciate that. If not to much to ask, could you also send me the template? Template:Condensed matter physics see: User:Harold Foppele/Quantum A Matter Of Size.
::Did you read the disucussion with Dan Polansky? I think its rather weird. I answered all his questions truthfully, since i have nothing to hide. (see my user page) And than he started some trivia about the double slit expiriment, went on without listening. Like the article was a sort of explosive that must be removed ASAP. That is not the way a curator should behave (my opinion). 
::I could acctually use a mentor physics to avoid mistakes in the future.
::I know both my articles have flaws but i can fix that in time.
::Do you maybe have suggestions?
::Last but not least, thanks again for the time you took to help me !!! Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 09:14, 12 October 2025 (UTC)
: @James: To reduce or eliminate further risk that I am abusing my curator priviledges in relation to suspected copyright violation (I don't think I am, but my point of view can be skewed), I can start tagging material for copyright violation using a template (does not require curator privileges). That should address concerns? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:01, 13 October 2025 (UTC)
::@[[User:Dan Polansky|Dan Polansky]] As long as you remove the insulting (in my opinion) remarks on both articles and remove the tag -since it does not violate '''[[creativecommons:by-sa/3.0/|CC-BY-SA 4.0]] license'''- i will be satisfied. As i explained, Wikipedia use a free-to-use policy. Also could you please clarify this code: &lt;nowiki&gt;{{subst:&lt;/nowiki&gt;[[Template:No thanks|no thanks]]|pg=User:Harold Foppele/Quantum A Matter Of Size|url=&lt;nowiki&gt;{{{url}}}&lt;/nowiki&gt;}} [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • . After this is resolved i'm willing to consider this complaint closed. Maybe we can start over with a new and different conversation, since I strongly believe in AGF.  You have a way much longer experience on Wikiversity than I do, so perhaps you could help me in a friendly and constructive way? It seems we have a lot in common and I shall gladly listen to any comments.
::CC @[[User:Jtneill|Jtneill]] Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 09:16, 13 October 2025 (UTC)
::: The page [[User:Harold Foppele/Quantum A Matter Of Size]] currently features multiple sentences from a CC-BY-SA source without using quotation marks. My determination is that the page shows copyright violation (failure to ''attribute'') of CC-BY-SA and should therefore be deleted.
::: If you, James, remove the copyright violation tagging, I will understand it as you taking responsibility for a possible copyright violation and I will probably disengage (or do I have a duty to take more pains and try to override your assessment?) --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:31, 13 October 2025 (UTC)
::: As for &quot;As i explained, Wikipedia use a free-to-use policy&quot;: that seems to be a misunderstanding or too vague understanding; Wikipedia uses CC-BY-SA copyright license, which requires proper ''attribution'' of authorship, which could have been done in the edit summary that created the article, but was not done. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:35, 13 October 2025 (UTC)
::::@[[User:Dan Polansky|Dan Polansky]] It has already been added, as you would have seen upon checking. I would still appreciate a response to the other points I mentioned earlier, if you are willing to continue the discussion. If not, your choise. CC:@[[User:Jtneill|Jtneill]] Cheers[[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 10:08, 13 October 2025 (UTC)
: James, as my mentor in my role of a custodian, if you want me to do something, or if you have a recommendation for me, please let me know on my talk page. I am struggling to figure out how to navigate these waters. You can also use email if it seems better from some perspective. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:21, 13 October 2025 (UTC)
::@[[User:Dan Polansky|Dan Polansky]] Why not take a step back? I offered you a solution and a possibility to cooperate instead of continuing a conflict. I still believe that working together is more productive than arguing over small details. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 10:26, 13 October 2025 (UTC)
:::The discussion at this talk page ended not very fruitfully.
:::Pitty, i really tried to make piece.
:::Yet I am not the only one complainting about Dan’s behaviour.
::: 
:::Anything I can do (or you) ?
:::Am I free to remove remarks and/or tags?
:::I dont want to end up in an editwar.
::: 
:::Sorry to have asked so much of your time [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 15:54, 13 October 2025 (UTC)

Thanks, both. May I suggest:
* {{ping|Harold Foppele}}: Any text you don't write yourself needs appropriate attribution or removal, otherwise it runs the risk of copyright violation. For example, this message appears on each edit source screen underneath the edit summary box: &quot;Do not copy text from other websites without permission. It will be deleted.&quot; If text is copied from Wikipedia it needs to be acknowledged as such because it is licensed under CC-by-SA which allows re-use but requires acknowledgement. Such acknowledgement could be made in the edit summary when the contribution is first made. If not, then the next best could be to put quotation marks around copied text and a link to the source(s) of the text.
* {{ping|Dan Polansky}}: Appreciate your administrative work. Let's try to AGF and work constructively with new users who are learning how to contribute. Wikiversity is a learning environment.
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 20:42, 13 October 2025 (UTC)

:@[[User:Jtneill|Jtneill]] Thank you very much. I hope it will work out since Dan does not respond, to me that is. Could you find time to look at the revised [[User:Harold Foppele/Quantum A Matter Of Size]] i made additions to it, but since it is a mix of WP, other sources and OR, it is alomost impossible to keep quoting. So i made a general intro. Is that enough? Also 99% of the [[]] refer directly to WP since WV does not have most of the words/pages. I also recreated the template so that it shows all original text/items. The new section ==Tunneling== is not cited yet, but it wiil be when I have time. Can I remove the tags myself? Thanks again [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 21:21, 13 October 2025 (UTC)
::Looks like a solid chunk is copied from Wikipedia: https://www.copyscape.com/view.php?o=4829&amp;u=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FMesoscopic_physics&amp;t=1760433515&amp;s=https%3A%2F%2Fen.wikiversity.org%2Fwiki%2FUser%3AHarold_Foppele%2FQuantum_A_Matter_Of_Size&amp;w=66&amp;i=1&amp;r=10
::without appropriate acknowledgement.
::Some ways to deal with this appropriately include:
::# Acknowledge the source in the edit summary when content is added to the page
::# Using quotation marks and citations to indicate the source of any content which you haven't authored yourself
::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:02, 14 October 2025 (UTC)
:::The &quot;chunk&quot; is correct :) I took that since it fits perfect to the article. At the top of the page I quoted:
:::{Wikipedia [[wikipedia:Mesoscopic_physics|Mesoscopic physics]]&lt;nowiki&gt;}}&lt;/nowiki&gt;
:::[[creativecommons:by-sa/4.0/|License CC-BY-SA 4.0]]
:::In Edit summary: The first section of this article is copied from Wikipedia &quot;Mesoscopic physics&quot;
:::Is that sufficient ?
:::I did cite almost everything what is not so much requested in Wikiversity as far as i found out, but is a first requirement in Wikipedia.
:::Is it OK if I remove the tags ? Thanks [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 10:51, 14 October 2025 (UTC)
::::I think it would be more transparent and demonstrate greater academic integrity to use quotation marks for text which is copied from elsewhere, especially because there was no appropriate edit summary when the text was added to the page.
::::[https://en.wikiversity.org/w/index.php?title=User%3AHarold_Foppele%2FQuantum_A_Matter_Of_Size&amp;diff=2760582&amp;oldid=2760574 Example of how this might be done].
::::I don't suggest removing the copyright tag until copied text is more clearly quoted and cited and there is consensus that it [[wikt:pass muster|passes muster]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:52, 14 October 2025 (UTC)
:::::Thank you SO MUCH !!  I had no idea that a &lt;blockquote existed nor what it does. This is the first time i used a Wikipedia copy into Wikiversity. So a simple explanation, as you gave me now, would have prevented all this. :) I changed the layout a bit to make it view nicer. Is this required also for my own publications on Wikipedia? Thanks again!! and a goodnight to you [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 12:28, 14 October 2025 (UTC)
::::::I decided to re-write the copyrighted text in my own words. It feels better this way, what do you think? [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 13:07, 14 October 2025 (UTC)
:::::::Great, I think that makes a big difference to rewrite in your own words. I've removed the copyright tag.
:::::::Let me know if I can do anything else as you go along. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:03, 15 October 2025 (UTC)
:::::::: The page still contains copyright violation. I am starting to track problems at [[User:Dan Polansky/Problem reports (about Wikiversity problems)]]. I will disengage from Harold Foppele; this is not being productive and can lead to my harm and thereby harm to the English Wikiversity. I have seen this kind of people elsewhere: I explained a class/type of a problem to the person and pointed to an example for clarity and the person corrected just the single item I gave as an example. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:17, 15 October 2025 (UTC)
:::::::::@[[User:Dan Polansky|Dan Polansky]] Since you want to take this personally instead of having a civilized conversation, I will not engage in a mud-throwing contest or labeling people as “this kind of people&quot;. I saw your problem report and I seriously question your objectivity as a science debater. You took ONE paragraph from an article—a paragraph that had been modified (as your question mark even shows)—plus a scientific debate over a previously accepted article on Wikipedia. You completely ignored the accepted contributions I have made to Wikipedia. Yet this alone is enough for you to request that a contributor be blocked.
:::::::::What do I gain from spending hours and hours doing research for a new article? Hours and hours searching for proper references? Hours writing and rewriting the text? How much do I get paid? Nothing. How much honor or credit do I receive? None. So what &quot;kind of people&quot; am I? [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 08:21, 15 October 2025 (UTC)
:::::::::: DFX. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:26, 15 October 2025 (UTC)
:::::::::::Exactly my point. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 09:19, 15 October 2025 (UTC)
:Thanks [[User:Harold Foppele|Harold]] and [[User:Dan Polansky|Dan]] — I appreciate your considerations and communications. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:51, 15 October 2025 (UTC)

== Peer review ==

@[[User:Jtneill|Jtneill]] Hello James, I hope you are doing well. The 2 articles I wrote are now ready to be published. Is there some kind of peer review possible? I tried to find some help at [[Portal:Particle physics]] but all data there is very old. How can we move forward from this? Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 09:52, 16 October 2025 (UTC)

:Perhaps try [[Wikiversity:Colloquium]] - that's the general way to communicate with English Wikiversity users/editors. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:08, 17 October 2025 (UTC)

== Hello James, I need your help. ==

Could join the discussion with us in [[Wikiversity:Colloquium#Concern regarding curator conduct User:Dan Polansky]]

We would like to solicit your input on this matter.  [[User:Tomlovesfar|Tomlovesfar]] ([[User talk:Tomlovesfar|discuss]] • [[Special:Contributions/Tomlovesfar|contribs]]) 03:54, 17 October 2025 (UTC)

== Quantum ==

Hello James, If you have time could you lease look at [[Quantum]]. An essay like page with simple information, that might attract students. I Know its not your field, but maybe it appeals to you. Thanks, [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 23:39, 18 October 2025 (UTC)

== ShakespeareFan00 ==

Goodevening, please, if you have time, take a look at the edits made by this user. A few hundred in 2 days ! Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 20:35, 31 October 2025 (UTC)

== When is a quote or blockquote needed? ==

Hi James, I hope you are doing well. I did wrote some articles and parts off them at Wikipedia. If i want to use parts of it at Wikiversity do i still need to quote that parts? Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:19, 2 November 2025 (UTC)

:Basically, if you didn't author text which is being added, then the genesis of the text needs to be made clear (e.g, edit summary, quotation etc.) It is also possible to import pages (e.g., from Wikipedia) which brings in the full edit history. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:38, 3 November 2025 (UTC)

== Publishing transcripts ==

Hi James, Is it allowed to publish a transcript in Wikiversity as per my example at [[User:Harold Foppele/sandbox-2]]. If not, then I remove the page ofcourse. I think it could be nice if I edit it to make it easy accessible in various Wikipages.
But again, if its not allowed, i remove it. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:28, 6 November 2025 (UTC)

== User:Dan Polansky ==

@Jtneill , Hi James, You are a curator/bureaucrat, if i'm not mistaken. Please look at: [[User:Dan Polansky/Problem reports (about Wikiversity problems)]] I feel outright insulted and ask you (if you can) to put an end to it. Thanks [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:59, 6 November 2025 (UTC)
: I wrote: &quot;The user account created articles in the subject of quantum mechanics that use wiki-voice and do not state the author. Since it is very likely that he does not understand quantum mechanics as per evidence in the revision history of his user talk page, it is also likely that they contain countless errors. The articles are presented to the reader as valid referenced content, not as one person's exercise in who-knows-what. Preventing the user account from creating new pages and moving all his articles to user space would address the issue.&quot;
: I think it is accurate. By now, we have enough evidence I think that the user account is a troll account, an intentional disruptor. There are multiple behavioral signs, both in Wikipedia and in Wikiversity.
: I propose an indef block of the user account. An alternative is not to feed into this troll account. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:03, 6 November 2025 (UTC)
::Well well here we go again [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 18:18, 6 November 2025 (UTC)
::: I opened [[Wikiversity:Request custodian action#Indefinite block for Harold_Foppele]]. I fear it will be in vain. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:26, 6 November 2025 (UTC)
::::You are allowed to hope [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 18:42, 6 November 2025 (UTC)

== Moving to personal namespace ==

What are the policies or customs on Wikiversity for moving pages to personal userspace? Isn't there a risk that Wikiversity will turn into a blogging platform where many users will cultivate pages in their userspace and the outside world will not benefit from it?

I see moving to ns user as a frequent suggestion in Requests for deletion (RFD). I would understand moving to ns Draft, which is clearly defined and there is a chance that the resource will then get into the main ns, thus serving the community. I would understand the suggestion to move to another wikiproject, where the text will serve the community. But I don't really understand the frequent moves to personal ns. Since it's in the RFD, it should either be kept or deleted. If someone contributes to Wikiversity, they automatically agree to its policies and also to the fact that they don't own the pages and someone can put them up for deletion. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:36, 22 November 2025 (UTC)

I personally don't need a free website to host my pages. How would I get rid of the unfinished [[Pomology]] meta course if it was moved to my NS? ([https://en.wikiversity.org/wiki/Wikiversity:Requests_for_Deletion#c-Dan_Polansky-20251121091100-Juandev-20251120220900 Moving it to my own NS is suggested in RFD]). I'm putting it in the Request for deletion because, even though I started it, it looks like other editors had significant input there. Will I have the right to request speedy deletion if the pages are moved to my user ns?

I think this tactic of moving to personal space is poorly thought out, but it has become the norm.

Is there any guideline or discussion from before? If something appears in a deletion request, the majority decides that it should be moved to user ns, how can the person in question defend themselves that they don't want it in their own ns? It seems the community is pressuring the original author to agree to deletion. It seems that the user ns is an untouchable territory into which the community has the right to throw whatever it thinks from the main ns. So why aren't those pages deleted when the community decides that they don't belong in the main ns? --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:30, 22 November 2025 (UTC)

{{ping|Juandev}} I replied on your talk page. But here's another version: Personally, in general, I try to keep my notes etc. in user space. Then if I have something more developed to share and collaborate on, then main space. Draft could be helpful to keep main space tidy, but is very quiet/unused, so in reality most drafts are in main space. But if the content is dubious, underdeveloped, lacking citation/peer review etc. then delete, or user space if it could still be developed. That's roughly how I see it. But everyone has a slightly different view/preference, so discuss to develop consensus. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:48, 22 November 2025 (UTC)

== Ninefold Resonance Theory ==
Dear Jtneill, I noticed that when you deleted [[Ninefold Resonance Theory]], you accidentally deleted the article in my own user space as well. However, I got the impression that most users felt that it should be allowed to exist in my own user space. I thought long and hard about my theory and I'm disappointed that it's gone now... Could you move the article back to my own user space, so not in the main space? I look forward to hearing from you! Kind regards, [[User:S. Perquin|S. Perquin]] ([[User talk:S. Perquin|overleg]] • [[Special:Contributions/S. Perquin|bijdragen]]) 06:22, 28 November 2025 (UTC)

:Nevermind. I will move all my ideas to everybodywiki.com. 😄 Kind regards, [[User:S. Perquin|S. Perquin]] ([[User talk:S. Perquin|overleg]] • [[Special:Contributions/S. Perquin|bijdragen]]) 06:36, 28 November 2025 (UTC)
::Could you please e-mail me the source code of the deleted page? Kind regards, [[User:S. Perquin|S. Perquin]] ([[User talk:S. Perquin|overleg]] • [[Special:Contributions/S. Perquin|bijdragen]]) 06:42, 28 November 2025 (UTC)
:[[User:S. Perquin|S. Perquin]]: Apologies, the user page version was accidentally deleted. It has now been restored. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:25, 29 November 2025 (UTC)
::Thank you! ☺️ Kind regards, [[User:S. Perquin|S. Perquin]] ([[User talk:S. Perquin|overleg]] • [[Special:Contributions/S. Perquin|bijdragen]]) 06:58, 29 November 2025 (UTC)
:::All pages in my user space have been moved to EverybodyWiki. Could you perhaps delete all the pages with the {{tl|speedy}} template on it? Kind regards, [[User:S. Perquin|S. Perquin]] ([[User talk:S. Perquin|overleg]] • [[Special:Contributions/S. Perquin|bijdragen]]) 07:08, 29 November 2025 (UTC)
::::[[User:S. Perquin|S. Perquin]]: The main space redirects and all your user sub-pages have been deleted. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:25, 1 December 2025 (UTC)
:::::Thank you! Kind regards, [[User:S. Perquin|S. Perquin]] ([[User talk:S. Perquin|overleg]] • [[Special:Contributions/S. Perquin|bijdragen]]) 08:24, 1 December 2025 (UTC)

== Vandalism ==

{{ping|Jtneill}} May I draw your attantion to this!
==== 6 December 2025 ====

* cur[https://en.wikiversity.org/w/index.php?title=Chaos_Theory_Extended&amp;diff=prev&amp;oldid=2778412 prev] &lt;bdi&gt;[https://en.wikiversity.org/w/index.php?title=Chaos_Theory_Extended&amp;oldid=2778412 13:15, 6 December 2025]&lt;/bdi&gt; [[User:Revolving Doormat|&lt;bdi&gt;Revolving Doormat&lt;/bdi&gt;]] [[User talk:Revolving Doormat|discuss]] [[Special:Contributions/Revolving Doormat|contribs]]  75,351 bytes +279  request speedy delete under CSD1 [https://en.wikiversity.org/w/index.php?title=Chaos_Theory_Extended&amp;action=edit&amp;undoafter=2777042&amp;undo=2778412 undo][[Special:Thanks/2778412|thank]] [[Special:Tags|Tag]]: [[Wikiversity:VisualEditor|Visual edit: Switched]]

[[User:Revolving Doormat|&lt;bdi&gt;Revolving Doormat&lt;/bdi&gt;]] account created today 
at the same time as = &lt;bdi&gt;~2025-38873-79&lt;/bdi&gt; =

So I assume they are all the same.
Am I allowed to remove the delete template by myself?
Greetings [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 16:41, 6 December 2025 (UTC)

:We are not the same person. I came here from an AfD on Wikipedia and your page creation ban here: https://en.wikipedia.org/wiki/Wikipedia:Administrators%27_noticeboard/Incidents#c-Ldm1954-20251205133800-Requesting_page_creation_block_of_User:Harold_Foppele
:The temp user already identified that I notified WP about the same activity on WV, and that brought them here. [[User:Revolving Doormat|Revolving Doormat]] ([[User talk:Revolving Doormat|discuss]] • [[Special:Contributions/Revolving Doormat|contribs]]) 17:08, 6 December 2025 (UTC)
::Its so coincidental that you all share the same IP range isn't it? Using an empty account? [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:19, 6 December 2025 (UTC)
:::The user already identified their WP account and my WP user id is the same one I have here. I don't believe you have access to our IP addresses, but but based on their WP biography, that would also be impossible. I will not be engaging with you further. [[User:Revolving Doormat|Revolving Doormat]] ([[User talk:Revolving Doormat|discuss]] • [[Special:Contributions/Revolving Doormat|contribs]]) 17:25, 6 December 2025 (UTC)
::::What you believe or not is up to you [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:41, 6 December 2025 (UTC)

== User Dan Polansky ==

I want to draw your attention to the edits (mainly copy/paste) by [[user:Dan Polansky|Dan Polansky]] today. Still trying to act as curator? They continue their previous harassment. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:07, 12 December 2025 (UTC)

== Happy New Year, Jtneill! ==
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{{Center|{{resize|179%|'''''[[New Year|Happy New Year]]!'''''}}}}
'''Jtneill''',&lt;br /&gt;Have a prosperous, productive and enjoyable [[New Year]], and thanks for your contributions to Wikiversity.
&lt;br /&gt;[[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:10, 2 January 2026 (UTC)&lt;br /&gt;&lt;br /&gt;
&lt;/div&gt;
&amp;nbsp;&amp;nbsp;&amp;nbsp;''{{resize|88%|Send New Year cheer by adding {{tls|Happy New Year fireworks}} to user talk pages.}}''
{{clear}}&lt;!-- From template:Happy New Year fireworks --&gt;

== Please delete [[MediaWiki:Gadget-WikiSign.js]] ==

Reason: This is a request by the author (major contributor). Custodians don't have interface admin rights, so custodians cannot delete this page. Bureaucrats can delete this page by temporarily adding themselves to the interface admin user group ([[User_talk:Jtneill/Archive/2024#Please_delete_MediaWiki:Wikidebate.js]]). Thank you for your attention. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 09:11, 11 February 2026 (UTC)

== DELETE request ==

Please DELETE [[Creating Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future]] to [[Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future]]. I created the article with an erroneous name. I will recreate it with the name I want. Thanks, [[User:DavidMCEddy|DavidMCEddy]] ([[User talk:DavidMCEddy|discuss]] • [[Special:Contributions/DavidMCEddy|contribs]]) 20:15, 11 February 2026 (UTC)
: {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:12, 13 February 2026 (UTC)

== Archiving ==

Hi and hello @[[User:Jtneill|Jtneill]] I did some archiving from Colloquium and RCA. If you have time that I'm on the right track? It where only a few, so if I did wrong, its easily undone, otherwise I continue as per request. Thanks [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 19:21, 12 February 2026 (UTC)

:@[[User:Harold Foppele|Harold Foppele]] Please remember to user &lt;nowiki&gt;{{archive|Wikiversity:Colloquium}}&lt;/nowiki&gt; instead of &lt;nowiki&gt;{{archive}}&lt;/nowiki&gt; so that people who find themselves in the archives know where to go if they are unsure of anything. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 07:12, 13 February 2026 (UTC)
::@[[User:PieWriter|PieWriter]] I have literally no idea what you are talking about. So elaborate please. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 08:53, 13 February 2026 (UTC)
:::Ahhh I see what you mean. Strange that you comment on MY edits only. NONE of the archive templates at WC archive have that. Did you overlook that?[[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 09:13, 13 February 2026 (UTC)
::::That’s why the discussion parameter is red linked, I am working on that. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:22, 13 February 2026 (UTC)
:::::Well, you could have said that instead. I think it's a bit overdone, since the page title is reads already Archive. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 09:26, 13 February 2026 (UTC)
::::::New users will click on the red linked, which brings them to create the talk page, which is not watched so they won’t receive a response to their question. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:15, 13 February 2026 (UTC)
:::::::That is true [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 12:58, 13 February 2026 (UTC)

== Email ==

I sent you an email about a private abuse filter, feel free to take a look. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:39, 15 April 2026 (UTC)

== AI slop, ownership, and wikilawyering. ==

Using AI images is worse than no images. Your constant reverting of reasonable edits removing images you prompted on pages you wrote would be considered [[w:wp:OWN]]ership on Wikipedia; even if there is no general guideline on Wikiversity the spirit of not having the final say because just you made the page is applicable to all Wikimedia wikis. Reverting a reasonable edit because it lacks an image seems like [[w:wp:WIKILAWYER]]ing— I don’t know if edit summaries are ''required'' here, but I doubt it, and on most wikis they are simply recommended. Not having one doesn’t invalidate the edit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 05:27, 26 April 2026 (UTC)

:I understand that you don't like many AI images because you consider them slop. My view is that some of these AI images can be useful for educational purposes.
:I understand that you think an alternative or no image is better than some AI images. My view is that some AI images are better than no image and are either useful in addition to alternative images or more useful than some alternatives.
:May I suggest deciding first on Commons whether to keep an image, rather than removing from Wikiversity and then nominating for deletion on Commons because of no use.
:I have no interest in edit warring. I'll invite [[WV:RCA]] to review your recent edits. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:20, 26 April 2026 (UTC)

== You may be an eligible candidate for the U4C election ==

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

The [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee (U4C)]] seeks candidates for the 2026 election. The U4C is the global committee responsible for overseeing enforcement of the [[foundation:Special:MyLanguage/Policy:Universal Code of Conduct|Universal Code of Conduct]]. Elections are held annually, if elected a committee member serves for two years.

This year the U4C requires candidates to hold administrator rights on at least one wiki, which is why you are being contacted as you appear to hold this right. There are other requirements, such as candidates must be at least 18 years old and may not be employed by the Wikimedia Foundation or other related chapters and affiliates. You can find more information in the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026#Call_for_Candidates|call for candidates on Meta-wiki]]. Additionally, the committee's working language is English; some ability to communicate in English is required.

The election opens on 18 May, if you are eligible and interested you have until 10 May to submit your candidacy. There will week between for candidates to answer questions from the community. Voting takes place privately in [[m:Special:MyLanguage/SecurePoll|SecurePoll]], successful candidates must receive at least 60% support. More information is available on [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|the 2026 Elections page]], including timelines and other candidacy information. If you read over the material and consider yourself qualified, please consider submitting your name to run for the committee. If you think someone else in your community might be interested and qualified, please encourage them to run.

In partnership with the U4C -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User_talk:Keegan (WMF)|talk]]) 18:32, 28 April 2026 (UTC) &lt;/div&gt;
&lt;!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Keegan_(WMF)/test&amp;oldid=30471751 --&gt;

== Thoughts about Wikinews closure ==

I think Wikiversity could bring in Wikinews users possibly. Thoughts? @[[User:Jtneill|Jtneill]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 23:05, 13 May 2026 (UTC)

:Welcome. Sorry for the loss of Wikinews. I hope WN editors can find their way into contributing to WMF sister projects most aligned with their interests and skills, including Wikiversity. For me, the key here is alignment with [[Wikiversity:Mission]]. It may take some time to work out what's possible. As @[[User:Koavf|koavf]] suggests, a good place to start could be building on [[:Category:Journalism]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:22, 13 May 2026 (UTC)
::Thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 23:23, 13 May 2026 (UTC)

== Hi. Would it be ok to post on your talk page using &quot;AI&quot;/LLMs? ==

Hello! Would it be ok if I posted some future messages that were generated by an &quot;AI&quot;/AI/LLM? If yes, would you prefer the generated message to be ie. max 100 words, less words or the talk message to include both original and generated message? Any other preferences/requirements? So far, 1 user has responded to this type of inquiry. They prefer 100 words max of generated talk page message. Best wishes [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 21:04, 22 June 2026 (UTC)
: You are welcome to post directly to my talk page if you think that is a good place for a conversation. Personally, I don't much care whether or not content is AI-generated, but note the principles suggested by [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:09, 23 June 2026 (UTC)

== Resources suitable for the main namespace ==

I noticed that we have a Draft namespace on Wikiversity and that there were discussions about moving pages to Draft. Some colleagues also hold the opinion that pages should be moved to the user ns. However, I did not understand what the criteria are for such transfers, in other words, what page deserves to be on Wikiversity, but cannot be in the main ns.

Now that I have [[:cs:Wikiverzita:Diskusní prostor#Ukončení činnosti na projektuh Wikiversity|finally left the Czech Wikiversity]], I am wondering if it is worth cloning my resources to the English one, or continuing on a personal wiki. For example, due to the resistance against AI-generated files on Commons, which has also spilled over to en.wv, I decided that I would not continue with [[Audio-visual German language materials]], because I wanted to generate the missing recordings and files in AI. This means that I will finish this course on my PC, rather than falling into eternal conjectures about why the AI ​​illustration of cherries is bad or good.

And I have a similar concern with my other creations, where there was already pressure about a year ago to move them to a personal ns. What I have been creating in recent years has been education/learning through research. A person interested in a given topic asks a question and then researches the literature, or experiments and writes down the answer. Another person interested does the same, or as part of the training, looks for answers to other people's questions. The system may resemble Stack Overflow and the like, but the goal is not to create full texts together, but to go through the process of searching for information and learning from that. Of course, if the page is then too long, it can be turned into full-text study material and, for example, a new page of a similar nature can be founded.

An example of such a project is needed [[Sweet Home 3D|here]] or [[User:Juandev/R/Compression stocking|here]], but there was an arumentation, they are underdeveloped and they should be moved to user ns. So that's why I'm asking what the evaluation criteria are, so that it doesn't end up in a way that the pages are moved away from the main ns and I end up finding out that I have to move it to my own wiki anyway. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:37, 3 July 2026 (UTC)
: Oh, sorry to hear that you are finished with Czech Wikiversity, but maybe that is good for en.wv.
: I guess we'll never really have any guarantees about anything placed on a publicly editable wiki because practices and users can change.
: I share your concerns about actual, or threats of, rather blunt approaches to educational use of AI. Of course, AI can be educational useful, and of course we are capable of finding nuanced, reasonable ways to include and use it. But as we see e.g., on Commons, there is a strong, simplistic anti-AI sentiment within the Wikimedia community.
: I don't recall much discussion about, or use of the Draft ns on en.wv. I think it was probably created very early on, to replicate Wikipedia, where a draft article makes sense before being moved to main space. I think the Draft or User space is welcome to be used for almost anything within scope, without much tension or debate. Then there is the issue around what some users consider acceptable or not for the main space on en.wv. Personally, I'm quite open. en.wv is still in early days of experimentation and trying things is needed, so I'm included to be inclusive and accepting, rather than shunting projects into Draft ns.
: I don't use Draft:, but I do use User: subpages and of course main space. I haven't had any issues with others asking me to justify main space content or proposals to move content to Draft or User.
: I'm sorry this doesn't provide any guarantees, except I guess to say I feel good about using en.wv as a working environment.
: Sincerely,&lt;br&gt; James
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:25, 3 July 2026 (UTC)
::Well, yeah. I was just wondering if there was a debate around Draft, but if werent its about the opinion of the future community. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:51, 17 July 2026 (UTC)

== LLM-generated content: Talk post about my neurodiversity draft ==

Hello Jtneill — does the topic I’ve brought up interest you? If so, would you be willing to spend about 5 minutes skimming through my “idea” on Wikiversity (“[[Draft:The Neurodiversity-inspired Idea]]”)? No need to overthink it, and I’m not expecting a reply soon—tomorrow, in a month, or even later would all be welcome. Thank you.

Metadata: Since edit summaries are not present on talk pages like this one, I'll link to the LLM interaction history that I saved in my Wikiversity user space here: [[User:ThinkingScience/All_General_AI_Prompt_History_Archive#Goal:_Interact_with_User:Jtneill_at_July_5,_2026]] so that I follow [[Wikiversity:Artificial intelligence]]. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 04:38, 5 July 2026 (UTC)

== Nodal Current Splitter ==

Hello Jtneill  ( I have been E-Mailed by you a few times in the past )

I had placed a page called Nodal Current Splitter on Wikiversity &amp; it was a way to solve unbalanced bridge circuits using Ohms Law rather then having to use Kirchhoff's mesh equation method.  It greatly reduced the amount of work that was required to solve such problems.  It once showed up on your site when searching Wikiversity as Nodal Current Splitter.  Again it allowed you to use Ohms Law instead of the harder method laid out by Kirchhoff. Can you tell me why the page was removed please ?   I have another page that's still on Wikiversity called Trirectangular Tetrahedrons &amp; I use the user name Tet-Math, 1 through 5 because Wikiversity prevents me from simply using Tet-Math more than once.  Thank You.

lm@start.ca [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 05:21, 20 July 2026 (UTC)

== Please change the name for me. ==

Hello again Jtneill

I just thought of something I wished I had done in the 1st place.  I think the Page Name should have been &quot;Nodal Current Divider&quot; all along because nobody searches for &quot;Splitter&quot; although that's what it actually does.

Thank You again

lm@start.ca [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 05:37, 20 July 2026 (UTC)</text>
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    <title>User:Jtneill</title>
    <ns>2</ns>
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    <revision>
      <id>2818578</id>
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      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Back at work</comment>
      <origin>2818578</origin>
      <model>wikitext</model>
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      <text bytes="3287" sha1="oou338yo7ukuq4217tc9n4wb341geeo" xml:space="preserve">__NOTOC__
&lt;!-- {{notice|I'm currently on leave and will be back later in July.}} --&gt;
&lt;div style=&quot;background:white; border:2px SteelBlue solid; padding:12px;&quot;&gt;
My name is James Neill (''he/him''). I'm an Assistant Professor in the [https://www.canberra.edu.au/about-uc/faculties/health/study/psychology Discipline of Psychology] at the [[University of Canberra]], Australia.

I'm passionate about [[open academia]]—I like to share knowledge openly.

On English Wikiversity, I'm a [[WV:Custodianship|custodian]] and [[WV:Bureaucratship|bureaucrat]]&lt;small&gt;&lt;sup&gt;[https://en.wikiversity.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Jtneill (verify)]&lt;/sup&gt;&lt;/small&gt;. Since 2005, I've made:
* ~[https://xtools.wmcloud.org/ec/en.wikiversity.org/Jtneill 80,000 edits] on [[Main page|Wikiversity]]
* ~[https://xtools.wmcloud.org/ec/en.wikipedia/Jtneill 4,900 edits] on [[w:|Wikipedia]]
* ~[https://xtools.wmcloud.org/ec/commons.wikimedia.org/Jtneill 2,200 edits] on [[c:|Wikimedia Commons]].

My [[User:Jtneill/Teaching/Philosophy|teaching philosophy]] is based on experiential learning. [[/Teaching|I teach]] a 3rd-year undergraduate [[psychology]] unit, [[motivation and emotion]], and a 4th-year Honours unit about [[research methods in psychology]].

{{/Research}}

[[/Presentations|I also present]] about open education, wikis in higher education, and collaborative development of [[open educational resources]]. 

Currently, I'm working on:
[[User:Jtneill/Presentations/Open wiki assignments for authentic learning|Open wiki assignments for authentic learning]].
&lt;!--
Most recently, I presented on:
[[User:Jtneill/Presentations/Interactive classroom exercises using Google Forms and Sheets|Interactive classroom exercises using Google Forms and Sheets]].
--&gt;

I like exploring outdoors, including [[w:guerilla gardening|guerilla gardening]] — which is much like wiki editing.

[[/Contact|Feel free to connect.]]
&lt;/div&gt;
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[[Category:User en|{{PAGENAME}}]]
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  <page>
    <title>The necessities in Numerical Methods</title>
    <ns>0</ns>
    <id>119778</id>
    <revision>
      <id>2818595</id>
      <parentid>2818449</parentid>
      <timestamp>2026-07-20T06:53:22Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Non-linear Equations */</comment>
      <origin>2818595</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2914" sha1="ef729bahazww3wmeow35h0iqgsjhzuf" xml:space="preserve">== Calculus == 
=== Numerical Differentiation ===
* Background on Differentiation ([[Media:NM.Diff.1Background.20240625.pdf |pdf]])
* Continuous Function Differentiation ([[Media:NM.Diff.1ContDiff.20241021.pdf |pdf]])
* Discrete Function Differentiation ([[Media:NM.Diff.1Discrete.20241116.pdf |pdf]])
* Forward, Backward, Central Divided Difference
* High Accuracy Differentiation
* Richardson Extrapolation
* Unequal Spaced Data Differentiation
* Numerical Differentiation with Octave
&lt;/br&gt;

=== Non-linear Equations ===
* Bisection Method ([[Media:NM.NLE.1Bisection.20241130.pdf |pdf]])
* Newton-Raphson Method ([[Media:NM.NLE.2Newton.20260720.pdf |pdf]])
* Secant Method
* False-Position Method
&lt;/br&gt;

=== Numerical Integration ===
* Trapezoidal Rule
* Simpson's 1/3 Rule
* Romberg Rule
* Gauss-Quadrature Rule
* Adaptive Quadrature
&lt;/br&gt;

=== Roots of a Nonlinear Equation ===
&lt;/br&gt;

=== Optimization ===
&lt;/br&gt;
&lt;/br&gt;

== Matrix Algebra == 

=== Simultaneous Linear Equations === 
* A system of linear equations ([[Media:SystemLinearEq.20240521.pdf |pdf]])


&lt;/br&gt;

=== Gaussian Elimination === 
&lt;/br&gt;

=== LU Decomposition === 
&lt;/br&gt;

=== Cholesky Decomposition === 
&lt;/br&gt;

=== LDL Decomposition === 
&lt;/br&gt;

=== Gauss-Seidel method === 
&lt;/br&gt;

=== Adequacy of Solutions === 
&lt;/br&gt;

=== Eigenvalue and Singular Value ===
&lt;/br&gt;

=== QRD ===
&lt;/br&gt;

=== SVD ===
&lt;/br&gt;

=== Iterative methods ===
&lt;/br&gt;
&lt;/br&gt;

== Regression ==

=== Linear Regression ===
&lt;/br&gt;

=== Non-linear Regression ===
&lt;/br&gt;

=== Linear Least Squares ===
&lt;/br&gt;
&lt;/br&gt; 


== Interpolation ==

=== Polynomial Interpolation ===
&lt;/br&gt;

=== Linear Splines ===
&lt;/br&gt;

=== Piecewise Interpolation ===
&lt;/br&gt;
&lt;/br&gt; 


== Ordinary Differential Equation ==
&lt;/br&gt;

== Partial Differential Equation ==
&lt;/br&gt;

== FEM (Finite Element Method) ==
&lt;/br&gt;
&lt;/br&gt; 


&lt;/br&gt;


== Using Symbolic Package in Octave ==

* Visit http://octave.sourceforge.net/index.html
* Download symbolic-1.0.9.tar.gz

* In Ubuntu, using the Ubuntu Software Center, I installed GiNac and CLN related software and symbolic package for Octave. But it did not properly installed.

* After extracting files from symbolic-1.0.9.tar.gz, I followed the following steps.
 ./configure 
 ./make 
 ./make INSTALL_PATH=/usr/share/octave/packages/3.2/symbolic-1.0.9 

* While doing this, I got an error message related to mkoctfile. So, I used the following command: sudo apt-get install ocatve3.2-headers. Then I was able to install the symbolic packages in the Ubuntu.

== Read some tutorials about symbolic computation ==

* Symbolic Mathematics in Matlab/GNU Octave (http://faraday.elec.uow.edu.au/subjects/annual/ECTE313/Symbolic_Maths.pdf)

* Symbolic Computations (http://www.math.ohiou.edu/courses/math344/lecture7.pdf)

[[Category:Numerical methods]]

== Using SymPy ( a Python library for symbolic mathematics) == 
&lt;/br&gt;


&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>ef729bahazww3wmeow35h0iqgsjhzuf</sha1>
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  </page>
  <page>
    <title>VHDL programming in plain view</title>
    <ns>0</ns>
    <id>121359</id>
    <revision>
      <id>2818561</id>
      <parentid>2818107</parentid>
      <timestamp>2026-07-19T19:46:32Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Data */</comment>
      <origin>2818561</origin>
      <model>wikitext</model>
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      <text bytes="2278" sha1="3h29fj4q7gspgvhuc0aznb4yuh5n57l" xml:space="preserve">&lt;!----------------------------------------------------------------------&gt;

== Flip Flop and Latch ==
* FFLatch.Overview.1.A ([[Media:FFLatch.Overview.1.A.20111103.pdf|pdf]])
* Counter.74LS193.1.A ([[Media:Counter.74LS193.1.A.20111108.pdf|pdf]])
* Clock.Overview.1.A ([[Media:Clock.Overview.1.A.20111108.pdf|pdf]])
* Function.Overview.1.A ([[Media:Function.Overview.1.A.20111201.pdf|pdf]])
&lt;br&gt; 

== Versions of VHDL ==
* VHDL Versions ([[Media:VHDL.1.A.Versions.20120619.pdf|pdf]])
* VHDL Libraries ([[Media:VHDL.1.A.Libraries.20140219.pdf|pdf]])
&lt;br&gt;

== Basic Features of VHDL ==
==== Data ====
* Data Objects ([[Media:Data.Object.1A.20260713.pdf|A]], [[Media:Data.Object.1B.20260602.pdf|B]])
* Data Types ([[Media:Data.Type.2A.20260602.pdf|A]], [[Media:Data.Type.2B.20260602.pdf|B]])
* Packages ([[Media:Data.Package.3A.20251206.pdf|pdf]])
* Signal Types ([[Media:Signal.Type.1A.20250614.pdf|pdf]])
* Attributes ([[Media:Data.4.A.Attribute.20251021.pdf|pdf]])
&lt;br&gt;

==== Signals &amp; Variables ====
* Signals &amp; Variables ([[Media:Signal.1A.SigVar.20250614.pdf|pdf]])
* Sequential Signal Assignments ([[Media:Signal.4A.Sequential.20250612.pdf|pdf]])
* Concurrent &amp; Sequential Signal Assignments ([[Media:Signal.1.A.ConSeq.20120611.pdf|pdf]])
* Inertial &amp; Transport Delay Models ([[Media:Signal.2.A.InertTrans.20120704.pdf|pdf]])
* Simulation &amp; Synthesis ([[Media:Signal.3.A.SimSyn.20120504.pdf|pdf]])


&lt;br&gt;

==== Structure ====
* Component ([[Media:Struct.1.A.Component.20120804.pdf|pdf]])
* Configuration ([[Media:Struct.1.A.Configuration.20121003.pdf|pdf]])
* Generic ([[Media:Struct.1.A.Generic.20120802.pdf|pdf]])
&lt;/br&gt;

==== Entity and Architecture ====

&lt;br&gt; 
==== Block Statement ====

&lt;br&gt; 
==== Process Statement ====

&lt;br&gt;
==== Operators ====

&lt;br&gt;
==== Assignment Statement ====

&lt;br&gt; 
==== Concurrent Statement ====

&lt;br&gt; 
==== Sequential Control Statement ====

&lt;br&gt;
==== Function ====
* Function.1.A Usage ([[Media:Function.1.A.Usage.20120611.pdf|pdf]])
* Function.2.A Conversion Function ([[Media:Function.2.A.Conversion.pdf|pdf]])
* Function.3.A Resolution Function ([[Media:Function.3.A.Resolution.pdf|pdf]])

&lt;br&gt;
==== Procedure ====


&lt;br&gt;
==== Package ====
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:VHDL]]
[[Category:FPGA]]</text>
      <sha1>3h29fj4q7gspgvhuc0aznb4yuh5n57l</sha1>
    </revision>
    <revision>
      <id>2818563</id>
      <parentid>2818561</parentid>
      <timestamp>2026-07-19T19:47:58Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Data */</comment>
      <origin>2818563</origin>
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      <format>text/x-wiki</format>
      <text bytes="2278" sha1="46626sxvdmiy90w4ikzzhkonyb1kzpy" xml:space="preserve">&lt;!----------------------------------------------------------------------&gt;

== Flip Flop and Latch ==
* FFLatch.Overview.1.A ([[Media:FFLatch.Overview.1.A.20111103.pdf|pdf]])
* Counter.74LS193.1.A ([[Media:Counter.74LS193.1.A.20111108.pdf|pdf]])
* Clock.Overview.1.A ([[Media:Clock.Overview.1.A.20111108.pdf|pdf]])
* Function.Overview.1.A ([[Media:Function.Overview.1.A.20111201.pdf|pdf]])
&lt;br&gt; 

== Versions of VHDL ==
* VHDL Versions ([[Media:VHDL.1.A.Versions.20120619.pdf|pdf]])
* VHDL Libraries ([[Media:VHDL.1.A.Libraries.20140219.pdf|pdf]])
&lt;br&gt;

== Basic Features of VHDL ==
==== Data ====
* Data Objects ([[Media:Data.Object.1A.20260714.pdf|A]], [[Media:Data.Object.1B.20260602.pdf|B]])
* Data Types ([[Media:Data.Type.2A.20260602.pdf|A]], [[Media:Data.Type.2B.20260602.pdf|B]])
* Packages ([[Media:Data.Package.3A.20251206.pdf|pdf]])
* Signal Types ([[Media:Signal.Type.1A.20250614.pdf|pdf]])
* Attributes ([[Media:Data.4.A.Attribute.20251021.pdf|pdf]])
&lt;br&gt;

==== Signals &amp; Variables ====
* Signals &amp; Variables ([[Media:Signal.1A.SigVar.20250614.pdf|pdf]])
* Sequential Signal Assignments ([[Media:Signal.4A.Sequential.20250612.pdf|pdf]])
* Concurrent &amp; Sequential Signal Assignments ([[Media:Signal.1.A.ConSeq.20120611.pdf|pdf]])
* Inertial &amp; Transport Delay Models ([[Media:Signal.2.A.InertTrans.20120704.pdf|pdf]])
* Simulation &amp; Synthesis ([[Media:Signal.3.A.SimSyn.20120504.pdf|pdf]])


&lt;br&gt;

==== Structure ====
* Component ([[Media:Struct.1.A.Component.20120804.pdf|pdf]])
* Configuration ([[Media:Struct.1.A.Configuration.20121003.pdf|pdf]])
* Generic ([[Media:Struct.1.A.Generic.20120802.pdf|pdf]])
&lt;/br&gt;

==== Entity and Architecture ====

&lt;br&gt; 
==== Block Statement ====

&lt;br&gt; 
==== Process Statement ====

&lt;br&gt;
==== Operators ====

&lt;br&gt;
==== Assignment Statement ====

&lt;br&gt; 
==== Concurrent Statement ====

&lt;br&gt; 
==== Sequential Control Statement ====

&lt;br&gt;
==== Function ====
* Function.1.A Usage ([[Media:Function.1.A.Usage.20120611.pdf|pdf]])
* Function.2.A Conversion Function ([[Media:Function.2.A.Conversion.pdf|pdf]])
* Function.3.A Resolution Function ([[Media:Function.3.A.Resolution.pdf|pdf]])

&lt;br&gt;
==== Procedure ====


&lt;br&gt;
==== Package ====
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:VHDL]]
[[Category:FPGA]]</text>
      <sha1>46626sxvdmiy90w4ikzzhkonyb1kzpy</sha1>
    </revision>
    <revision>
      <id>2818565</id>
      <parentid>2818563</parentid>
      <timestamp>2026-07-19T19:48:49Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Data */</comment>
      <origin>2818565</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2278" sha1="6wb4c035721lraumdgmya9193pnbru2" xml:space="preserve">&lt;!----------------------------------------------------------------------&gt;

== Flip Flop and Latch ==
* FFLatch.Overview.1.A ([[Media:FFLatch.Overview.1.A.20111103.pdf|pdf]])
* Counter.74LS193.1.A ([[Media:Counter.74LS193.1.A.20111108.pdf|pdf]])
* Clock.Overview.1.A ([[Media:Clock.Overview.1.A.20111108.pdf|pdf]])
* Function.Overview.1.A ([[Media:Function.Overview.1.A.20111201.pdf|pdf]])
&lt;br&gt; 

== Versions of VHDL ==
* VHDL Versions ([[Media:VHDL.1.A.Versions.20120619.pdf|pdf]])
* VHDL Libraries ([[Media:VHDL.1.A.Libraries.20140219.pdf|pdf]])
&lt;br&gt;

== Basic Features of VHDL ==
==== Data ====
* Data Objects ([[Media:Data.Object.1A.20260720.pdf|A]], [[Media:Data.Object.1B.20260602.pdf|B]])
* Data Types ([[Media:Data.Type.2A.20260602.pdf|A]], [[Media:Data.Type.2B.20260602.pdf|B]])
* Packages ([[Media:Data.Package.3A.20251206.pdf|pdf]])
* Signal Types ([[Media:Signal.Type.1A.20250614.pdf|pdf]])
* Attributes ([[Media:Data.4.A.Attribute.20251021.pdf|pdf]])
&lt;br&gt;

==== Signals &amp; Variables ====
* Signals &amp; Variables ([[Media:Signal.1A.SigVar.20250614.pdf|pdf]])
* Sequential Signal Assignments ([[Media:Signal.4A.Sequential.20250612.pdf|pdf]])
* Concurrent &amp; Sequential Signal Assignments ([[Media:Signal.1.A.ConSeq.20120611.pdf|pdf]])
* Inertial &amp; Transport Delay Models ([[Media:Signal.2.A.InertTrans.20120704.pdf|pdf]])
* Simulation &amp; Synthesis ([[Media:Signal.3.A.SimSyn.20120504.pdf|pdf]])


&lt;br&gt;

==== Structure ====
* Component ([[Media:Struct.1.A.Component.20120804.pdf|pdf]])
* Configuration ([[Media:Struct.1.A.Configuration.20121003.pdf|pdf]])
* Generic ([[Media:Struct.1.A.Generic.20120802.pdf|pdf]])
&lt;/br&gt;

==== Entity and Architecture ====

&lt;br&gt; 
==== Block Statement ====

&lt;br&gt; 
==== Process Statement ====

&lt;br&gt;
==== Operators ====

&lt;br&gt;
==== Assignment Statement ====

&lt;br&gt; 
==== Concurrent Statement ====

&lt;br&gt; 
==== Sequential Control Statement ====

&lt;br&gt;
==== Function ====
* Function.1.A Usage ([[Media:Function.1.A.Usage.20120611.pdf|pdf]])
* Function.2.A Conversion Function ([[Media:Function.2.A.Conversion.pdf|pdf]])
* Function.3.A Resolution Function ([[Media:Function.3.A.Resolution.pdf|pdf]])

&lt;br&gt;
==== Procedure ====


&lt;br&gt;
==== Package ====
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:VHDL]]
[[Category:FPGA]]</text>
      <sha1>6wb4c035721lraumdgmya9193pnbru2</sha1>
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  </page>
  <page>
    <title>User:Atcovi/to do</title>
    <ns>2</ns>
    <id>145726</id>
    <revision>
      <id>2818581</id>
      <parentid>2816049</parentid>
      <timestamp>2026-07-20T02:17:00Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>/* Atcovi/to do */ +[[User:Atcovi/to do/Current Projects/2026]] to archives</comment>
      <origin>2818581</origin>
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      <format>text/x-wiki</format>
      <text bytes="3005" sha1="gfak8m5dhxo4d34pwcxrdpnq1qp5dcc" xml:space="preserve">==Atcovi/to do==

=== Current Projects (2026) ===

* [[Intuitive Calculus]]
* [[User:Atcovi/OGM &amp; Suicide/The Paper]] - ''[moved]''
* [[User:Atcovi/Journey to Clinical PhD]] - figuring this out; current life goal.

* [[WikiJournal Preprints/Mental health in Sri Lanka]] (and later in August: [[User:Atcovi/APA2026 Abstract]])
** [[User:Atcovi/WikiJournal Preprints/Mental health in Sri Lanka/Future Outlook]].
====Future Endeavors====
* [[WikiJournal Preprints/Suicide amongst refugees in Sweden]] [https://scholar.google.com/scholar?hl=en&amp;as_sdt=0%2C47&amp;as_ylo=2020&amp;as_yhi=2025&amp;q=Suicide+in+Sweden+refugees&amp;btnG=]
* Get [[User:Atcovi/Spring2024]] &amp; [[User:Atcovi/Psychopathology]] into the mainspace. Develop [[Child psychology]] &amp; [[User:Atcovi/PSYC318W]] into a complete course. Merge [[Validity]] into [[User:Atcovi/PSYC318W|PSYC318W]].
* Develop resources related to [[suicidology]] (3 stress response systems? effects of catecholamines on suicidal ideation? neurobiology of suicidal ideation? relation between autobiographical memory and suicide?), expand [[wikipedia:Suicidology#Theories_of_suicide|Suicidology#Theories_of_suicide]] either through [[WikiJournal of Science]] or WP editing.
=====Wikiversity-Related Works=====
* Promote [[Help:Project boxes]], something very useful and unique to Wikiversity. Focus on trying to not only create more project boxes, but to define resource types used in project boxes.
**Ex, what is a [[:Category:Workshops|workshop]]? What differentiates between an [[Help:Essay|essay]] and a [[Help:Paper|paper]]? What differentiates between a [[Template:Notes|notes resource]] (that may be ''derived'' from a homework assignment) and a [[Help:Assignment|homework assignment]] [small note: this page seems to be created by accident and may need a revamp]?
* [[Wikiversity:Original research and scholarly standards]] &amp; improvements/proposals for [[Wikiversity:Original research]] (ex, [[Template:Original research]] should be a mandatory addition to original research on WV + a notice letting readers know that the work is not established science). Develop other pages related to research ethics, including [[Wikiversity:Research]] &amp; [[Wikiversity:Research ethics]].
** [[Wikiversity:Review board]] - should this be Wikiversity 'crats that review original research proposals? 
* [[Wikiversity:Verifiability]] - start heavily scrutinizing pages that don't meet this criteria. 
* [[Wikiversity:Artificial intelligence]] - &quot;substantial&quot;? What defines &quot;substantial&quot;? 
* Expand [[Wikiversity:Differences between Wikiversity and Wikipedia]]. 
{{Archive box|
{{center top}}'''[[User:Atcovi/to do|To do list]]'''{{center bottom}}
----
{{center top}}'''Archives'''{{center bottom}}
*[[User:Atcovi/to do/Current Projects/2026]]
*[[User:Atcovi/to do/Current Projects/2023]]
*[[User:Atcovi/to do/Current Projects/January 4, 2022]]
*[[User:Atcovi/to do/Current Projects/September 2017 - January 2018]]
*[[User:Atcovi/to do/Current Projects/2015]]
----
}}
[[Category:Atcovi's Work]]</text>
      <sha1>gfak8m5dhxo4d34pwcxrdpnq1qp5dcc</sha1>
    </revision>
    <revision>
      <id>2818583</id>
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      <timestamp>2026-07-20T02:17:39Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* Atcovi/to do */</comment>
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      <text bytes="2924" sha1="oxrykyxlajzpol9x5vdwaxim2hm3t2l" xml:space="preserve">==Atcovi/to do==

=== Current Projects (2026) ===

* [[User:Atcovi/Journey to Clinical PhD]] - figuring this out; current life goal.

* [[WikiJournal Preprints/Mental health in Sri Lanka]] (and later in August: [[User:Atcovi/APA2026 Abstract]])
** [[User:Atcovi/WikiJournal Preprints/Mental health in Sri Lanka/Future Outlook]].
====Future Endeavors====
* [[WikiJournal Preprints/Suicide amongst refugees in Sweden]] [https://scholar.google.com/scholar?hl=en&amp;as_sdt=0%2C47&amp;as_ylo=2020&amp;as_yhi=2025&amp;q=Suicide+in+Sweden+refugees&amp;btnG=]
* Get [[User:Atcovi/Spring2024]] &amp; [[User:Atcovi/Psychopathology]] into the mainspace. Develop [[Child psychology]] &amp; [[User:Atcovi/PSYC318W]] into a complete course. Merge [[Validity]] into [[User:Atcovi/PSYC318W|PSYC318W]].
* Develop resources related to [[suicidology]] (3 stress response systems? effects of catecholamines on suicidal ideation? neurobiology of suicidal ideation? relation between autobiographical memory and suicide?), expand [[wikipedia:Suicidology#Theories_of_suicide|Suicidology#Theories_of_suicide]] either through [[WikiJournal of Science]] or WP editing.
=====Wikiversity-Related Works=====
* Promote [[Help:Project boxes]], something very useful and unique to Wikiversity. Focus on trying to not only create more project boxes, but to define resource types used in project boxes.
**Ex, what is a [[:Category:Workshops|workshop]]? What differentiates between an [[Help:Essay|essay]] and a [[Help:Paper|paper]]? What differentiates between a [[Template:Notes|notes resource]] (that may be ''derived'' from a homework assignment) and a [[Help:Assignment|homework assignment]] [small note: this page seems to be created by accident and may need a revamp]?
* [[Wikiversity:Original research and scholarly standards]] &amp; improvements/proposals for [[Wikiversity:Original research]] (ex, [[Template:Original research]] should be a mandatory addition to original research on WV + a notice letting readers know that the work is not established science). Develop other pages related to research ethics, including [[Wikiversity:Research]] &amp; [[Wikiversity:Research ethics]].
** [[Wikiversity:Review board]] - should this be Wikiversity 'crats that review original research proposals? 
* [[Wikiversity:Verifiability]] - start heavily scrutinizing pages that don't meet this criteria. 
* [[Wikiversity:Artificial intelligence]] - &quot;substantial&quot;? What defines &quot;substantial&quot;? 
* Expand [[Wikiversity:Differences between Wikiversity and Wikipedia]]. 
{{Archive box|
{{center top}}'''[[User:Atcovi/to do|To do list]]'''{{center bottom}}
----
{{center top}}'''Archives'''{{center bottom}}
*[[User:Atcovi/to do/Current Projects/2026]]
*[[User:Atcovi/to do/Current Projects/2023]]
*[[User:Atcovi/to do/Current Projects/January 4, 2022]]
*[[User:Atcovi/to do/Current Projects/September 2017 - January 2018]]
*[[User:Atcovi/to do/Current Projects/2015]]
----
}}
[[Category:Atcovi's Work]]</text>
      <sha1>oxrykyxlajzpol9x5vdwaxim2hm3t2l</sha1>
    </revision>
    <revision>
      <id>2818586</id>
      <parentid>2818583</parentid>
      <timestamp>2026-07-20T02:19:10Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* Atcovi/to do */</comment>
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      <format>text/x-wiki</format>
      <text bytes="2941" sha1="cx4tmanae7ytdux09yqcwx7uknj5alk" xml:space="preserve">==Atcovi/to do==

=== Current Projects (2026) ===

* [[User:Atcovi/Journey to Clinical PhD]] - figuring this out; current life goal.

* [[WikiJournal Preprints/Mental health in Sri Lanka]] (and later in August: [[User:Atcovi/APA2026 Abstract]])
** [[User:Atcovi/WikiJournal Preprints/Mental health in Sri Lanka/Future Outlook]].
====Suicidology/Psychopathology Works====
* [[WikiJournal Preprints/Suicide amongst refugees in Sweden]] [https://scholar.google.com/scholar?hl=en&amp;as_sdt=0%2C47&amp;as_ylo=2020&amp;as_yhi=2025&amp;q=Suicide+in+Sweden+refugees&amp;btnG=]
* Get [[User:Atcovi/Spring2024]] &amp; [[User:Atcovi/Psychopathology]] into the mainspace. Develop [[Child psychology]] &amp; [[User:Atcovi/PSYC318W]] into a complete course. Merge [[Validity]] into [[User:Atcovi/PSYC318W|PSYC318W]].
* Develop resources related to [[suicidology]] (3 stress response systems? effects of catecholamines on suicidal ideation? neurobiology of suicidal ideation? relation between autobiographical memory and suicide?), expand [[wikipedia:Suicidology#Theories_of_suicide|Suicidology#Theories_of_suicide]] either through [[WikiJournal of Science]] or WP editing.
=====Wikiversity-Related Works=====
* Promote [[Help:Project boxes]], something very useful and unique to Wikiversity. Focus on trying to not only create more project boxes, but to define resource types used in project boxes.
**Ex, what is a [[:Category:Workshops|workshop]]? What differentiates between an [[Help:Essay|essay]] and a [[Help:Paper|paper]]? What differentiates between a [[Template:Notes|notes resource]] (that may be ''derived'' from a homework assignment) and a [[Help:Assignment|homework assignment]] [small note: this page seems to be created by accident and may need a revamp]?
* [[Wikiversity:Original research and scholarly standards]] &amp; improvements/proposals for [[Wikiversity:Original research]] (ex, [[Template:Original research]] should be a mandatory addition to original research on WV + a notice letting readers know that the work is not established science). Develop other pages related to research ethics, including [[Wikiversity:Research]] &amp; [[Wikiversity:Research ethics]].
** [[Wikiversity:Review board]] - should this be Wikiversity 'crats that review original research proposals? 
* [[Wikiversity:Verifiability]] - start heavily scrutinizing pages that don't meet this criteria. 
* [[Wikiversity:Artificial intelligence]] - &quot;substantial&quot;? What defines &quot;substantial&quot;? 
* Expand [[Wikiversity:Differences between Wikiversity and Wikipedia]]. 
{{Archive box|
{{center top}}'''[[User:Atcovi/to do|To do list]]'''{{center bottom}}
----
{{center top}}'''Archives'''{{center bottom}}
*[[User:Atcovi/to do/Current Projects/2026]]
*[[User:Atcovi/to do/Current Projects/2023]]
*[[User:Atcovi/to do/Current Projects/January 4, 2022]]
*[[User:Atcovi/to do/Current Projects/September 2017 - January 2018]]
*[[User:Atcovi/to do/Current Projects/2015]]
----
}}
[[Category:Atcovi's Work]]</text>
      <sha1>cx4tmanae7ytdux09yqcwx7uknj5alk</sha1>
    </revision>
    <revision>
      <id>2818587</id>
      <parentid>2818586</parentid>
      <timestamp>2026-07-20T02:19:38Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>/* Suicidology/Psychopathology Works */ rearrange</comment>
      <origin>2818587</origin>
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      <text bytes="2943" sha1="3jhjaw6bydls37vgct4b3gsovno5cu6" xml:space="preserve">==Atcovi/to do==

=== Current Projects (2026) ===

* [[User:Atcovi/Journey to Clinical PhD]] - figuring this out; current life goal.

* [[WikiJournal Preprints/Mental health in Sri Lanka]] (and later in August: [[User:Atcovi/APA2026 Abstract]])
** [[User:Atcovi/WikiJournal Preprints/Mental health in Sri Lanka/Future Outlook]].
====Suicidology/Psychopathology Works====

* Develop resources related to [[suicidology]] (3 stress response systems? effects of catecholamines on suicidal ideation? neurobiology of suicidal ideation? relation between autobiographical memory and suicide?), expand [[wikipedia:Suicidology#Theories_of_suicide|Suicidology#Theories_of_suicide]] either through [[WikiJournal of Science]] or WP editing.

* [[WikiJournal Preprints/Suicide amongst refugees in Sweden]] [https://scholar.google.com/scholar?hl=en&amp;as_sdt=0%2C47&amp;as_ylo=2020&amp;as_yhi=2025&amp;q=Suicide+in+Sweden+refugees&amp;btnG=]
* Get [[User:Atcovi/Spring2024]] &amp; [[User:Atcovi/Psychopathology]] into the mainspace. Develop [[Child psychology]] &amp; [[User:Atcovi/PSYC318W]] into a complete course. Merge [[Validity]] into [[User:Atcovi/PSYC318W|PSYC318W]].
=====Wikiversity-Related Works=====
* Promote [[Help:Project boxes]], something very useful and unique to Wikiversity. Focus on trying to not only create more project boxes, but to define resource types used in project boxes.
**Ex, what is a [[:Category:Workshops|workshop]]? What differentiates between an [[Help:Essay|essay]] and a [[Help:Paper|paper]]? What differentiates between a [[Template:Notes|notes resource]] (that may be ''derived'' from a homework assignment) and a [[Help:Assignment|homework assignment]] [small note: this page seems to be created by accident and may need a revamp]?
* [[Wikiversity:Original research and scholarly standards]] &amp; improvements/proposals for [[Wikiversity:Original research]] (ex, [[Template:Original research]] should be a mandatory addition to original research on WV + a notice letting readers know that the work is not established science). Develop other pages related to research ethics, including [[Wikiversity:Research]] &amp; [[Wikiversity:Research ethics]].
** [[Wikiversity:Review board]] - should this be Wikiversity 'crats that review original research proposals? 
* [[Wikiversity:Verifiability]] - start heavily scrutinizing pages that don't meet this criteria. 
* [[Wikiversity:Artificial intelligence]] - &quot;substantial&quot;? What defines &quot;substantial&quot;? 
* Expand [[Wikiversity:Differences between Wikiversity and Wikipedia]]. 
{{Archive box|
{{center top}}'''[[User:Atcovi/to do|To do list]]'''{{center bottom}}
----
{{center top}}'''Archives'''{{center bottom}}
*[[User:Atcovi/to do/Current Projects/2026]]
*[[User:Atcovi/to do/Current Projects/2023]]
*[[User:Atcovi/to do/Current Projects/January 4, 2022]]
*[[User:Atcovi/to do/Current Projects/September 2017 - January 2018]]
*[[User:Atcovi/to do/Current Projects/2015]]
----
}}
[[Category:Atcovi's Work]]</text>
      <sha1>3jhjaw6bydls37vgct4b3gsovno5cu6</sha1>
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  </page>
  <page>
    <title>In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Nicholas II (1894-1917)</title>
    <ns>0</ns>
    <id>160777</id>
    <revision>
      <id>2818541</id>
      <parentid>2669669</parentid>
      <timestamp>2026-07-19T18:03:41Z</timestamp>
      <contributor>
        <username>ABDULHAMIRAGO</username>
        <id>3102072</id>
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      <text bytes="233469" sha1="36qk71lz3wkz7ou1nge538mj60ohxby" xml:space="preserve">__NOTOC__
==11. REIGN OF NICHOLAS II (1894-1917)==

[[File:Tsar_Nicholas_II_-1898.jpg|thumb|Fig. 45 Tsar Nicholas II (1898), photograph by A.A. Pasetti.]]

 
'''See also: '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander II (1855-1881)#I51|I51]]''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander II (1855-1881)#I79|I79]]''', '''1000501639951''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander II (1855-1881)#I143|I143]]''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander II (1855-1881)#I146|I146]]''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander II (1855-1881)#I167|I167]]''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander II (1855-1881)#I173|I173]]''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander II (1855-1881)#I179|I179]]''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander II (1855-1881)#I180|I180]]''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J40|J40]]''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J43|J43]]''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J44|J44]]''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J65|J65]]''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J76|J76]]''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J77|J77]]''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J82|J82]]''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J87|J87]]''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J88|J88]]''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J94|J94]]''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J120|J120]]''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J122|J122]]''', '''[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J140|J140]]


======K1======
'''‘Viator’, '''''Overland to Persia. ''London: John and Edward Bumpus, 1906. xii+169pp.
::Anonymous account of a journey undertaken during a summer sometime in the 1890s. In a breezy narrative the English author describes his journey by train from Warsaw to Odessa and his travels with a Russian companion through the Crimean peninsula and along the Black Sea coast. They take the train through Georgia to Tiflis and proceed by horse into Persian territory. The text is enlivened with illustrations by Ambrose Dudley from sketches by the author (pp. 1-105).

 
======K1a======
'''Bishop, Isabella Lucy Bird''', ''Korea and her neighbors: a narrative of travel, with an account of the recent vicissitudes and present position of the country''. With a preface by Sir Walter C. Hillier. London: John Murray, 1898. 2 vols.
::The author of four books of travel under her maiden name of Bird by the time of her marriage to Dr John Bishop (d. 1886) in 1881, Isabella (1831-1904), in 1892 the first woman to be elected F.R.G.S., was in Vladivostok from early November to 10 December 1894 between visits to Korea. In Vladivostok she became close friends with Eleanor Pray (see J138). Bishop visited Korean settlements in Siberia and journeyed along the Trans-Siberian railway as far as Ussuri (vol. I, pp. 213-41).


======K2======
'''Harris, Walter Burton,''' ''From Batum to Baghdad via Tiflis, Tabriz, and Persian Kurdistan''. Edinburgh and London: William Blackwood and Sons, 1896. xii+335pp.
::Harris (1866-1933), F.R.G.S., ''Times'' correspondent and Moroccan specialist, who had been in Archangel some years earlier, sailed from Constantinople for Batumi in April 1895 and went by train first to Tiflis and then by carriage to Erevan, and on into Persia, without incident or much of note (pp. 25-84).

 
======K3======
'''Muir, Hal Moncreiff,''' ''A tour in Russia''. Leith: printed for the author by Mackenzie and Storrie, 1898. 76pp.
::A “humble account of travel” by an Edinburgh Scot, author of similar efforts for Switzerland and the Pyrenees, who sailed with three friends from Grangemouth to Cronstadt “several years ago”, presumably c.1895. A conventional description of the sights of Petersburg and its environs and of an excursion to Moscow suddenly changes tack and is followed by a string of “stories” illustrating Russian religious, social and village life (pp. 26-70).

 
======K4======
'''Perris, George Herbert, '''''Russia in revolution. ''London: Chapman &amp;amp; Hall, 1905. xvi+359pp.
::Offered as “a review of the last thirty-five years of Russian public life” and making copious use of printed sources and the active contribution of many Russian revolutionaries in exile, this account, dated 15 April 1905 and thus published before the tragic end of that year, is suffused with Perris’s own memories from numerous visits to Russia during the first decade of Nicholas’s reign. Perris (1866-1920), who had published ''Leo Tolstoy, the grand mujik'' (1898), contributed articles while in Russia to various papers including the ''Daily Chronicle''.

 
======K5======
'''Dillon, Emile Joseph,''' ''The eclipse of Russia.'' London &amp;amp; Toronto: J.M. Dent and Sons, 1918. viii+420pp.
::Dedicated to the Russian premier Count Witte for whom Dillon had worked as private adviser from 1903 to 1914, this work, representing Dillon’s bleak view of Russia under Nicholas II, also includes a chapter entitled ‘Some personal recollections’ (pp. 61-82). (See also '''I157''' and, under pseudonym of Lanin, [[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J89|J89]].)

 
======K6======
'''Hodgetts, Edward Arthur Brayley,''' ''Round about Armenia: the record of a journey across the Balkans through Turkey, the Caucasus and Persia, in 1895.'' London: Sampson Low, Marston &amp;amp; Co., 1896. xii+296pp.
::Hodgetts (see [[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander II (1855-1881)#I79|I79]], [[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J113|J113]]) was later to call this trip on behalf of the ''Daily Graphic'' to report on the Turkish massacres of Armenians “the most interesting of the many journeys I had ever undertaken”. It took him to Tiflis and Baku and into both Russian and Turkish Armenia.

 
======K7======
'''Jefferson, Robert Louis,''' ''Awheel to Moscow and back: the record of a record cycle ride''. With a preface by A.R. Savile. London: Sampson Low, Marston &amp;amp; Co., 1895. xii+172pp.
::Cycling enthusiast Jefferson’s first venture into Russia, making a round trip from Warsaw to Moscow in fifty days in April-June 1895 (pp. 61-159). (See also [[#K30|K30]], [[#K48|K48]].)

 
======K8======
'''Pearson, Henry John, “'''''Beyond Petsora eastwards”: two summer voyages to Novaya Zemlya and the islands of the Barents Sea.'' With appendices on the botany and geology by Col. H.W. Feilden. London: R.H. Porter, 1899. xiv+335pp.
::The Nottingham foundry-owner and dedicated ornithologist (1850-1913), together with his brother Charles, the Rev. Slater, and Col. Feilden, sailed on a small steamer, the ''Saxon,'' to Russian Lapland and the islands of Kolguev and Novaia Zemlia in the summer of 1895, recording and collecting birds and their eggs. In 1897, on a larger steamer, the ''Laura'', he and Feilden went as far as the Kara Sea. Their observations were first published in ''Ibis'' in 1896 and 1897.

 
======K9======
'''Demidov, Elim Pavlovich, '''''Hunting trips in the Caucasus''. London: Rowland Ward, 1898. xvi+319pp.
::The immensely rich Demidov (1868-1943), 3rd Prince of San Donato, born in Vienna and dying in Athens, provided a “faithful account of three shooting trips to three different parts of the Caucasus”. The first took him to the Kuban in the autumn of 1895; the second along the Russo-Persian border in the summer of 1896; the third, again to the Kuban, in the autumn of 1896. The last two accounts were in fact written by '''Dr H.D. Levick''', who accompanied Demidov. The book was dedicated to St. George Littledale, who was also on the third expedition and who had shot in the Caucasus several times in earlier years. (For further expeditions, see [[#K31|K31]] and [[#K69|K69]]).

 
======K10======
'''Patterson, John Edward,''' ''My vagabondage, being the intimate autobiography of a nature’s nomad''. London: William Heinemann, 1911. xix+373pp.
::Patterson (1866-1919), Yorkshire-born merchant seaman and later novelist, recounts his adventures at sea, including disastrous shore-leave in St Petersburg at the end of the nineteenth century (pp. 252-66).

 
======K11======
'''Kenworthy, John Coleman,''' ''A pilgrimage to Tolstoy: being letters written from Russia, to the ‘New Age’, January 1896''. Croydon: Brotherhood Publishing Co., 1896. 45pp.
::Kenworthy (1863-1946), leader of the Croydon Brotherhood Church, came under the spell of Tolstoi’s writings in 1890 and soon thereafter began a correspondence that led to his visit to Moscow at the end of 1895 and early 1896. In between his meetings with Tolstoi he paid a visit to Kostroma to see the “real” Russia of the countryside.

 
======K12======
'''Palmer, Francis H.E., '''''Russian life in town &amp;amp; country.'' London: George Newnes, 1901. xii+271pp.
::Based on a residence of several years in Russia (the author mentions incidents in 1895 and 1900), Palmer’s contribution to the Newnes’s ‘Our neighbours’ series is a detailed, informed and sympathetic account of the social and domestic life of the Russians in town and village.

 
======K13======
'''Joubert, Carl, '''''[https://archive.org/details/russiaasitreall00joubgoog Russia as it really is]''. London: Eveleigh Nash, 1904. xii+300pp.
::Joubert (d. 1906)'', ''an Englishman of Huguenot descent, who claimed to have visited as a tramp (''brodiaga'') almost every part of Russia including Sakhalin during nine years from c.1881, travelled there several times thereafter. It is a journey in 1896 that he recalls in particular in this first of three books he wrote in the space of eighteen months, condemning tyranny, the penal system, and anti-semitism, and sensing revolution.

======K14======
'''Olufsen, Axel Frits Olaf Henrik,''' ''Through the unknown Pamirs: the second Danish Pamir expedition 1898-99. ''London: William Heinemann, 1904. xxii+238pp.
::Lt. Olufsen (1865-1929) of the Danish army led two expeditions to the Pamirs in the 1890s. The first left Copenhagen on 25 March 1896 and returned on 1 March of the following year, travelling from St Petersburg via Georgia to Baku, across the Caspian, and by the Transcaspian railway to Samarkand, thereafter by tarantas and horse into the Pamirs. This journey was seen as essentially one of reconnoitring for the second expedition that lasted from 23 March 1898 to 22 November 1899. It was this second expedition, following the same outward route as far as Osh and then proceeding deep into south Pamir that produced the material for the book.

 
======K15======
'''Olufsen, Axel Frits Olaf Henrik,''' ''The emir of Bokhara and his country: journeys and studies in Bokhara (with a chapter on my voyage on the Amu Darya to Khiva). ''London: William Heinemann, 1911. xii+599pp.
::In a complementary volume to his 1904 work, Olufsen, now retired from the army, a professor and secretary to the Royal Danish Geographical Society, concentrates on a comprehensive study of Bokhara, both before and after it became a vassal state of Russia. Both Danish expeditions spent extended stays as guests of the emir in Bokhara.

 
======K16======
'''Bigham, Clive, '''''A ride through Western Asia.'' London: Macmillan and Co., 1897. xii+284pp.
::Anxious to get to Armenia, Bigham left England on 22 June 1895 by what seemed, given the political situation, the most difficult route via Constantinople. He traversed Persia and eventually entered Russian territory on 20 April 1896. He travelled through Russian Turkestan and visited Bokhara and Samarkand, briefly entered China, and then across the steppe to Omsk and homewards by the Trans-Siberian to St Petersburg, which he reached on 26 June. He calculated that he had travelled in Asia over 8,000 miles, half of which were on horseback (pp. 205-69).

 
======K17======
'''Gordon, Samuel,''' ''A handful of exotics: scenes and incidents, chiefly of Russo-Jewish life. ''London: Methuen &amp;amp; Co., 1897. x+297pp.
::In his preface, dated September 1896, Gordon offers, as an amateur ethnographer, a series of “light sketches endeavour[ing] to depict the Russian Jew in his native surroundings”. Two of the ten tales are devoted to non-Jewish subjects, “illustrating the environments in which the Russian Jew moves”.

 
======K18======
'''Malcolm, Ian Zachary, '''''Trodden ways 1895-1930''. London: Macmillan and Co., 1930. xii+288pp.
::Sir Ian (1868-1944), 17th chieftain of the clan Malcolm and a M.P., was attached to the British embassy for three weeks at the time of the coronation of Nicholas II in May 1896 and he records his impressions in an essay entitled ‘The last coronation’ (pp. 44-68). In 1916 he returned to Russia as British Red Cross commissioner, mainly in Petrograd but also visiting Kiev. He includes a description of his audience with the tsar at Tsarskoe selo (pp. 69-92).

 
======K19======
'''Logan, John Alexander, Jr., '''''In joyful Russia.'' London: C. Arthur Pearson, 1897. x+275pp.
::The American army officer Logan (1865-99) records “a thoroughly delightful” trip and defends his “rose-coloured” view of a country “in holiday attire” for the coronation of Nicholas II in Moscow on 26 May 1896. He even plays down the tragedy at the festival at Khodynskoe pole on 30 May for “it is not best to let the unthinking brood too deeply over the irretrievable” and what happened only underlined the “sympathy” between the people and the throne. Logan, his mother, and his friend G left Moscow for St Petersburg on 7 June, where they enjoyed further delights before “departing the land of the Great White Tsar with regret”.

 
======K20======
'''Grenfell, Francis Wallace,''' ''Three weeks in Moscow''. London: for the author by Harrison and Sons, 1896. iv+152pp.
::Lt-General Sir Francis (1841-1925), afterwards 1st Baron Grenfell, was in the suite accompanying the Duke and Duchess of Connaught to attend the coronation of Nicholas II. In a series of letters to E. (Evelyn, his first wife), he describes their departure from Sheerness on 11 May on board the ''Victoria and Albert ''and arrival a week later at the English Embankment in St Petersburg, proceeding to Moscow by train. Five days after the coronation he was also present at the public festival at Khodynskoe pole and reports on the tragedy that ensued. They sailed from St Petersburg on 9 June (pp. 10-116).

 
======K21======
'''Grenfell, Francis Wallace, '''''Memoirs of Field Marshall Lord Grenfell.'' London: Hodder &amp;amp; Stoughton, 1925. xv+236pp.
::Grenfell recalls his visit in 1896 (pp. 127-42).

 
======K22======
'''Creighton, Mandell, '''''Life and letters. ''By his wife [Louise Creighton]. London: Longmans, Green, &amp;amp; Co., 1904. 2 vols.
::Creighton (1843-1901), Bishop of Peterborough, represented the Anglican church at the coronation, travelling with W.J. Birkbeck, the acknowledged British authority on the Russian church. He describes his visit in letters to his wife and in a notebook, detailing his meetings with, among others, the patriarch and Pobedonostsev (vol. II, pp. 148-64).

 
======K23======
'''Sykes, Arthur Alkin,''' ''The coronation cruise of the ‘Midnight Sun’ to Russia, Whitsuntide, 1896: a record''. London: for the author, 1896. 118pp.
::Sykes (1861-1939), contributor to ''Punch'' and translator of Gogol’s ''Revizor'', joined a three-week tour (11 May-6 June) of five northern capitals that was designed to coincide with Nicholas II’s coronation. Grenfell (see [[#K20|K20]]) mentions the “cheering” British tourists (168 in fact) on board the ''Midnight Sun'', who made their way to Moscow for the great event but left before the tragedy of Khodynka (pp. 49-82).

 
======K24======
'''Davis, Richard Harding, '''''A year from a correspondent’s note-book.'' New York and London: Harper &amp;amp; Brothers, 1898. x+305pp.
::Journalist, novelist, playwright, and F.R.G.S., Davis (1864-1916), later renowned for his war reporting, describes his visit to Russia for the coronation in May 1896 in an article previously published in ''Harper’s Magazine'', but makes no mention of the Khodynka tragedy (pp. 3-65).

 
======K25======
'''[Maude, Aylmer], '''''The tsar’s coronation as seen by “De Monte Alto” resident in Moscow.'' London and Croydon: Brotherhood Publishing Co., 1896. 128pp.
::A healthy counterblast to accounts that “present nothing but the conventional, superficial laudations of a spectacle which enlightened conscience and sober reason must see in a wholly different light”. Maude (see [[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J86|J86]]), the “resident” concealing his identity, follows the preparations for the event , the pageant itself, and in graphic detail the “catastrophe” at Khodynka on 18 May 1896, before penning an epilogue, suffused with the teachings of Tolstoi and railing against the perils of ultra-patriotism that affects equally the British.

 
======K26======
'''Addams, Jane, '''''Twenty years at Hull-House, with autobiographical notes. ''New York: Macmillan, 1910. 462pp.
::Social activist, founder of the U.S. Settlement House movement and recipient of the Nobel Peace Prize, Addams (1860-1935) travelled with wealthy debutante Mary Rozet Smith to Russia in July 1896. Addams was yet another enthusiast for Tolstoi’s social writings, but her meeting with him proved awkward and confrontational (pp. 266-74).

 
======K27======
'''Simpson, James Young,''' ''[https://archive.org/details/sidelightsonsib01simpgoog Side-lights on Siberia: some account of the great Siberian railroad, the prisons and the exile system]''. Edinburgh and London: William Blackwood and Sons, 1898. xvi+383pp.
::A journey undertaken in the summer of 1896 that took Simpson (1873-1934), who had recently graduated from Edinburgh University, along the Trans-Siberian. Offering not so much a travelogue as an investigation into the potential of Siberia and much about the penal system, Simpson sought “the truth” mid-point between the views of Kennan and De Windt, but predicted that revolution was inevitable.

======K28======
'''Mavor, Sam, '''''Memories of people and places. ''London, Edinburgh and Glasgow: William Hodge and Co., 1940. iv+327pp.
::Russia played a large role in the life of the electrical engineer and industrialist, whose memoirs, gathered together when he was seventy six, comprise mainly articles he had written at various times for his firm’s ''Apprentices magazine.'' Mavor (b. 1863) first went to Russia in the summer of 1896 to inspect the installation of electric lighting at the Thornton woollen mills in St Petersburg (pp. 5-9). He also describes a momentous journey to deliver a Tyne-built steamer from Newcastle to St Petersburg and then by the waterways to Astrakhan (pp. 163-86). In 1899 he made a pilgrimage to Solovetsk from Norway (pp. 187-213). His final visit to St Petersburg was in 1912.

 
======K29======
'''Fraser, John Foster, '''''[https://archive.org/details/cu31924023252707 Round the world on a wheel: being the narrative of a bicycle ride of nineteen thousand two hundred and thirty-seven miles through seventeen countries and across three continents, by John Foster Fraser, S. Edward Lunn, and F.H. Lowe]''. London: Methuen, 1899. xii+532pp.
::Cycling trip by three friends across the world that began on 17 July 1896 and took 774 days was chronicled by Sir John (1868-1936). They entered Russia from Romania in the autumn en route for Odessa and thereafter cycled through the Crimea to the Caucasus and through Georgia to Erevan. Passing Ararat, they left Armenia and entered Persia (pp. 28-90). (See also [[#K88|K88]], [[#K90|K90]], [[#K301|K301]])

======K30======
'''Jefferson, Robert Louis,''''' Roughing it in Siberia; with some account of the Trans-Siberian railway, and the gold-mining industry of Asiatic Russia''. London: Sampson Low, Marston &amp;amp; Co., 1897. 252pp.
::Forsaking for once his bike for the railway, Jefferson (see [[#K7|K7]]) accompanies three business associates from Moscow to Krasnoiarsk, the then terminus of the railway, and then down the Enisei in January-April 1897. Visited gold-mines, interviewed owners and miners, gathered samples of ore.

 
======K31======
'''Demidov, Elim Pavlovich, '''''After wild sheep in the Altai and Mongolia''. London: Rowland Ward, 1900. xii+324pp.
::Demidov (see [[#K9|K9]]) travelled with his wife and the great travellers St. George Littledale (1851-1931) and his wife Teresa (1839-1928) from London in April 1897 to shoot the wild sheep (''ovis ammon''), and much else.

 
======K32======
'''Gillis, Charles J., '''''A summer vacation in Iceland, Norway, Sweden and Russia''. New York: Printed for private distribution, 1898. 55pp.
::Seasoned traveller and author of a number of privately printed travel accounts, Gillis joined a party sailing from New York on 26 June 1897 for Scandinavia and Russia. Very brief notes with photographs on their visit to St Petersburg, Moscow, and then Peterhof, before they sailed off for Southampton towards the end of August (pp. 36-52).

 
======K33======
'''Dana, Charles A., '''''Eastern journeys: some notes of travel in Russia, in the Caucasus, and to Jerusalem'''''. '''New York: D. Appleton and Co., 1898. iv+146pp.
::A three-month round trip from New York in the summer of 1897 that takes the American tourists Dana (1819-97) and his wife first to Odessa by boat from Marseilles and then on to Batumi, through Georgia to Rostov and the railway link to Nizhnii and Moscow (pp. 17-101).

 
======K34======
'''Symons, Arthur William, '''''Cities. ''London: J.M. Dent &amp;amp; Co., 1903. xii+261pp.
::In this collection of essays, printed previously in journals, the prolific literary scholar and author Symons (1865-1945) included ‘Moscow’, which he visited in the hot summer of 1897 and which he “hated”, as much as Naples, in contrast to St Petersburg, which had “nothing to say” to him (pp. 155-85).

 
======K35======
'''Ridley, James Cartmell,''' ''Reminiscences of Russia: the Ural mountains and adjoining Siberian district in 1897. ''Newcastle-upon-Tyne: A. Reid &amp;amp; Co., 1898. 100pp.
::Prior to attending the International Geological Congress, held in St Petersburg in August 1897, Ridley and another unnamed delegate from Newcastle embarked on a long journey that took them by train via Warsaw to Moscow, where they arrived on 27 July. They then travelled by various forms of transport via Samara as far as Ekaterinburg and looped back to Moscow via Perm and Kazan, before proceeding to the capital for the conference. Highly impressed by “a great country”, they sailed for home on 6 September.

 
======K36======
'''Hayes, Matthew Horace,''' ''Among horses in Russia''. London: R.A. Everett &amp;amp; Co., 1900. xiv+214pp.
::Capt Hayes''', '''a leading authority and author on all things equine, paid four visits to Russia. The first in July-September 1897, at the invitation of the imperial guards stationed at Krasnoe selo outside St Petersburg, was quickly followed by a second in October with further horses and a visit to an imperial stud at Dubrovka in Ukraine. Two further visits followed in March and August 1898, during the second of which he was later joined by his wife, a formidable horsewoman. Much valuable information on the famous Orlov stud and Russian horse-breeding.

 
======K37======
'''Honeyman, Abraham Van Doren, and Mason, Abbie Ranlett''', ''From America to Russia in summer of 1897. ''Edited by A.V.D. Honeyman. Plainfield, New Jersey: Honeyman &amp;amp; Co., 1897. 167pp.
::Two chapters in this third collection of travel accounts by members of Honeyman tourist groups from New York and New Jersey are devoted to their visit to Russia in early August 1897: Mason describes their arrival in St Petersburg from Finland, their tour of the city sights, and a visit to the military review at Krasnoe selo (pp. 88-97); Honeyman (1849-1936) recounts the party’s less enjoyable trip to Moscow, where they note the greater level of poverty and social disorder compared to St Petersburg (pp. 98-114).

 
======K38======
'''Miles, Nelson Appleton, '''''Military Europe: a narrative of personal observation and personal experience. ''New York: Doubleday &amp;amp; McClure Co., 1898. x+112pp.
::U.S. major-general (1839-1925), veteran of the Civil War and Indian wars, arrived in Russia on 15 August 1897, was received by the tsar at Peterhof, and observed the annual manoeuvres of the Russian army at Krasnoe selo, before proceeding to Germany and France (pp. 73-94).

======K39======
'''Renshaw, Charles Jeremiah, '''''Travels in Russia''. London: National Union Publishing Co., 1900. 32pp.
::Dr Renshaw of Ashton-on-Mersey travelled by train to Moscow with a party of thirty men and ten women to attend the 12th International Medical Congress in August 1897. He briefly describes their outward journey, their stay in Moscow, their visit to Petersburg and homewards via Finland.

 
======K40======
'''Kerr, John, '''''Leaves from an inspector’s notebook''. London: Thomas Nelson &amp;amp; Sons, [1913]. 278pp.
::Kerr (1830-1916), senior chief inspector of schools in Scotland, travelled to St Petersburg via Sweden and Finland en route for Moscow, where he also attended the Medical Congress that opened on 19 August in the Bolshoi theatre. He also went to the fair at Nizhnii Novgorod (pp. 158-76).

 
======K41======
Entry Omitted

======K42======
'''Flint, Josiah Frederick,''' ''Tramping with tramps: studies and sketches of vagabond life''. With prefatory note by Hon. Andrew D. White. New York: The Century Co., 1899. xvi+398pp.
::In part II of his book in which he details his experiences with tramps in various parts of the world, Flint (b. 1850) includes ‘With the Russian goriouns [unfortunates]’, describing his visit to Russia in 1897, when he not only went to see Tolstoi but also “tramped” for some days in Vitebsk district (pp. 200-28).

 
======K43======
'''Perowne, John Thomas Woolrych, '''''Russian hosts and English guests in Central Asia.'' London: The Scientific Press, 1898. xvi+198pp.
::Perowne, Cambridge graduate and translator from French and German, does “nothing more than describe a journey, made in November and December last, over the Transcaspian Military Railway”. One of a party of twenty-five English who left Constantinople on 6 November 1897, Perowne, styling himself “something of a Russophil”, charts their progress to Batumi and on to Tiflis and Baku, where they take the steamer to Krasnovodsk. There they board the Transcaspian railway for a three-week journey to Samarkand with various stops en route, including a reception by Kuropatkin, the governor-general, on their return to Ashabad.

 
======K44======
'''Phibbs, Isabelle Mary,''' ''A visit to the Russians in Central Asia''. London: Kegan Paul, Trench, Trübner &amp;amp; Co., Ltd., 1899. viii+238pp.
::Travel writer in the same party as Perowne, although neither names the other, offers her account of the “marvellously interesting journey”, which for her was marred only by a bout of influenza that made the return by train “almost entirely a blank”.

 
======K45======
'''Loch, Emily, '''''The memoirs of Emily Loch: discretion in waiting, Tsarina Alexandra and the Christian family''. Edited by Judith Poore. Kinloss: Librario, 2007. 394pp.
::Emily (d. 1932), lady-in-waiting to Helena, Princess Christian, accompanied Princess Helena, Princess Christian’s eldest daughter, on visit to the Russian imperial family during the winter of 1897-98 (pp. 183-252).

 
======K46======
'''Cobbold, Ralph Patteson, '''''Innermost Asia: travel &amp;amp; sport in the Pamirs. ''London: William Heinemann, 1900. xviii+354pp.
::With the modest aim of making his book “the standard work of reference on its subject”, Cobbold charts his travels and hunting through the Pamirs in 1897-98. Coming from the Chinese side, he crossed the Russian border at the beginning of January 1898 and travelled to lake Balkash, where he shot his first tiger. He returned to Kashgar before receiving permission to travel in the Russian Pamirs. At one stage detained as a spy, Cobbold is ultimately glad to make his way to “freedom” in British Kashmir (pp. 92-210).


======K46a======
'''Salter, John Henry,''' ''Dr Salter of Tolleshunt D’Arcy in the county of Essex, medical man, freemason, sportsman, sporting-dog breeder and horticulturalist. His diary and reminiscences from the year 1849 to the year 1932.'' Compiled by J.O. Thompson with an appreciation by the rt. hon. the earl of Lonsdale, K.G.  London: John Lane, 1933. xviii+404pp.
::It was his love of sporting dogs that brought Salter (1841-1932), vice-president of the Kennel Club, to Russia for the first time in January 1898, where he was a judge at the imperial dog show in Moscow. He was to make eight further trips up to 1912, all meticulously recorded in the diaries he kept for some eighty years (pp. 107-11, 115-18, 120-2, 125-6, 129-31, 142-3, 149-50). In addition to the diary entries there are ten ‘reminiscences’, sympathetic essays on various aspects of his Russian visits, concerned mainly with the judging of dogs, the hunting of wolves, and the shooting  of birds  (pp. 338-69).


======K47======
'''Colquhoun, Archibald Ross, '''''The ‘overland’ to China.'' London: Harper &amp;amp; Brothers, 1900. xii+465pp.
::Much travelled in China over the previous twenty years, Colquhoun, F.R.G.S., was convinced of the importance of the nearly completed Trans-Siberian railway beyond the boundaries of Russia itself and undertook a journey from St Petersburg in late 1898 that took him to the temporary terminus of the railway at Baikal before travelling through Mongolia to Pekin. An attempt at a serious historico-geographical assessment, based on his own observations and “original sources” (pp. 1-149).

 
======K48======
'''Jefferson, Robert Louis,''''' A new ride to Khiva''. London: Methuen &amp;amp; Co., 1899. xii+352pp.
::Intent on replicating Burnaby’s famous 1875-76 ride to Central Asia but purely for sporting reasons, Jefferson (see [[#K7|K7]], [[#K30|K30]]) set out from England on the six-thousand mile trek in April 1898, entering Russia via Galicia. He made his way to Moscow and accompanied by Russian cycling friends, went to Nizhnii Novgorod and followed the Volga and on across the Kirghiz steppe until his reception by the khan of Khiva. He returned to England by ship via Constantinople and by train from Marseille (pp. 71-312).

 
======K49======
'''Vanderlip, Washington Baker,''' ''In search'' ''of a Siberian Klondike. As told to Homer B. Hulbert.'' London: T. Fisher Unwin, 1906. 315pp.
::American prospector Vanderlip (1863-1949) seached for gold and copper deposits in Siberia, Kamchatka and Sakhalin in 1898-89 and again in 1900. He describes wildlife and the indigenous peoples and narrates his encounters with rival prospectors and foreign adventurers.

 
======K50======
'''Stadling, Jonas Jonsson,''' ''[https://archive.org/details/throughsiberia00guilgoog Through Siberia].'' Edited by F.H.H. Guillemard. London: Archibald Constable &amp;amp; Co., 1901. xvi+315pp.
::Stadling (see [[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J113|J113]]) set out from Stockholm on 20 April 1898 on an unsuccessful expedition to find the Swedish explorer Andrée lost in northern Siberia. They travelled deep into Siberia, reaching the Enisei, before turning back, arriving in Stockholm (after another visit to Iasnaia Poliana) at the end of December after a journey of 15,500 miles.

======K51======
'''Bookwalter, John Wesley, '''''Siberia and Central Asia.'' New York: Frederick A. Stokes, 1899. xxxi+548pp.
::The American manufacturer (1837-1915) records in detail and in numerous photographs his extensive travels throughout Siberia, the Caucasus, and Central Asia during the summer and autumn of 1898. He showed particular interest in the transport infrastructure in the regions he visited as well as in their industries and commercial potential.

 
======K52======
'''Russell-Cotes, Annie Nelson, '''''Letters from Russia''. London: for the author, 1899. iv+50pp.
::Undemanding tourist’s letters sent by Lady Russell-Cotes, née Clark, to her daughter E., beginning in St Petersburg on 28 August 1898 and ending in Göteborg on 22 September. From the capital they journeyed to Moscow and then to Nizhnii Novgorod, returning to Moscow and back to Petersburg (pp. 1-39).

 
======K53======
'''Reid, Arnot, '''''From Peking to Petersburg. ''London: Edward Arnold, 1899. viii+300pp.
::An unremarkable account of a journey on the Trans-Siberian by a traveller, an “average indoors-man”, who having arrived in Singapore, wanted to travel home to England by a different route. Entered Siberia at Kiakhta on 21 September 1898 and reached St Petersburg exactly fifty days after leaving Pekin (pp. 131-274).

 
======K54======
'''Leroy-Beaulieu, Pierre Paul,''' ''The awakening of the East: Siberia—Japan—China''. Translated from the French by Richard Davey. With a preface by [Sir] Henry Norman. William Heinemann, 1900. xxvii+298pp.
::French economist (1843-1916), after extensive travels in 1898-99 through Siberia, Japan and China, writes a comparative study, drawing particular attention to the significance of the almost completed Trans-Siberian railway (pp. 1-80).

 
======K55======
'''Hammond, John Hays''','' The autobiography of John Hays Hammond''. New York: Farrar &amp;amp; Rinehart, 1935. 2 vols.
::American mining engineer and diplomat (1885-1936) records his experiences of three trading missions to Russia between 1898 and 1912. In 1898, invited by the Minister of Finance Witte and accompanied by the English financier L. Hirsch, he surveyed mines and mineral resources in the Ural and Altai Mountains. He went again in 1910, meeting the tsar at Tsarskoe selo and negotiating an agreement over the use of American capital to finance Russian industries (vol. II, pp. 454-78). During a third visit to St Petersburg and Moscow in mid-May 1912 Hammond held talks with the prime minister and Witte (pp. 603-06).

 
======K56======
'''Oudendyk, William J., '''''Ways and by-ways in diplomacy.'' London: Peter Davies, 1939. xii+386pp.
::The Dutch diplomat Oudendyk (b. 1874) received his first posting to the Dutch legation in China at the age of nineteen and spent much of his first long spell of leave from late 1898 to early 1900 in Moscow, where he learnt Russian. He visited Kiev and St Petersburg before taking the Trans-Siberian to Irkutsk and then proceeding to Vladivostok (pp. 78-92). In the summer of 1907 he arrived in St Petersburg as head of the Dutch legation, remaining until May 1908, when he was again appointed to China (pp. 133-47). After further service in China and Persia, whence he visited Russian Turkestan (pp. 183-90), he was in Russia at least three times in 1914-16 (pp. 201-10), before returning to Petrograd as temporary minister in the spring of 1917 and remaining until the end of 1918 (pp. 212-312).

 
======K57======
'''Beeby-Thompson, Arthur, '''''Oil pioneer: selected experiences and incidents associated with sixty years of world-wide petroleum exploration and oilfield development.'' With a foreword by Herbert Hoover. London: Sidgwick and Jackson, 1961. 544pp.
::Newly employed by the European Petroleum Company, Beeby-Thompson (b. 1872) travelled out to Baku in November 1898. He combines technical data with his impressions of life in the area, where he remained until 1903. He returned to London to begin his career as a consultant oil engineer, published in 1904 a solid tome entitled ''The oil fields of Russia and the Russian petroleum industry'', and was to pay one further short visit to the oilfields in 1905, when he witnessed violence between Armenians and Tartars (pp. 52-76).

 
======K58======
'''Pares, Bernard, '''''My Russian memoirs. ''London: Jonathan Cape, 1931. 623pp.
::Sir Bernard (1867-1949), the pioneer of Russian studies at the universities of Liverpool and London, first visited Russia in 1898-99, when he attended lectures at Moscow University, and thereafter visited the country in many guises and on many occasions through WWI and finally left from Siberia in 1919. (See also [[#K59|K59]], [[#K123|K123]], [[#K124|K124]], [[)#K309|K309]].)

 
======K59======
'''Pares, Bernard, '''''A wandering student: the story of a purpose. ''Utica, New York: Syracuse University Press, 1948. xv+448pp.
::In this version of his memoirs, completed in 1947, Pares includes and reworks material on Russia included in his earlier books.

 
======K60======
'''Eagar, M., '''''Six years at the Russian court. ''London: Hurst and Blackett, 1906. xvi+283pp.
::Irish Miss Eagar arrived in Petersburg in February 1899 as governess to the infant Grand Duchesses Olga and Tatiana and remained with the imperial family until the end of 1904. Her “slight sketches of life in the Palaces”, offered as “plain, unvarnished truth”, are naïve and undemanding but nonetheless provide intimate and unusual glimpses of the family in its Petersburg palaces and on holiday in the Crimea.

 
======K61======
'''Ossendowski, Ferdynand Antoni, '''''Man and mystery in Asia. ''In collaboration with Lewis Stanton Palen. London: Edward Arnold &amp;amp; Co., 1924. xii+295pp.
::The first forty years of the colourful and adventurous life of the Pole Ossendowski (1878-1945) were inextricably bound with the fortunes of Russia from his birth in Russian Poland, through his education in Petersburg, extensive travels throughout the empire, political activity, imprisonment, and much else. In a first volume ''Beasts, men and gods'' (1923), he wrote of his escape from the Bolsheviks into Mongolia; here he returns to his earlier adventures in Siberia as far as Sakhalin from about 1899.

 
======K62======
'''Ossendowski, Ferdynand Antoni,''' ''The shadow of the gloomy East. ''Translated [from the Polish] by F.B. Czarnomski. London: George Allen &amp;amp; Unwin, 1925. 223pp.
::His self-styled “sketches” represent an attempt “to lay bare before the civilised world the true face of the Russian people”, interesting above all for his personal memories, his three meetings with Rasputin, and service with Kolchak.

 
======K63======
'''Morton, Rosalie Slaughter, '''''A woman surgeon: the life and work of Rosalie Slaughter Morton. ''London: Robert Hale &amp;amp; Co., 1937. 355pp.
::Shortly after graduating from medical school, Dr Morton (1876-1968), who was to become one of the most distinguished female doctors in America, went abroad to continue her studies in Germany. She spent the Christmas vacation of 1899 in St Petersburg, appalled by the contrasts of wealth and degrading poverty and with hindsight concluding that “here was Red Russia in the making, her garments dyed in blood”. She then travelled to Moscow, where she visited Tolstoi on three occasions at his home in Khamovniki, recording in detail their conversations (pp. 75-86).

 
======K64======
'''Müller, Max, '''''Reminiscences of a roving life'''''. '''Exeter: William Pollard &amp;amp; Co., 1906. xii+125pp.
::Müller, not the famous Prof. Müller of Oxford University as he points out in his preface, recounts “the life of a wanderer – a vagabond – and nothing more”. He made many cruises on the s.y. ''Argonaut'', one of which that took him in 1899 to St Petersburg, whence he travelled to Moscow and Nizhnii Novgorod, and a later one to the Crimea, but he says little of any interest (pp. 118-23).

 
======K65======
'''Pearson, Henry J., '''''Three summers among the birds of Russian Lapland, with history of Saint Triphon’s monastery and appendices.'' London: R.H. Porter, 1904. xvi+216pp.
::In his second book, Pearson (see [[#K8|K8]]) describes three visits to Russian Lapland in 1899, 1901, and 1901 to observe, record, and photograph bird life (pp. 1-169). There follows a translated version of a Russian history of the monastery by the Pechenga river.

 
======K66======
'''Curtin, Jeremiah, '''''A journey in southern Siberia: the Mongols, their religion and their myths.'' Boston: Little, Brown, and Co., 1909. xiv+319p.
::Curtin (see [[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander II (1855-1881)#I51|I51]]) journeyed from Moscow into southern Siberia to study the language, religion, and customs of the Buriats, living around Lake Baikal and on the only island within it. The account of his travels, from his arrival in Irkutsk on 9 July 1900 to his departure on 15 September, occupies pp. 18-91.

 
======K67======
'''Hill, Elizabeth, '''''In the mind’s eye:'' ''the memoirs of Dame Elizabeth Hill.'' Edited by Jean Stafford Smith. Lewes: The Book Guild Ltd., 1999. viii+520pp.
::Dame Elizabeth (1900-96), first professor of Slavonic studies at the University of Cambridge from 1948 to 1968, recalls her upbringing in St Petersburg in an Anglo-German family before their departure for England at the end of 1917 (pp. 3-55).

 
======K68======
'''Kenworthy, John Coleman,''' ''Tolstoy: his life and works. ''London and Newcastle-on-Tyne: Walter Scott Publishing Co., 1902. 255pp'''.'''
::In 1900 Kenworthy (see [[#K11|K11]]) went to Russia for a second time to see Tolstoi at Prince Obolenskii’s country house outside Moscow for five days before proceeding alone to Iasnaia Poliana to absorb the atmosphere (pp. 210-28). Also included is the previously published description of his 1895 visit (pp. 47-98).

 
======K69======
'''Demidov, Elim Pavlovich,''' ''A shooting trip to Kamchatka.'' London: Rowland Ward, 1904. xvi+304pp.
::Travelling again with his fellow passionate sportsman St George Littledale, Demidov (see [[#K9|K9]] and [[#K31|K31]]) completed the round trip from London across Siberia to Vladivostok by train and by steamer to Kamchatka in April-September 1900, shooting whatever moved whenever the opportunity arose.

 
======K70======
'''Clark, Francis Edward, '''''A new way around an old world. ''New York &amp;amp; London: Harper &amp;amp; Brothers, 1901. xiv+212pp. [also published as ''The Great Siberian Railway: what I saw on my journey.'' London: S.W. Partridge and Co., 1904.]
::President of the United Society of Christian Endeavor and the World’s Christian Endeavor Union, Dr Clark (1851-1927), his wife, and son reached Vladivostok from Japan on 31 May 1900 and began the six-week journey by the newly opened Trans-Siberian and by steamer to Moscow and home via St Petersburg – the first Americans “to go around the world by the new route”.

 
======K71======
'''Clark, Francis Edward, '''''Memories of many men in many lands.'' Boston: United Society of Christian Endeavor, 1922. 704pp.
::Succinct description of his 1900 journey across Siberia to Moscow (pp. 239-56).

 
======K72======
'''Roberts, James Hudson, '''''A flight for life and an inside view of Mongolia. ''Boston: Pilgrim, 1903. 402pp.
::A missionary for the American Board of Commissioners for Foreign Missions based in Tientsin, Roberts (1851-1945) describes his and his mission’s escape from China during the Boxer Rebellion in the summer of 1900. Travelling through Mongolia, he arrives in Russia at Kiakhta on 13 August and makes his way to Irkutsk, where he boards the Trans-Siberian railway. He reaches St Petersburg in mid-September (pp. 279-331).

 
======K73======
'''Meakin, Annette M.B., '''''A ribbon of iron. ''London: Archibald Constable &amp;amp; Co., 1901. 320pp.
::Anthropologist, biographer, translator, Meakin (1867-1959), F.R.G.S., travelled with her mother along the Trans-Siberian railway from Moscow to Vladivostok in 1900.

 
======K74======
'''Benn, Edith Annie Fraser Parker, '''''An overland trek from India by side-saddle, camel, and rail; the record of a journey from Baluchistan to Europe''. London: Longmans, Green and Co., 1909. 343pp.
::The Russian stage of her journey took Mrs Benn through the Caucasus mountains and on to Georgia during late 1900 and early 1901 (pp. 274-96).

 
======K75======
'''Thwing, Charles Franklin, '''''Universities of the world. ''New York: Macmillan Co., 1911. xvi+284pp.
::The president of Western Reserve University (1853-1937) visited over “years not a few” nineteen of the twenty universities he featured in a series of essays, including the university of St Petersburg. The year of his visit is not specified but would seem to be from the turn of the century (pp. 167-78).

 
======K76======
'''Norman, Henry, '''''All the'' ''Russias: travels and studies in contemporary European Russia, Finland, Siberia, the Caucasus, &amp;amp; central Asia'': London: William Heinemann, 1902. xvi+476pp.
::Norman (see [[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J95|J95]]), recently elected to Parliament, nevertheless undertook four journeys throughout the Russia empire in 1900-01 “to present a picture of the aspects of contemporary Russia of most interest to foreign readers”, including the almost obligatory visit to Tolstoi at Iasnaia Poliana (pp. 47-63).

 
======K77======
'''Holmes, Burton,''' ''The Burton Holmes lectures. With illustrations from photographs by the author. ''Battle Creek, Michigan: The Little-Preston Company, 1901. 10 vols.
::The famous and successful American traveller and innovative travel lecturer (1870-1958) visited St Petersburg for the first time in April 1901 and then moved on to Moscow in May, taking a trip to Iasnaia Poliana, where he met and filmed Tolstoi. On 19 June he began his nine-day journey on the Trans-Siberian to Irkutsk. Crossing Baikal by ferry, he proceeded by train to the Cossack settlement of Stretensk, arriving in early July (vol. VIII, pp. 1-336). He then undertook a hazardous and seemingly endless journey along the Amur from Stretensk to Khabarovsk, before finally reaching Vladivostok (vol. IX, pp. 3-112). In his later guide to the Soviet Union, ''The traveler’s Russia'' (1934), Holmes was to recall his earlier visits, particularly the meeting with Tolstoi.

 
======K78======
'''Beveridge, Albert Jeremiah, '''''The Russian advance. ''New York and London: Harper &amp;amp; Brothers, 1904. ix+486pp.
::American historian and senator for Indiana, Beveridge (1862-1927) travelled through Russian and Siberia in 1901, intent on assessing the changing role of Russia in international politics. Much of his material was published in the Philadelphia ''Saturday Evening Post'' in the same year.

 
======K79======
'''Baring, Maurice, '''''The puppet show of memory.'' London: Heinemann, 1922. ix+457pp.
::Baring (1874-1945), poet, dramatist, novelist, and translator with a profound love of Russia and its people, went to Russia for the first time in July 1901 as guest of Count Konstantin Benkendorf at his Tambov estate of Sosnovka (pp. 219-24). Many further visits, as family guest and as newspaper correspondent, ensued, all described in his autobiography, which covers the period up the outbreak of WWI (pp. 260-390). (See also [[#K134|K134]], [[#K146|K146]], [[#K174|K174]]-[[#K175|75]], [[#K231|K231]], [[#K274|K274]].)

 
======K80======
'''Morgan, Christopher A., '''''From China by rail: an account of a journey from Shanghai to London via the Trans-Siberian Railways.'' Edinburgh: Ballantyne, Hanson and Co., 1902. 139pp.
::British tourist describes his journey on the Trans-Siberian between 19 June and 20 August 1901, concentrating on the section between Vladivostok and Irkutsk.

 
======K81======
'''Senn, Nicholas, '''''Around the world via Siberia. ''Chicago: W.B. Conkey Co., 1902. 402pp.
::Articles originally published in the ''Chicago Tribune'' by the professor of surgery at Rush Medical College and surgeon-general of Illinois (1844-1908), some of which describe his journey from St Petersburg to Vladivostok between 19 July and 25 August 1901 (pp. 36-198).

 
======K82======
'''Landor, Arnold Henry Savage, '''''Across coveted lands; or, a journey from Flushing (Holland) to Calcutta, overland.'' London: Macmillan, 1902. 2 vols.
::Grandson of the poet, Landor (1865-1924), painter, traveller, and author, travelled from Warsaw to Kiev in 1901 and then on to Rostov and Baku, where he embarked on the mail steamer for Persia (vol. I, pp. 1-28).

 
======K83======
'''Meakin, Annette M.B., '''''In Russian Turkestan: a garden of Asia and its people. ''London: George Allen, 1903. 304pp.
::Meakin’s (see [[#K73|K73]]) journey through Russian central Asia in 1901 before her return to England in March 1902 takes her from Krasnovodsk on the Caspian via Askhabad and Merv to Bokhara (which she had first visited in 1896), Samarkand and Tashkent.

 
======K84======
'''Hawes, Charles Henry, '''''[https://archive.org/details/cu31924023036258 In the uttermost East: being an account of investigations among the natives and Russian convicts of the island of Sakhalin, with notes of travel in Korea, Siberia, and Manchuria]. ''London: Harper, 1903. xxviii+478pp.
::British anthropologist and associate director of Boston museum of fine art, Hawes (1867-1943) sailed from Japan to Vladivostok in August 1901, travelled to Khabarovsk, then went by boat along the Amur to Nikolaevsk before crossing to Sakhalin, “the island of punishment”, where he observed the lives of both native tribes and imperial prisoners. After fifty adventure-filled days on the island, he returned by boat to Vladivostok and thence by train to Moscow and on to London, which he reached at the end of the year (pp. 15-464). Hawes was later to write the ''Handbooks ''on Eastern Siberia and on Sakhalin for the Foreign Office and published in 1920.

======K85======
'''Gerrare, Wirt, [pseudonym of Greener, William Oliver], '''''Greater Russia. The continental empire of the old world. ''London: Heinemann, 1903. 310pp.
::Greener (1862-1935), author of ''The story of Moscow'' (1900), uses two journeys east and west to see “greater” Russia, i.e. particularly the areas east of Baikal and Russia’s “port-hole on to the Pacific”, and also slips in disguise into Manchuria. (See also [[#K136|K136]].)

 
======K86======
'''Adams, Henry,''' ''The education of Henry Adams: an autobiography''. Cambridge, Mass.: Riverside Press, 1918. 519pp.
::Adams (1838-1918), American journalist and member of the famous Adams political family, visited St Petersburg and Moscow between 17 August and 7 September 1901 (pp. 406-10).

 
======K87======
'''Adams, Henry, '''''Letters of Henry Adams, 1892-1918. ''Edited by Worthington Chauncey Ford. Boston: Houghton Mifflin, 1938. 2 vols.
::Letters Adams sent to his friends Elizabeth Cameron and John Hay, describing his trip to Russia in 1901 (vol. II, pp. 339-50).

 
======K88======
'''Fraser, John Foster, '''''[https://archive.org/details/realsiberiatoge02frasgoog The real Siberia; together with an account of a dash through Manchuria].'' London: Cassell &amp;amp; Co., 1902. 279pp.
::“A mission of curiosity” at the behest of the owner of the ''Yorkshire Post'' saw Fraser (see [[#K29|K29]]) leave Moscow on 22 August 1901 by train, but not the Trans-Siberian. He travelled as far as Stretensk, from where he sailed by steamer down the Shilka and Amur rivers to Blagoveshchensk and then on to Khabarovsk. After his “dash” into Manchuria (pp. 208-55), he returned to Irkutsk at the end of October and boarded the Trans-Siberian for Moscow.

======K89======
'''Palmer, Frederick, '''''With my own eyes: a personal story of battle years''. London: Jarrolds, 1934. 350p.
::The experienced American war correspondent (1873-1958) travelled with senators Beveridge and Cabot Lodge from Tokyo to Moscow on the Trans-Siberian en route for London in 1901, noting in particular the threat the railway brought to Japan and China (pp. 188-93). Palmer, very sympathetic to the Japanese cause, was to publish in 1904 ''With Kuroki in Manchuria''.

 
======K90======
'''Fraser, John Foster, '''''[https://archive.org/details/realsiberiatoge02frasgoog Life’s contrasts]. ''London: Cassel &amp;amp; Co., 1908. 339pp.
::There was inevitably much in Fraser’s earlier works ([[(title of the correspondent chapter)#K29|K29]], [[(title of the correspondent chapter)#K87|K87]]) about convicts and exiles in Siberia and in this book of essays he recounts an encounter at Irkutsk with a political prisoner ‘Ivan Ivanovitch’, returning to his native Moscow after fifteen years’ exile (pp. 33-57). In another sketch on ‘the cloaks of religion’ he recalls his visit to Kiev and its cathedral (pp. 76-82).

======K91======
'''Morgan, Wilma, '''''Glimpses of four continents, being an account of the travels of Richard Cope Morgan''. London: Morgan &amp;amp; Scott, 1911. xii+388pp.
::As the “constant companion during the last eleven years” of her husband (1827-1908), founder of ''The Christian'', Mrs Morgan was able to supplement extracts from his diaries and writings with her own personal observations. Unlike Morgan, who was exclusively concerned with visiting missions, preaching, and writing, she remarks on the places and countryside they saw. In September 1901 they were persuaded by Dr Baedeker (see [[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander II (1855-1881)#I143|I143]]) to accompany him to Russia. They went first to Rostov-on-Don and travelled through Georgia and on to Baku, Batumi, and the Crimea, visiting prison and meeting believers of various sects (pp. 123-36).

 
======K92======
'''De Windt, Harry, '''''[http://www.gutenberg.org/ebooks/26007 From Paris to New York by land]. ''London: George Newnes, 1904. 311pp.
::After failing in 1896 to travel by land from New York to Paris, De Windt succeeded in the opposite direction, leaving Paris on 19 December 1901 and reaching New York on 25 August 1902. The primary aim of the expedition was to assess the possibility of a Paris-New York rail link, but De Windt’s three-man expedition travelled by rail, sled and foot to reach the Bering Strait at the end of April 1902 (pp. 1-193).

======K93======
'''Swenson, Olaf, '''''Northwest of the world: forty years’ trading and hunting in Northern Siberia.'' London: Robert Hale, 1951. 221pp.
::Swenson (1883-1938), born in Michigan to Swedish parents, sailed from Alaska to Siberia for the first time in 1902 with a group of prospectors from the North-eastern Siberian Company and made many further trips as a trader from 1905 through the 1930s. He came to know and admire the native Chukchi.


======K93a======
'''Macdonell, Aeneas Ranald,''' ''…And nothing long.'' London: Constable &amp; Co., 1938. vii+328pp.
::In 1902 Macdonell (1875-1941), by then 21st chief of Clan Ronald of Knoydart and Glengarry, accepted a position with an unidentified British company in Russia and after a few months in Moscow arrived in turbulent Baku, where he was later to become British vice-consul and remain during the years of revolution and civil war (pp. 61- 325). Employed by the Foreign Office, he was awarded the OBE in 1919 and CBE the following year for his services in often dangerous circumstances . On his return to England he became a journalist and later a grocer in Swanage. His memoirs, penned from memory many years after the events, contained, according to his eldest son, many fascinating stories based on other people’s experiences rather than his own. 


======K94======
'''Shoemaker, Michael Myers,''' ''The great Siberian railway from St. Petersburg to Pekin''. New York and London: G.P. Putnam’s Sons (The Knickerbocker Press). 1903. x+243pp.
::The Kentucky traveller and author Shoemaker (1853-1924) left St Petersburg in April 1902 to travel the Trans-Siberian as far as Chita (pp. 1-121), before crossing into Manchuria and on to Pekin. His aim was to present a description of the railway and the country it passes through, without worrying about prisons or politics.

 
======K95======
'''Cary, Clarence, '''''The Trans-Siberian route or notes of a journey from Pekin to New York in 1902''. New York: Evening Post Job Printing House, 1902. 53pp.
::American journalist Cary provides a discursive commentary on a journey taken along the Chinese Eastern and then Trans-Siberian railways between 5 August and 18 August 1902, detailing his own experiences and offering tips for future travellers.

 
======K96======
'''Lynch, George, '''''The path of empire. ''London: Duckworth &amp;amp; Co. 1903. xx+257pp.
::Journalist and explorer Lynch (1868-1928) evaluates the potential political and economic impact of the Trans-Siberian following a journey along it during late 1902. Two early chapters describe his travels around the commercial terminus at Dalnii and the military terminus at Port Arthur (pp. 50-74). After discussing his travels in China, he returns to his westwards journey on the railway to Moscow (pp. 152-257).

 
======K97======
'''Edwards, William Seymour, '''''Through Scandinavia to Moscow. ''Cincinnati: Robert Clarke Co., 1906. xiv+237pp.
::The Virginia lawyer Edwards (1856-1915) and his wife visited Russia as part of their honeymoon trip, arriving in St Petersburg from Stockholm early in September 1902. In a series of letters to his father, beginning on 16 September, he describes their short stays in the capital and Moscow and their exit via Smolensk on 21 September. He was highly aware of the vast abyss between rich and poor, predicting “a saturnalia of blood and tears, a squaring of ten centuries’ accounts, more fraught with human anguish and human joy than ever dreamed a Marat and a Robespierre” (pp. 136-213).

 
======K98======
'''Polhill[-Turner], Arthur Twistleton, '''''Across Siberia with a baby, &amp;amp; a visit to a Chinese prison.'' Edited with a preface by Robert Skinner, D. D. Cambridge: Deighton, Bell &amp;amp; Co., 1904. xii+84pp.
::Rev. Polhill (1863-1935) was one of the “Cambridge Seven” missionaries who worked for the China Island Mission from 1885 until he fled during the Boxer Rebellion in 1900. In the autumn of 1902 he returned to China, travelling on the Trans-Siberian Railway from Moscow on 5 October and departing from Port Arthur on 29 October. His account is based on a series of letters he sent to his brother and fellow missionary Cecil (pp. 1-54).

 
======K99======
'''Shoemaker, Michael Myers,''' ''The heart of the Orient: saunterings through Georgia, Armenia, Persia, Turkomania, and Turkestan, to the vale of Paradise. ''New York and London: G.P. Putnam’s Sons (The Knickerbocker Press). 1904. xiv+416pp.
::In the winter of 1902 Shoemaker (see [[#K94|K94]]) travelled from Constantinople to Georgia and on to Baku, before reaching Persia (pp. 14-89). Leaving Persia once more for Baku, he then travels through Russian Central Asia, visiting Bokhara and Samarkand (from where he travels by ''tarantas''). Returning finally to Baku, he takes the train to Moscow and on to St Petersburg (pp. 211-409).

 
======K100======
'''Miles, Nelson Appleton, '''''Serving the republic: memoirs of the civil and military life of Nelson. A. Miles. ''New York and London: Harper and Bros., 1911. x+340pp.
::Shortly before his retirement from the army, Miles (see [[#K38|K38]]) by then a lieutenant-general, paid an official visit to China and Japan at the end of 1902, visiting also Port Arthur, where he met General Alekseev, and travelling on the Trans-Siberian on his way home via Paris and London (pp. 308-09).

 
======K101======
'''Vay de Vaya and Luskod, Peter,''' ''Empire and emperors of Russia, China, Korea, and Japan: notes and recollections''. Preface by John Murray. London: John Murray, 1906. xxxii+399pp.
::Hungarian aristocrat and later bishop, Count Vay (1863-1948) was sent by Pope Leo XIII to investigate Catholic institutions in the east and was received by the tsar and tsaritsa at Peterhof, prior to his departure from St Petersburg for Siberia, Manchuria and beyond in 1902 (pp. 1-62).

 
======K102======
'''Turner, Samuel,''' ''Siberia: a record of travel, climbing, and exploration. ''With an introduction by Baron Heyking. London: T. Fisher Unwin, 1905. 361pp.
::Turner (1869-1929), F.R.G.S., went to Siberia to assess the dairy industry, but then, an expert climber, he explored the Altai mountains and climbed Mount Belukha during a visit lasting from March to May 1903.

 
======K103======
'''Turner, Samuel, '''''My climbing adventures in four continents. ''London: T. Fisher Unwin, 1911. 283pp.
::Turner recapitulates his climbing expedition of 1903, but now including “quite a lot of details” previously omitted (pp. 91-166).

 
======K104======
'''Overton, Kathleen (Toni), '''''An odious child? memories 1903-1932. ''Edited by Catherine Archer. Hertford: privately printed, 2000. ii+75pp.
::Mrs Overton, née Ward (1903-98) was born to British parents in St Petersburg, where her father was an engineer. She describes her childhood years in the Russian capital which they quitted in October 1916 during WWI to move to England (pp. 3-19).

 
======K105======
'''Fell, Edward Nelson, '''''Russian and nomad: tales of the Kirghiz steppes.'' London: Duckworth &amp;amp; Co., 1916. xviii+201pp.
::American director of a London mining company that acquired coal and copper mines in the midst of the steppes near the headwaters of the river Ishim, Fell (b. 1857) spent several years between 1903 and 1908 working and mixing on friendly terms with Russian and Kirghiz. One ‘tale’ and the concluding ‘Eagle’s song’ were written by his young daughter '''Marian''' (pp. 155-69, 200-01).

 
======K106======
'''Ronaldshay, Dundas, Lawrence John Lumley, Earl of, '''''On the outskirts of empire in Asia.'' Edinburgh and London: William Blackwood and Sons, 1905. xxii+408pp.
::Ronaldshay (1876-1961), later 2nd Marquess of Zetland, F.R.G.S., politician, and author, was aide-de-camp to Lord Curzon, viceroy of India, during the years he travelled extensively in Asia. In April 1903 his travels took him from Persia to Baku and then to Krasnovodsk and the Transcaspian railway by which he went to Bokhara and Samarkand. He travelled across Turkestan to shoot wild sheep in the Altai. He completed the Russian part of his travels on the Trans-Siberian and by steamer eastwards towards Kharbin, which he reached at the beginning of October (pp. 155-315).

 
======K107======
'''Ronaldshay, Dundas, Lawrence John Lumley, Earl of, '''''An eastern miscellany. ''Edinburgh and London: William Blackwood and Sons, 1911. xiv+422pp.
::A collection of essays and speeches, mostly already published and concerned principally with India and Japan, contains ‘A Siberian mystery’, on a visit to Tomsk and the legend of Alexander I/Fedor Kuzmich (pp. 36-44), an essay on Baku in 1905 (pp. 73-88), and ‘Notes on a journey across Asia’, partly in Russia (pp. 143-63).

 
======K108======
'''Swayne, Harold George Carlos, '''''Through the highlands of Siberia''. London: Rowland Ward, 1904. xiv+259pp.
::A major in the Royal Engineers, F.R.G.S., and inveterate hunter and photographer, Swayne (b. 1860) used three months of a year’s furlough from service in India to travel in June 1903 to St Petersburg and Moscow with his wife and then on with his companion Seton Karr to Siberia and the Altai mountains to shoot wild rams.

 
======K109======
'''Pumpelly, Raphael, Davis, William Morris, Pumpelly, Raphael Welles, and Huntington, Ellsworth, '''''Explorations in Turkestan with an account of the basin of Eastern Persia and Sistan: expedition of 1903, under the direction of Raphael Pumpelly''. Washington, D.C.: Carnegie Institution of Washington, 1905. xii+324pp.
::The volume consists of five contributions from the four members of the Carnegie-sponsored expedition that left America in the spring of 1903 with the aim of investigating “the past and present physico-geographical conditions and archaelogical remains” of Turkestan. The members pursued somewhat different itineraries and it is ‘A journey across Turkestan’, the extensive contribution of Davis (1850-1934), professor of geology at Harvard, whose travels extended into Siberia, that is of general interest (pp. 23-119).

 
======K110======
'''Cockerell, Sydney Carlyle, '''''Friends of a lifetime: letters to Sydney Carlyle Cockerell.'' Edited by Viola Meynell. London: Jonathan Cape, 1940. 384pp.
::In July 1903 Sir Sydney (1867-1962), then a process engraver in partnership with Emery Walker and later director of the Fitzwilliam Museum in Cambridge, paid a visit with two American friends to Tolstoi at Iasnaia Poliana (pp. 78-86).

 
======K111======
'''Thomas, Joseph B., Jr., '''''Observations on borzoi called in America Russian wolfhounds, in a series of letters to a friend. ''Foreword by Henry T. Allen. Boston and New York: Houghton Mifflin Co., 1912. viii+123pp.
::All one needs to know about the borzoi, as communicated in letters to Major Allen, who had been American military attaché in St Petersburg. In letters 4 and 5 Thomas, Boston architect and financier and wolfhound fanatic and breeder, recalls his visits to St Petersburg, Tula and Moscow in August 1903 and in 1904 to visit kennels and to hunt (pp. 39-70).

 
======K112======
'''Story, Douglas, '''''To-morrow in the East. ''London: Chapman &amp;amp; Hall, 1907. x+267pp.
::The Scottish journalist (1872-1921) offered his book as “the result of ten years” observation as war-correspondent and special correspondent in the countries of the East”, but he also included a chapter devoted to the powerful tsarist minister Count Witte, whom he interviewed in St Petersburg in August 1903 (pp. 227-42). For Story’s reporting of the Russo-Japanese war, see [[#K133|K133]].

 
======K112a======
'''Grafton, Charles Chapman,''' ''The works of the Rt. Rev. Charles C. Grafton, S.T.D., LL.D. ''Edited by B. Talbot Rogers. New York and London: Longmans, Green, and Co., 1914. 8 vols.
::The bishop of Fond du Lac, Wisconsin (1830-1912) left New York on 22 August 1903 and returned on 8 November after a brief visit to St Petersburg, Moscow and Sergiev posad to promote “fraternal relations between the Eastern Church in Russia and the Church in America” (vol. IV, pp. 252-70). He was accompanied by the English scholar W.J. Birkbeck (see [[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J82|J82]]). Original American edition ''A journey Godward'' (Milwaukee: Young Churchman Co., 1910).

 
======K113======
'''Spring-Rice, Cecil Arthur, '''''The letters and friendships of Sir Cecil Spring Rice: a record. ''Edited by Stephen Gwynn. London: Constable &amp;amp; Co., 1929. 2 vols.
::A career diplomat, Sir Cecil (1859-1918) served in the British embassy in St Petersburg from September 1903 until April 1906, when he was appointed minister in Persia. He was in St Petersburg throughout the Russo-Japanese war and the revolutionary events of 1905. His letters provide an informed view of events, not least those addressed to his close friend Mrs Roosevelt, wife of the American president (vol. I, pp. 362-504, II, pp. 1-76).

 
======K114======
'''Weale, Bertram Lenox Putnam [pseudonym of Simpson, Bertram Lenox], '''''Manchu and Muscovite: being letters from Manchuria written during the autumn of 1903, with an historical sketch entitled ‘Prologue to the crisis’, giving a complete account of the Manchurian frontiers from the earliest days and the growth and final meeting of the Russian and Chinese empires in the Amur regions''. London: Macmillan and Co., 1904. xx+552pp.
::Commissioned to write a series of articles on Manchuria, Simpson (1877-1930) travelled extensively through the country from September to November 1903. He was highly critical of Russian presence in the area and dedicated his book to the “gallant Japanese nation”. (see also [[#K163|K163]].)

 
======K115======
'''Ready, Oliver George, '''''Through Siberia and Manchuria by rail. ''London: Chapman &amp;amp; Hall, 1904. 26pp.
::Following the declaration of war between Russia and Japan, Ready (1864-1940), English travel writer and author of ''Life and Sport in China,'' decided to publish his travel diary that was originally intended for private circulation in typescript. He travelled from London to Shanghai via the Trans-Siberian, boarding the train in Moscow on 21 October 1903, and reaching Dalnii on 4 November, where he pronounced “the railway in its entirety is flimsy and liable to collapse almost everywhere”.

 
======K116======
'''Onslow, Richard William Alan,''' ''Sixty-three years: diplomacy, the Great War and politics, with notes on travel, sport and other things. ''London: Hutchinson, 1944, 204pp.
::British diplomat and civil servant, Earl Onslow (1876-1945), then Viscount Cranley, arrived in Petersburg to become personal secretary to the ambassador, Sir Charles Scott, and his immediate successors, Sir Charles Hardinge and Sir Arthur Nicolson, between January 1904 and January 1906 and between May and September 1906. His recollections cover social and diplomatic life in the capital, momentous events like the Bloody Sunday massacre and the Russo-Japanese war, and fond memories of hunting and fishing trips (pp. 88-132).

 
======K117======
'''Ganz, Hugo Markus, '''''The downfall of Russia: behind the scenes in the realm of the czar. ''London: Hodder and Stoughton, 1904. 320pp. [In the same year there appeared another version: ''The land of riddles: Russia of to-day. ''Translated from the German by H. Rosenthal. New York and London: Harper &amp;amp; Brothers, 1904. 330pp.]
::The German-born journalist (1862-1922) travelled from Vienna via Warsaw to St Petersburg at the very beginning of 1904 and was present, for instance, at the funeral of Nikolai Mikhailovskii on 10 February 1904, before travelling on to Moscow (pp. 33-320). His book contains much of interest, not least an assessment of the work of Repin (pp. 147-62), and includes a long account of his visit to to see Tolstoi at Iasnaia Poliana (pp. 274-320). Translated from the German original ''Vor der Katastrofe ''(1904).

 
======K117a======
'''Maud, Renée Elton, '''''One year at the Russian court: 1904-1905''. London: John Lane, 1918. vii+222pp.
::In memoirs written in 1917 Mrs Maud (née Gaudin de Villaine) recalls her visit as a young woman to Russia from early summer 1904 to the summer of 1905. Through her French and Russian connections (her maternal grandmother was Baroness Nikolay) she frequented court and diplomatic circles in St Petersburg, visited the Nikolay estate of Monrepos near Vyborg and stayed in Tiflis with other relatives. Back in the capital, she was unsumpathetic to the events of Bloody Sunday. The fourth and final part of her book (pp. 1107ff.) is devoted to Rasputin, based on what she later heard and read.

 
======K118======
'''Villari, Luigi, '''''Russia under the great shadow.'' London: T. Fisher Unwin, 1905. 330pp.
::The Italian historian, traveller and diplomat (1876-1959) offered in part a record of observations noted during travels in Russia in the summer and autumn of 1904, but in the main an impersonal analysis of the state of Russian government, society and economy during the Russo-Japanese War. Arriving in St Petersburg in July 1904, he travelled on to Moscow and the fair at Nizhnii Novgorod, before taking the steamer down the Volga to Saratov, proceeding overland through Ukraine to Odessa and back to Kiev.

 
======K119======
'''Harper, Samuel Northrup, '''''The Russia I believe in: the memoirs of Samuel N. Harper, 1902-1941. ''Edited by Paul V. Harper with the assistance of Ronald Thompson. Chicago: University of Chicago Press, 1945. xiv+279pp.
::Harper (1882-1943), who was to become professor of Russian language and institutions at the University of Chicago, where he taught from 1912, and be recognized as the first American-born scholar to devote an academic career to the study of Russia, made eighteen trips to Russia between 1904 and 1939. In the opening chapters of his autobiography he describes his early studies at the University of Moscow, his observation of the working of the Dumas between 1906 and 1910, his extensive travels through Russia when he was a lecturer at the University of Liverpool (1910-12), and finally his experiences during WWI and the subsequent Revolution, initially as adviser to Ambassador Francis in 1916 and then in 1917 as adviser and interpreter to the Root Mission (pp. 1-108).

 
======K120======
'''Bryan, William Jennings, '''''Under other flags: travels, lectures, speeches. ''Lincoln, Nebraska: Woodruff-Colliers Printing Co., 1904. 397pp.
::Three-times defeated Democrat candidate for the American presidency and speech-maker extraordinary, Bryan (1860-1925) and his wife took a European sabbatical that included a visit to Russia in 1904. They travelled from Warsaw to Moscow, from where they made a trip to Iasnaia Poliana to see Tolstoi, whose “colossal strength lies in his heart more than in his mind” (pp. 96-108). Proceeding to St Petersburg, they were taken to Tsarskoe selo by the American ambassador to meet the tsar (pp. 77-85).

 
======K121======
'''Garnett, David, '''''The golden echo. ''London: Chatto and Windus, 1935. 272pp.
::Twelve-year old David (1892-1981) accompanied his mother, the famous translator Constance Garnett, on her second visit to Russia in the summer of 1904. Arriving in St Petersburg on 16 May they moved after two weeks to Moscow and thence to stay with the novelist and landowner Aleksandr Ertel at his estate in Tambov province. They travelled back to England overland and arrived on 13 August (pp. 74-93).

 
======K122======
'''Joubert, Carl,''' ''The truth about the tsar and the present state of Russia.'' London: Eveleigh Nash, 1905. 265pp.
::In his second book, finished in December 1904, Joubert (see [[#K13|K13]]) writes of the Russo-Japanese conflict, but also takes issue with British apologists of the tsar, whom he dubs with prescience “the last of the Romanoffs”.

 
======K123======
'''Pares, Bernard,''' ''[https://archive.org/details/russiareform00pareiala Russia and reform]''. London: Archibald Constable, 1907. xiv+576pp.
::Pares (see [[#K58|K58]], [[#K59|K59]], [[#K124|K124]], [[#K309|K309]]) was in close and constant contact with key liberal figures in the Duma. In this study of Russia before the revolution of 1905 and its aftermath he offers an informed analysis events based on his studies and his diaries and notebooks.

======K124======
'''Pares, Bernard, '''''The fall of the Russian monarchy: a study of the evidence.'' London: Jonathan Cape, 1939. 510pp.
::This is Pares’s retrospective on the causes of the 1917 revolutions, based on both published evidence and on the numerous interviews with leading players that he conducted from 1904 to 1914.

 
======K125======
'''Meakin, Annette M.B., '''''Russia: travels and studies. ''London: Hurst and Blackett, 1906. xx+450pp.
::The third and weightiest work by the Russian-speaking anthropologist (see [[#K73|K73]], [[#K83|K83]]), based on her extensive travels in 1904 that took her from St Petersburg via Moscow and Kharkov to the Crimea, and then via Odessa and Kishinev and Kiev on to Georgia and the Caucasus. She was yet another visitor to Tolstoi.

 
======K126======
'''McCullagh, Francis, '''''With the Cossacks, being the story of an Irishman who rode with the Cossacks throughout the Russo-Japanese war''. London: Eveleigh Nash, 1906. xiv+392pp.
::In August 1903 McCullagh (1874-1956) gave up his job with the English-language ''Japan Times'' and moved to Port Arthur, staying briefly en route in the Russian port of Dalnii. He was working for the Russian paper ''Novyi krai'', when the Russo-Japanese war broke out in February 1904, and became “embedded” with the Russian forces until he was captured by the Japanese during the retreat from Mukden in March 1905 and taken to Japan as a prisoner of war. He was in Moscow by the end of the year, editing his book, which was based on dispatches he had sent to the ''New York Herald.'' He returned to Russia in 1920 during the Intervention and was captured by the Reds.

 
======K127======
'''McCormick, Frederick, '''''The tragedy of Russia in Pacific Asia.'' London: Grant Richards, 1909. 2 vols.
::American journalist McCormick (b. 1870) provides a comprehensive account of the key battles of the Russo-Japanese War, during which he was stationed with the Russian forces. Interspersing descriptive narrative with personal experiences, he begins before the outbreak of war in Port Arthur in January 1904 and describes events and battles until the cessation of hostilities. The second half of vol. II provides an account of the psychology, abilities and material situation of the average Russian soldier.

 
======K128======
'''McCully, Newton Alexander, '''''The Russo-Japanese War''. Edited by Richard von Doenhoff. Annapolis, Maryland: Naval Institute Press, 1977. xiv+338pp.
::Report of US navy Lt-Commander McCully (1867-1951), written in late 1905 and based on a diary he kept while assistant naval attaché in St Petersburg and subsequently with the Russian forces in the Far East during the Russo-Japanese War. It contains detailed accounts of his travels between 15 March 1904 and 18 July 1905 and his observations on the places and peoples he witnessed in Siberia and Manchuria.

 
======K129======
'''McKenzie, Frederick Arthur, '''''From Tokyo to Tiflis: uncensored letters from the war.'' London: Hurst and Blackett, 1905. x+340pp.
::Special correspondent of the ''Daily Mail'', the Canadian McKenzie (1869-1931) was with the Japanese army, sending his dispatches from the battlefields of the Russo-Japanese war. He also reported from Warsaw and Tiflis on the strikes and unrest in 1905 (pp. 285-327). McKenzie was later to write two accounts (1923, 1930) of his visits to Soviet Moscow under NEP.

 
======K130======
'''Kennard, Howard Percy, '''''The Russian peasant''. London: T. Werner Laurie, 1907. xvi+302pp.
::Dr Kennard (d. 1915) wrote the preface to his book in May 1907 from Samara, where he was helping in famine relief, and his work reflects his deep sympathies for the Russian peasantry among whom he had lived in many parts of European Russia since the time of the Russo-Japanese war. The first part of his book offers a comprehensive anthropological description of village life, focusing on customs, beliefs, family relationships and ceremonies. The second part is an historical overview of pre- and post-serfdom Russia, while the third is a sustained critique of the impact of bureaucracy, policing, censorship and surveillance, and the Church on the lives of the Russian peasantry.

 
======K131======
'''Gilliard, Pierre, '''''Thirteen years at the Russian court (a personal record of the last years and death of the Czar Nicholas II and his family). ''Translated from the French by F. Appleby Holt. London: Hutchinson &amp;amp; Co., 1921. 304pp.
::Gilliard (1879-1962) arrived in the Crimea in the autumn of 1904 as French tutor to Duke Sergei of Leuchtenberg and a year later, he became tutor to the tsar’s daughters Olga and Tatiana. He remained with the imperial family virtually until the end, following them into Siberia, but was himself separated from them at Tiumen on 22 May 1918. He eventually returned to France in September 1920.

 
======K132======
'''Wilton, Robert Archibald, '''''Russia’s agony. ''London: Edward Arnold, 1918. xii+356pp.
::Although born in Norwich, Wilton (1868-1925) was the son of a British mining engineer working in Russia and he dates his personal experience of the country back “nearly half a century” in the preface to his book (13 January 1918). It is, however, the last fourteen years he had been ''The Times''’s correspondent that provide the material for his tracing of Russia’s destiny through revolution and war “without fear or favour”. Subsequently Wilton went to Siberia, but following the fall of Kolchak, escaped to Paris.

 
======K133======
'''Story, Douglas,''' ''The campaign with Kuropatkin. ''London: T. Werner Laurie, 1904. xii+301pp.
::Story (see [[#K112|K112]]) travelled from Hong Kong via Tokyo to Mukden, where he was received by the Russian viceroy Alekseev and became the first foreign correspondent formally accredited to the Russian army on 24 April 1904. He wrote positively of the Russian officer and particularly of the ordinary soldier ‘Ivan Ivanovitch’. He travelled home via the Trans-Siberian and on to St Petersburg.

 
======K134======
'''Baring, Maurice, '''''With the Russians in Manchuria.'' London: Methuen &amp;amp; Co., 1905. xv+205pp.
::In April 1904 Baring (see [[#K79|K79]]) was appointed as the ''Morning Post''’s correspondent to cover the Russo-Japanese war and his book comprises the dispatches he sent to the paper. He travelled from Moscow on the Trans-Siberian railway and arrived in Kharbin on 19 May. He travelled on to Mukden and then nearer to the battlefields. He left Mukden for England at the beginning of December 1904. In August 1905 he left St Petersburg once more en route for Manchuria, where he was to remain until October. His book was dedicated to Guy Brooke (see [[#K135|K135]]).

 
======K135======
'''Brooke, Leopold Guy Francis Maynard Greville, '''''An eye-witness in Manchuria''. London: Eveleigh Nash, 1905. viii+312pp.
::Brooke (1882-1928), later 6th Earl of Warwick, the Reuter’s special correspondent covering the Russo-Japanese war, travelled with Baring from Moscow to Kharbin in May 1904. He remained for nine months with the Russian army, for which he expressed great admiration, and returned to England via St Petersburg.

 
======K136======
'''Greener, William Oliver, '''''A secret agent in Port Arthur. ''London: Archibald Constable &amp;amp; Co., 1905. viii+316pp.
::Greener, writing under his real name (cf. Wirt Gerrare, see [[#K85|K85]]) was sent to Port Arthur to report events of the Russo-Japanese war. He describes his journey from Moscow by rail, the ferry across Baikal, and on to Vladivostok before entering Manchuria (pp. 18-38). “The status of a secret agent is that of a special correspondent travelling incognito” and he reported back to both ''The Times ''and the'' China Times ''on what he witnessed.

 
======K137======
'''Henry, James Dodds,''' ''Baku: an eventful history.'' Introductory note by Sir Boverton Redwood. London: Archibald Constable &amp;amp; Co., 1906. xviii+256pp.
::Editor of the ''Petroleum World'', Henry (b. 1864) returned from the oilfields of Baku in February 1905 and attempted in his book to provide an informed update on Marvin’s work of 1884 ([[(title of the correspondent chapter)#J26|J26]]) and an objective assessment of the city and its industry’s potential.

 
======K138======
'''Joubert, Carl, '''''The fall of tsardom.'' London: Eveleigh Nash, 1905. 255pp.
::Joubert (see [[#K13|K13]], [[#K122|K122]]) welcomed the 1905 revolution and the inexorable progress, as he hoped, towards a constitution.

 
======K139======
'''Noble, Algernon, '''''Siberian days: an engineer’s record of travel and adventure in the wilds of Siberia. ''London: H.F. and G. Witherby, 1928. 223pp.
::Noble (d. 1975) provides a non-chronological account of his involvement in copper and coal mining in the Kirghiz steppe and in prospecting for gold in Siberia from Tomsk to Baikal, between 1905 and 1914.

 
======K140======
'''Meyer, George von Lengerke, '''''George von Lengerke Meyer, his life and public services.'' By Mark Antony DeWolfe Howe. New York: Dodd, Mead and Co., 1919. 556pp.
::Meyer (1864-1960) was American ambassador to Russia from April 1905 to January 1907, arriving in the midst of the Russo-Japanese War. His biographer includes generous selections from his letters (to President Roosevelt, Senator Lodge and his wife) and diary entries, particularly for 1905, but more selectively for 1906-07, where the focus is on the Algericas Conference and the first meeting of the Russian Duma on 10 May 1906 (pp. 137-335).

 
======K141======
'''Anet, Claude [pseudonym of Schopher, Jean], '''''Through Persia in a motor-car by Russia and the Caucasus. ''Translated by M. Beresford Ryley. London: Hodder &amp;amp; Stoughton, 1907. xvi+281pp.
::Arriving in Russia in April 1905, Schopher (1868-1931), a Swiss professional tennis player and successful writer, drove through Bessarabia, the Caucasus and the Crimea. In Yalta he met Maksim Gorkii. In Georgia he noted the social tensions, strikes, and civil unrest. He took the train from Tiflis to Baku and the boat from there for Persia at the end of April (pp. 1-83).

 
======K142======
'''Sarolea, Charles Louis-Camille, '''''Count L.N. Tolstoy, his life and work''. London: T. Nelson and Sons, 1912. viii+384pp.
::The Belgian scholar (1870-1953), later professor of French at Edinburgh University and author of several books on Russia, was another pilgrim to Iasnaia Poliana, visiting “the Master” in May 1905 (pp. 316-36).

 
======K143======
'''Sarolea, Charles Louis-Camille,''' ''Europe’s debt to Russia''. London: William Heinemann, 1916. x+251pp.
::“An attempt to give a systematic and co-ordinated survey of Russian history and policy”, following his further visit in 1915, Sarolea’s work includes, at the “insistence” of Tolstoi, his personal impressions of 1905 and “the tragic events of the Russian Annus Mirabilis” (pp. 188-228).

 
======K144======
'''Villari, Luigi, '''''Fire and sword in the Caucasus. ''London: T. Fisher Unwin, 1906. 347pp.
::Villari (see [[#K118|K118]]) provides an account of political unrest, ethnic tensions and violence in the Caucasus region during a stay in August-October 1905. His extensive travels took him to Batumi, Tiflis, Baku, and Erevan and he was in Vladikavkaz on the day the October Manifesto was published.

 
======K145======
'''Winter, Nevin Otto, '''''The Russian Empire of to-day and yesterday. The country and its peoples, together with a brief review of its history, past and present, and a survey of its social, political, and economic conditions''. London: Simpkin, Marshall &amp;amp; Co., 1913. xvi+487pp.
::Ohio lawyer and author, Winter (1869-1936) offers an extensive descriptive account of the Russian Empire, based to a large degree on personal observations made during visits in the early 1905 and 1912. In the first half he covers a variety of topics, including St Petersburg and Moscow, the status of Jews, the Russian character and social customs and structure, the education system and Russian literature, while in the second he provides a historical survey.

 
======K146======
'''Baring, Maurice, '''''A year in Russia. ''London: Methuen &amp;amp; Co., 1907. xx+319pp.
::The year he covered in more letters sent to the ''Morning Post'' was from 8 August 1905 to 6 August 1906. Begins with his departure again for Mongolia and return to Moscow by 3 November and then alternates between Moscow and St Petersburg. He was witness to the revolutionary events that unfolded in the old capital and attended meetings of the Duma in the Taurida Palace. His “collection of notes, a bundle of impressions” includes an item on the 25th anniversary of the death of Dostoevskii.

 
======K147======
'''Ular, Alexander, '''''Russia from within.'' London: Heinemann, 1905. xii+290pp.
::Ular (b. 1876) provides an account of the causes of the 1905 revolution which lays the blame primarily in the hands of the Russian political leadership and the tsar. Ular was in Russia at the time (the preface is dated May 1905), but his account is presented as a description of the basic facts, supplemented by the occasional reference to personal experiences and interviews he has undertaken.

 
======K148======
'''Walling, William English, '''''Russia’s message: the true world import of the revolution.'' London: A.C. Fifield, 1909. xviii+476pp.
::The prominent American socialist (1877-1936), who believed that “official Russia is in a land of lies”, arrived in Petersburg for the first time at the end of 1905. Over the next two years he spent many months there, latterly with his new wife Anna Strunskaia, sending back numerous articles to American papers and journals and intent on “gaining a rounded view”. He met and interviewed countless prominent Russian officials, ministers, and intellectuals, including Tolstoi, and concluded that the Russian revolution (of 1905) offered a message of hope for a new world civilization.

 
======K149======
'''Nevinson, Henry Woodd, '''''The dawn in Russia or scenes in the Russian revolution''. London and New York: Harper &amp;amp; brothers, 1906. xiv+349pp.
::Nevinson (1856-1941), sent to Petersburg by the ''Daily Chronicle'' as its special correspondent, offered his book as a description of “scenes which I witnessed in Russia during the winter of 1905-1906”, some published in the newspaper, but all re-arranged and re-structured to give a general view of events not only in the capital, but also in Tula, Moscow, Odessa and elsewhere.

 
======K150======
'''Nevinson, Henry Woodd,''' ''More changes more chances''. London: Nisbet &amp;amp; Co., 1925. xviii+427pp.
::In the second volume of his autobiography Nevinson recounts events already included in his 1906 book but also adds his subsequent travels through Georgia and the Crimea in 1906-07 (pp. 98-211). Later included in his condensed autobiography, ''Fire of life'' (1935), pp. 181-209.

 
======K151======
'''Hedin, Sven, '''''Overland to India. ''London: Macmillan and Co., 1910. 2 vols.
::On his way to India and sailing from Constantinople across the Black Sea, Hedin (see [[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J132|J132]]) is caught up with events of the 1905 revolution, when his ship puts into Batumi at the end of October 1905. He details strikes and unrest in Batumi and decides to travel to Poti and take the train to Tiflis, but is forced to turn back and sail to Trebizond (vol. I, pp. 1-21).

 
======K152======
'''Bullard, Arthur, '''''The Russian pendulum: autocracy-democracy-bolshevism.'' New York: Macmillan, 1919. xvi+256pp.
::American journalist and noted socialist Bullard (1879-1929) recalls in the opening chapters his first visit to Russia in the years 1905-08, when he sent back numerous articles to American journals, mostly under the pseudonym of Albert Edwards. In July 1917 he returned to Petrograd as head of the Russian branch of the American government’s Committee on Public Information. He left from Archangel in June 1918, but returned to Vladivostok until November.

 
======K153======
'''Henderson, Nevile Meyrick, '''''Water under the bridges.'' London: Hodder &amp;amp; Stoughton, 1945. 221pp.
::In the course of a long diplomatic career, Sir Nevile (1882-1942) served on two occasions in the British embassy in St Petersburg. He arrived in December 1905 and remained until April 1909 (pp. 24-48). After a spell in Tokyo, he left for St Petersburg again in January 1912, travelling via the Trans-Siberian. He served under Sir George Buchanan until April 1914 (pp. 61-68). His posthumously published memoir is predominantly concerned with the routines of embassy life.

 
======K154======
'''Washburn, Stanley, '''''The cable game: the adventures of an American press-boat in Turkish waters during the Russian revolution. ''London: Andrew Melrose, 1912. 222pp.
::The Minneapolis lawyer-turned-journalist Washburn (1878-1950), having just covered the Russo-Japanese war for the Chicago'' Daily News,'' was sent to Russia to report on the final stages of the 1905 revolution. He arrived in Odessa in December 1905 and then travelled on to Sevastopol and Batumi in search of newsworthy incidents of unrest and violence.

 
======K155======
'''Preston, Thomas, '''''Before the curtain''. London: John Murray, 1950. iv+313pp.
::Preston (1886-1976), later Sir Thomas, 6th baronet, “lived in Russia, off and on, since 1905, in almost every corner of this vast continent and amongst the most varied communities”. He first worked in Batumi and for a mining company at Dzhanzhul, prior to entering Cambridge University in 1907. He then returned to prospecting and mining in Siberia until his appointment in 1913 as British vice-consul, subsequently consul, at Ekaterinburg, where he was still in post at the time of the murder of the imperial family. He remained in Siberia under the Whites until October 1920, but was to return to Russia in 1922 and remained as British official agent in Leningrad until 1926 (pp. 14-230).

 
======K156======
'''Decle, Lionel, '''''The new Russia. ''London: Eveleigh Nash, 1906. 279pp.
::Stirred by the events of 1905, the British author Decle (1859-1907) arrived in St Petersburg in January 1906 to interview ministers and high officials in order to provide “a general ''aperçu'' of the present system under which the administration, the law, education, taxation are organized”. In his preface he prides himself that many of his predictions had been realized, although the final sentence of his book had suggested that “there is only one thing impossible in Russia, and that is to understand the Russians”.

 
======K157======
'''Durland, Kellogg,''' ''The red reign: the true story of an adventurous year in Russia. ''London: Hodder &amp;amp; Stoughton, 1908. xxvi+533pp.
::American journalist and adventurer Kellogg (1881-1911) provides a detailed account of his extensive travels throughout Russia between January and December 1906. His intention was to portray as accurately as possible a country ''in'' revolution and witnessed in different contexts and circumstances. Thus he travelled to the Caucasus with a group of Cossack officers; observed the effects of famine around Saratov; interviewed the “terrorist” Maria Spiridonova at Tambov; attended the opening session of the Duma in St Petersburg; was in Cronstadt during the August mutiny; visited the British-run model industrial town of Iuzovka, before travelling to see Tolstoi at Iasnaia Poliana. In December he left Odessa for Constantinople.


======K157a======
'''Bainbridge, Henry Charles,''' ''Twice seven: the autobiography of H.C. Bainbridge''. London: George Routledge &amp; Sons, 1933. xi+312pp.
::Bainbridge (1874-1954) was the manager of the Fabergé shop in London at 48 Dover Street and, after it re-located in 1911, at 173 New Bond Street until its closure in 1915. He paid the first of his annual visits to Russia in late 1906 and his last in 1913, spending most of his time with the Fabergés in St Petersburg (“He who not seen Petersburg in her hey-day has seen nothing”), but also frequenting the Fabergé summer house at Levashova near the Finnish border. Much given to name-dropping and philosophizing on the enigma of Russia, he offers surprisingly little of interest (pp. 24-34, 178-82, 247-58). In 1949 he published ''Peter Carl Fabergé, goldsmith and jeweller to the Russian imperial court'', incorporating material from his earlier autobiography.


======K158======
'''Niedieck, Paul, '''''Cruises in the Bering Sea, being records of further sport and travel.'' Translated from the German by R.A. Ploetz. London: Rowland Ward, 1909. xvi+252pp.
::Leaving London in March 1906, Niedieck, indefatigable hunter and specimen-collector, arrived in Kamchatka, via the USA and Japan, in April, remaining until July. He describes in detail his hunting of bears and walruses, but also provides descriptions of the lives of the various tribes and nomadic people he encounters (pp. 3-107). German original ''Kreutzfahrten in Beringmeer'' (1907).

 
======K159======
'''Bouillane de Lacoste, E.A. Henri de, '''''Around Afghanistan.'' London: Sir Isaac Pitman &amp;amp; Sons, 1909. Preface by M. Georges Leygues. Translated from the French by J.G. Anderson. xxxii+218pp.
::Major Bouillane de Lacoste (1894-1937) journeyed around Afghanistan’s perimeter in the summer of 1906, entering Russian territory at Gaudan on the Persian-Turkestan border on 18 May, travelling on to Askhabad, where he took the Transcaspian railway to Andijan. Accompanied by a Kirghiz family, he travelled through the Alai and Trans-Alai region, leaving Russian territory at the Beik Pass (pp. 34-75).

 
======K160======
'''Fraser, David, '''''The marches of Hindustan, the record of a journey in Thibet, Trans-Himalayan India, Chinese Turkestan, Russian Turkestan, and Persia. ''Edinburgh and London: William Blackwood and Sons, 1907. xvi+521pp.
::Setting out in January 1906 on a 5630-mile journey, Fraser entered Russian Turkestan in October. He travelled through the mountains to Osh and Tashkent, from where he took the railway to Askabad and then crossed into Persia. He provides a history of Russian penetration into the area and an assessment of its military strength (pp. 284-376).

 
======K161======
'''Fraser, John Foster, '''''Red Russia.'' London: Cassell and Co., 1907. xii+288pp.
::Fraser (see [[#K29|K29]], [[#K88|K88]], [[#K90|K90]], [[#K301|K301]]) travelled through large areas of the Russia empire throughout 1906, assessing the mood and situation after the revolutionary events of 1905. Various chapters reflect his visits to St Petersburg, Moscow, Nizhnii Novgorod, Samara, Kazan, Bessarabia, the Caucasus and the Crimea, as well as Warsaw and Finland.

 
======K162======
'''De Windt, Harry, '''''Through savage Europe; being the narrative of a journey (undertaken as special correspondent of the “Westminster gazette”), throughout the Balkan states and European Russia. ''London: T. Fisher Unwin, 1907. 300pp.
::De Windt departed from Trieste in the summer of 1906 by boat on a journey that was to take him through Montenegro, Bosnia, Bulgaria, and Romania, and finally into Russia, “the land of mystery, gloom, and death”, where the “red flag” flew after the events of 1905. He visited Odessa, Rostov, Vladikavkaz, and Baku (pp. 261-89).

 
======K163======
'''Weale, Bertram Lenox Putnam [pseudonym of Simpson, Bertram Lenox], '''''The coming struggle in eastern Asia. ''London: Macmillan and Co., 1908. xiv+656pp.
::Simpson offers his volume as the fourth and final in a series of “political treatises”, opened by ''Manchu and Muscovite ''(1904) (see [[#K114|K114]]), that sought to examine Russo-Japanese rivalry. In the autumn of 1906 he sailed from Korea to Vladivostok, of which he provides a detailed assessment, and devotes Part I of his book to ‘Russia beyond Lake Baikal’, including Russian Manchuria (pp. 1-322).

 
======K164======
'''Gerhardi[e], William Alexander, '''''Memoirs of a polyglot, ''London: Duckworth, 1931. 381pp.
::Son of the British industrialist Charles Alfred Gerhardi, the novelist and critic (1895-1977) describes his upbringing and education in St Petersburg between 1906 and 1912. Having moved to England, he found himself back in wartime and revolutionary Petrograd attached to the British embassy between 1916 and 1918. He was subsequently with the British intervention forces in Siberia until 1920 (pp. 1-153).

======K165======
'''Barrett, R.J., '''''Russia’s new era. Being notes, impressions and experiences – personal, political, commercial and financial – of an extended tour in the empire of the tsar. With statistical tables, portraits, snapshots and other illustrations''. London: The Financier and Bullionist, Ltd., 1908. 292pp.
::Barrett, F.R.G.S., travelled extensively through Russia in the spring and summer of 1907 and provided a detailed social and economic account of Russia, with the primary focus on the commercial and investment opportunities available to the British.

 
======K166======
'''Barzini, Luigi, '''''Pekin to Paris: an account of Prince Borghese’s journey across two continents in a motor-car.'' Translated by L.P. de Castelvecchio. London: E. Grant Richards, 1907. 645pp.
::In the summer of 1907 the Italian war correspondent Barzini (1874-1947) accompanied Prince Scipione Borghese (1871-1927) in a famous motor race from Pekin to Paris. They drove to victory an Itala car on a 10,000-mile journey that took them through China into Siberia. Entering Siberia from Mongolia on 25 June, they travelled via Irkutsk, Tomsk, Omsk and Nizhnii Novgorod to Moscow, and then to St Petersburg, which they left on 1 August for Poland (pp. 297-594). The book is illustrated with numerous unique photographs, showing memorable encounters and the many breakdowns of the car. The Italian original was entitled ''La metà del mondo vista da un’automobile da Pechino a Parigi in sessanta giorni.''

 
======K167======
'''Barrows, Isabel Chapin, '''''A sunny life: the biography of Samuel June Barrows. ''Boston: Little, Brown and Company, 1913. xii+323pp.
::In her biography of her husband (1845-1909), the American Republican congressman and prison reformer, Isabel (1845-1913) records the trip the couple made in the summer of 1907 to visit prisons in St Petersburg, Moscow, and Nizhnii Novgorod. They then went down the Volga to Samara to see the lingering effects of the famine, and on their return journey, visited Tolstoi at Iasnaia Poliana (pp. 192-96, 204-05). In the spring of 1909 Isabel went alone to St Petersburg in connection with the arrest of Ekaterina Bereshkovskaia (1844-1934), “the little grandmother of the Revolution” (pp. 240-41).

 
======K168======
'''Jackson, Abraham Valentine Williams, '''''From Constantinople to the home of Omar Khayyam: travels in Transcaucasia and Northern Persia for historic and literary research''. New York: Macmillan Co., 1911. xxxiv+317pp.
::Jackson (1862-1937), American traveller and professor of Indo-Iranian languages at Columbia University, provides a combination of archaeological and linguistic scholarship and travel account of trips made through the Caspian and Transcaspian regions in 1907, 1908 and 1910. It is a synthesis of observations and experiences from numerous research trips, but presented as a geographically consistent travel narrative that takes him into Russia at Sevastopol, through the Crimea and by steamer to Batumi, then via Tiflis to Baku. A second section focuses on the city of Baku; a third on a research trip to the city of Derbent in 1910 (pp. 12-84).

 
======K169======
'''Fischer, Emil Sigmund, '''''Overland via the Trans-Siberian railway: description of a trip from the Far East to Europe and the United States of America. ''Tientsin: Tientsin Press, 1908. ix+44pp.
::Fischer (1865-1945), best known for his later travels and writings on China and Japan, describes his return home from China in the summer of 1907.

 
======K170======
'''Foulke, William Dudley, '''''A random record of travel during fifty years.'' New York: Oxford University Press, 1925. 241pp.
::Noted journalist, author, and reformer, and from 1903 president of the American Society of Friends of Russian Freedom, Foulke (1848-1935) visited Finland and Russia from Norway in 1907. Routine tourist impressions of St Petersburg and Moscow (“a far more interesting place”) also include description of a visit to the Duma and meeting with Prof. Miliukov (pp. 94-111).

 
======K171======
'''Young, Charles Christian, '''''Abused Russia. ''New York: Devin-Adair Co., 1915. 109pp.
::Dr Young (b. 1875) travelled to Russia in 1907 and again between 1912 and 1914. He attempts in his book to refute some of the criticisms and misconceptions held by Americans with regard to Russian society and politics and to urge a renewal of close links with Russia. A final chapter deals with Young’s experiences when travelling as a sheep salesman through Turkmenistan between 1912 and 1914 and his observations on the region and the city of Bokhara.

 
======K172======
'''Lydekker, Richard, '''''A trip to Pilawin, the deer-park of Count Joseph Potocki in Volhynia Russia. ''With a preface by Count Joseph Potocki. London: Rowland Ward, 1908. xiv+115pp.
::Naturalist, geologist, and cataloguer of the Natural History Museum’s fossil mammals, birds, and reptiles, Lydekker (1849-1915) was invited to visit the Pilawin preserve near the Potocki palace of Antoniny in present-day Ukraine. Travelling from London with his daughter, he arrived on 22 August 1907 and left on 4 September. He describes in expert detail and with excellent photographs the animals in the extensive forest preserve.

 
======K173======
'''Bayne, Samuel Gamble, '''''Quicksteps through Scandinavia, with a retreat from Moscow. ''New York and London: Harper &amp;amp; Brothers, 1908. 64pp.
::Irish-born Bayne (1844-1924), having made his fortune in banking and oil in America, indulged a liking for travel and authorship, visiting St Petersburg and Moscow in the summer of 1907. His visit to the Russian capital coincided with the opening ceremony of the Church of the Resurrection of Christ (‘On the blood’) which he mistakenly calls of the Ascension (pp. 17-31).

 
======K174======
'''Baring, Maurice, '''''Russian essays and stories''. London: Methuen &amp;amp; Co., 1908. xvii+295pp.
::A collection of eleven essays and seven stories, mostly originally published in the ''Morning Post'', which cover a wide range of non-political (Baring’s description) topics – travels, conversations, incidents, many of them literary.

 
======K175======
'''Baring, Maurice, '''''What I saw in Russia. ''London: Nelson, 1913. 381pp.
::A compilation of selected chapters from his first three books about Russia, covering the period 1904-1907. The final three chapters (from ''Russian essays'') describe his journeys down the Volga in August-September 1907 and to Vologda in the north in November of the same year.

 
======K176======
'''Foulke, William Dudley, '''''A random record of travel during fifty years. ''New York: Oxford University Press, 1925. viii+241pp.
::The American travel writer recounts his trip to Russia, arriving from Finland in 1907, intermixing descriptions of life as a tourist in St Petersburg and Moscow with an assessment of the political climate and an account of a visit to the Duma while in St Petersburg (pp. 98-111).

 
======K177======
'''Murray, Robert H., '''''Around the world with Taft: a book of travel, description, history''. Detroit, Michigan: F.B. Dickerson Company, 1909. 412pp.
::Associated Press correspondent Murray was attached to then American secretary of war William Taft during a diplomatic mission to the Philippines that involved the party circumnavigating the world. Sailing from Manila, the Taft party arrived in Vladivostok on 17 November 1907 to board the Trans-Siberian that took them to Moscow. In early December they were in St Petersburg, where Taft met the tsar at Tsarskoe selo and they attended a military review (pp. 325-76).

 
======K178======
'''Wood, John Nicholas Price, '''''Travel &amp;amp; sport in Turkestan''. London: Chapman &amp;amp; Hall, 1910. xx+201pp.
::Wood, a captain in the 12th Royal Lancers, travelled from India in May 1907 to indulge a long-held dream of shooting along the borders of Mongolia and to return to England through Russian Turkestan. He was allowed into Russia on 23 November and travelled by train via Orenburg and Samara to Moscow, which he left for London on 20 December (pp. 179-94).

 
======K179======
'''Nostitz-Azabal, Madeleine, '''''Romance and revolutions''. London: Hutchinson &amp;amp; Co., 1937. 258pp.
::Three-times-married Iowa-born actress Madeleine Bouton, also known as Lilie, moved from her first husband, the German Count Guido von Nimptsch, to become the wife of Count Grigorii Nostitz, who took her to Russia in 1907, first to the family estate in Ukraine and then to Petersburg (pp. 66-89). There followed five years at the Russian embassy in Paris before they returned to Russia in 1914 and eventually escaped via Kiev and the Crimea at the end of 1918 (pp. 131-238).

 
======K180======
'''Tracey, Margot, '''''Red rose.'' Newton Abbot and London: David &amp;amp; Charles, 1978. 230pp.
::Mrs Tracey (b. 1907), née Girard, born in Moscow to a rich French industrialist and his Russian wife, recounts her childhood and her and her sister’s harrowing experiences during the revolution and the first years of Soviet rule until their eventual departure for France in 1921 (pp. 11-131). The second part of the book is devoted to an account of her visit as a tourist to Moscow in 1970. The book’s title refers to the Soviet name for her family’s Moscow factory.

======K181======
'''Scott, A. MacCallum, '''''Through Finland to St. Petersburg''. London: Grant Richards, 1908. 291pp.
::Essentially a guide-book to “this enterprising little country”, Finland, with three concluding chapters on St Petersburg, where “the English visitor may study institutions and ways of life so strangely different from those he knows at home”.

 
======K182======
'''Reynolds, Rothay, '''''My Russian year. ''London: Mills &amp;amp; Boon, 1913. xii+304pp.
::The year would seem, from vague internal evidence, to have been 1908, but there are references to 1905 and 1906. A fluent speaker of Russian, the British traveller and correspondent of the ''Daily News'', Reynolds offers a series of sketches of people and places, designed “to make the reader see Russia as I have seen it”, i.e. in all its variety and contradictoriness, as “the land of ideals”, “the home of melodrama”, “the land of liberty undreamt of by the shackled West”, and much else.

 
======K183======
'''Reynolds, Rothay, '''''My Slav friends. ''London: Mills &amp;amp; Boon, 1916. vii+312pp.
::A second volume, very much in the style and spirit of the first, without evidence of later visits. His aim was “to write of people I have met, of cities I have visited”, etc., using secondary sources to bolster his narration wherever necessary. There is a greater emphasis on the desirability of Anglo-Russian friendship and understanding without betraying the “truth” that the British public deserves.

 
======K184======
'''Calina, Josephine,''' ''Scenes of Russian life. ''London: Constable and Co., 1918. 302pp.
::Calina (c.1890-1962), born and bred in Poland, spent several years “walking in the small dirty villages of Russia” after release from a Russian prison c.1908 and offered her sketches of prison and peasant life as taken “from the very depth of Russian life with its sadness and its humour”. About the time of the October revolution she sought refuge in England, where she was to marry the eminent Shakespearean scholar Allardyce Nicoll and herself write ''Shakespeare in Poland'' (1923).

 
======K185======
'''Farmborough, Florence''','' Nurse at the Russian front: a diary 1914-18''. London: Constable, 1974. 422pp.
::Florence (1887-1978) originally went to Russia in 1908 as a governess and teacher of English, first in Kiev and later in Moscow. She trained as a nurse with the outbreak of WWI and served almost continuously at the front until 1917. When her unit was disbanded, she made her way back to England via the Trans-Siberian railway and steamers.

 
======K186======
'''Farmborough, Florence,''' ''Russian album 1908-1918''. Edited by John Joliffe. [Salisbury]: M. Russell, 1979. 96pp.
::Stunning photographs, covering the whole of Florence’s ten years in Russia.

 
======K187======
'''Zur Mühlen, Hermynia, '''''The end and the beginning: the book of my life. ''Translated, annotated and with an introduction by Lionel Gossman. Cambridge: Open Book Publishers, 2010. 297pp.
::Countess Zur Mühlen (1883-1951), the Austrian translator and author, lived six unhappy years on her husband’s estate of Eigstfer in Russian Estonia between 1908 and 1913. She paints a vivid picture of life among a prejudiced and intolerant German community and reveals her sympathies for the oppressed. She also spent a summer in Petersburg (pp. 98-150). The present translation is an improved version of the original by Frank Barnes that appeared under the title ''The runaway countess'' in New York in 1930.

 
======K188======
'''Craig-McKerrow, Margaret, '''''Distant journeys, 1908-1928. ''London: Baylis &amp;amp; Son, 1930. 369pp.
::In the spring of 1908 the German Mrs Craig-McKerrow (née Reibold) travelled from Japan to Russia, first via boat to Vladivostok, and then by the Trans-Siberian to Moscow. She describes her train journey and stay in Moscow (pp. 15-37). She re-visited Russia in 1928 (pp. 349-69).

 
======K189======
'''Close, Etta, '''''Excursions and some adventures. ''London: Constable, 1926. 296pp.
::In September 1908 Miss Close, F.R.G.S., and a companion after a short stay in St Petersburg proceded to Moscow, where they boarded the Trans-Siberian to Kharbin (pp. 220-37). Following visits to China, Japan and Korea, they returned by train across a now wintery Siberia (pp. 292-95)

 
======K190======
'''Gibbes, Charles Sydney,''' ''Tutor to the tsarevich: an intimate portrait of the last days of the Russian imperial family compiled from the papers of Charles Sydney Gibbes now in the possession of George Gibbes. ''By J.C. Trewin. London: Macmillan, 1975. 148pp.
::Cambridge-educated Gibbes (1876-1963) was English tutor to the tsarevich and teacher to the four grand duchesses from the autumn of 1908 virtually until the murder of the imperial family in July 1918 in Ekaterinburg. Trewin’s narrative incorporates long extracts from notes and diaries kept by Gibbes during the period.

 
======K191======
'''Browning, Oscar, '''''Memories of later years. ''London: T. Fisher Unwin, 1923. 223pp.
::Browning (1837-1923), following his retirement as history don at King’s Cambridge, was invited to lecture on English literature and education at Petersburg university in the autumn of 1908 (pp. 116-19).

 
======K192======
'''Austin, Herbert Henry, '''''A scamper through the Far East, including a visit to Manchurian battlefields. ''London: Edward Arnold, 1909. xvi+336pp.
::Major Austin (1868-1937), of the Indian army and author of several books of exploration and travel, returns from leave in London via the Trans-Siberian in September 1908 (pp. 1-27). He left the train at Kharbin in order to visit the Russo-Japanese battlefields (pp. 28-164).

 
======K193======
'''Graham, Stephen,''' ''Part of the wonderful scene: an autobiography.'' London: Collins, 1964. 320pp.
::Graham (1884-1974), the most prolific and influential of British writers on Russia in the first decades of the twentieth century and proponent of Holy Russia, recalls his obsession with Russia, his first brief holiday in 1906, and his numerous visits from 1908 to 1917 to almost all of the regions of a vast country that spawned no less than nine books over the same period (pp. 14-149). (See [[#K232|K232]]-[[#K234|34]], [[#K247|K247]], [[#K248|K248]], [[#K276|K276]]-[[#K277|77]], [[#K289|K289]], [[#K352|K352]].)

 
======K194======
'''Bates, Lindon Wallace, Jr., '''''The Russian Road to China''. London, Constable and Co., 1910. 391pp.
::An American engineer, who had first been to Russia in 1896, Bates (1883-1915) travels from Russia to China during 1909, first by the Trans-Siberian Railway and then by sledge through Transbaikalia (pp. 1-172). A subsequent chapter provides political and ethnological observations on Russia’s status in the world (pp. 273-321).

 
======K195======
'''Taft, Marcus Lorenzo''', ''Strange Siberia along the Trans-Siberian railway: a journey from the Great Wall of China to the skyscrapers of Manhattan''. New York: Eaton &amp;amp; Mains, 1911. 260pp.
::Following years as a missionary in China, Taft (1850-1936) describes a final journey on the Trans-Siberian with his wife and daughter in the spring of 1909. He draws attention to the large Lutheran communities encountered along the railroad and comments on such topics as the steppes, architecture, and differences in American and Russian entrepreneurs. Their journey ends with quarantine for cholera upon leaving St Petersburg on 19 June, and an apology regarding the removal by the censors of six pages on Russia’s policy towards the Jews.

 
======K196======
'''Loew, Charles E., '''''Reminiscences of the Nordland, or, glimpses of Scandinavia, Russia, Germany and the Netherlands''. New York: D.T. Bass, 1910. 322pp.
::Loew describes a tourist trip he and a group of friends made to Russia in August-September 1909 (?), visiting the sights of St Petersburg and Moscow, before moving on to Berlin (pp. 157-258).

 
======K197======
'''Aflalo, Frederick George, '''''An idler in the near East. ''London: John Milne, 1910. xvi+279pp.
::Angler and naturalist, Aflalo (1870-1918) took a cruise on the Black Sea in the summer of 1909 after a long stay in Turkey and the Holy Land. He disembarked at Batumi and went to Tiflis, which he regarded as “a wonderful monument of bluff” (pp. 241-60).

 
======K198======
'''Hubback, John,''''' Russian realities, being impressions gathered during some recent journeys in Russia.'' London: John Lane, The Bodley Head, 1915. xvi+279pp.
::Recollections of eleven short journeys to Russia undertaken by the author between 1909 and 1914, frequently accompanied by his wife. He travelled extensively through southern Russia – Ukraine, the Crimea, the Caucasus, and down the Volga.

 
======K199======
'''Hone, Joseph Maunsell and Dickinson, Page Lawrence''','' Persia in revolution. With notes of travel in the Caucasus''. London: Fisher Unwin, 1910. xvi+218pp.
::Account by Hone (1882-1959) and Dickinson (b. 1881) of their travels through Transcaucasia and Persia during 1909, largely written by the former. They describe their outward journey from Warsaw to Resht (pp. 1-13) and their return from Persia, reaching Baku by steamer from Enzeli, and travelling through western Georgia to Kutais and Batumi. They note in particular signs of political unrest and growing Georgian nationalism (pp. 144-218).

 
======K200======
'''Hoover, Herbert,''' ''The memoirs of Herbert Hoover: years of adventure 1874-1920''. New York: The Macmillan Company, 1951. xii+496pp.
::Hoover (1874-1964), thirty-first President of the United States, recalls the short yearly visits he paid to Russia between 1909 and 1914, when as a freelance mining engineer he was involved in projects in the Urals and later Altai mountains (pp. 102-09).

 
======K201======
'''Ingham, Ernest Graham, '''''From Japan to Jerusalem.'' London: Church Missionary Society, 1911. viii+232pp.
::Rt Rev. Ingham (1851-1926), secretary of the Church Missionary Society and formerly bishop of Sierra Leone, accompanied by his wife travelled from London via Berlin to Moscow, where they arrived on 23 August 1909 and joined the Trans-Siberian which took them to Vladivostok by 4 September (pp. 6-18). They were en route for Japan and China to visit church missions and then made their way to India and Ceylon before returning home via Palestine and Egypt.

 
======K202======
'''Sara, Muriel, '''''Russia remembered. ''With a foreword by Cuthbert Bardsley. Hayle, Cornwall: for the author, 1971. 62pp.
::Wife of a geologist-consultant to an oil company, Mrs Sara, née Tiack (1885-1975) joined her husband in St Petersburg and travelled to the oil fields at an unspecified place in the Caucasus, where they remained until the outbreak of WWI. They left Russia via Scandinavia in the winter of 1915 (pp. 7-48).

 
======K203======
'''Dukes, Paul, '''''The unending quest: autobiographical sketches''. London: Cassell &amp;amp; Co., 1950. 260pp.
::Famed and knighted for his exploits as a British secret agent in Russia following the October Revolution, recounted in his ''Story of “ST 25” ''(1938), Sir Paul (1889-1967) had arrived in the Russia empire for the first time in the summer of 1909, teaching English in Riga, before moving to St Petersburg, where he studied music at the conservatoire and became immersed in the musical and artistic life of the capital. He studied piano under Professor Anna Esipova and became close to the Petersburg-born conductor Albert Coates, as well as dabbling in fashionable spiritualism and working during WWI in the British embassy under Ambassador Buchanan (pp. 17-115).

 
======K204======
'''Harrison, Ernest John, '''''Peace or war, east of Baikal.'' Yokohama: Kelly &amp;amp; Walsh, 1910. 563pp.
::Leading English expert on judo and later author of anti-Soviet novels, Harrison (1873-1961) was sent to Russia as a special correspondent of the Yokohama ''Japan Herald'' in 1909 to investigate the likelihood of a future Russo-Japanese or American-Japanese conflict. He spent the month of September travelling through eastern Siberia from Chita to Khabarovsk, before returning to Japan from Vladivostok (pp. 63-211).

 
======K204a======
'''Bowra, Cecil Maurice,''' ''Memories, 1898-1939.'' London: Wiedenfeld and Nicholson, 1966. [vi]+369pp.
::In September 1909 the young Bowra (1898-1971) returned to China, where he had been born and where his father worked for Chinese Customs, travelling on the Trans-Siberian and delighting in the countryside (pp. 18-19). In May 1916 he accompanied his mother and siblings to Beijing, again on the Trans-Siberian, and returned by the same route in September, but stayed for three weeks in Petrograd in the flat of ''The Times'' correspondent Robert Wilton (see [[#K132|K132]]), gaining some knowledge of Russian and enjoying the rich musical and literary life of the capital (pp. 60-9). After service in France during WWI, Bowra began his illustrious career at Oxford, serving as Warden of Wadham and Vice-Chancellor of the University and knighted in 1951, a renowned and productive Classical scholar, who also edited two books of Russian verse (1943, 1948).


======K205======
'''Latimer, Robert Sloan, '''''With Christ in Russia''. London: Hodder and Stoughton, 1910. x+239pp.
::Latimer followed his books on Dr Baedeker (see [[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander II (1855-1881)#I143|I143]]) and ''Under three tsars'', a study of religious movements in the post-Crimean war period, with his own experiences in Russia in 1909, although offering it principally as a biography of the Russian Stundist Dr Wilhelm Fetler, his close friend and companion. Probing the religious situation in contemporary Russia, Latimer travelled extensively, visiting not only St Petersburg and Moscow but also Kiev, Tiflis and Baku.

 
======K206======
'''Etherton, Percy Thomas, '''''Across the roof of the world: a record of sport and travel through Kashmir, Gilgit, Hunza, the Pamirs, Chinese Turkistan, Mongolia and Siberia. ''London: Constable and Co., 1911. xvi+437pp.
::Lt. Etherton (b. 1879), Indian army and F.R.G.S., embarked on his four-thousand mile journey from Lansdowne in the Himalayas in the spring of 1909, drawn by the politics of the region but more by the opportunities of shooting the wild sheep that had attracted the guns of his compatriots for the past two decades. He eventually entered Russian territory from Mongolia, following the river Irtysh, early in January 1910. He proceeded across the steppes by sledge via Ustkamenogorsk towards Barnaul to join the Trans-Siberian at Novonikolaevsk on 17 February (pp. 395-429).

 
======K207======
'''Kemp, Emily Georgiana, '''''The face of Manchuria, Korea &amp;amp; Russian Turkestan''. London: Chatto &amp;amp; Windus, 1910. xvi+248pp.
::Miss Kemp (b. 1860), F.R.G.S. and author of a book on China, accompanied by a friend, Miss MacDougall, decided to observe the present state of Manchuria and Korea under the twin threats of Japan and Russia. They travelled out on the Trans-Siberian on 1 February 1910 and back again four months later, when they decided to make a detour from Samara down to Turkestan, visiting Bokhara and Samarkand before travelling home across the Caspian to Baku and Tiflis and Vienna (pp. 151-240).

 
======K208======
'''Bax, Arnold, '''''Farewell, my youth''. London: Longmans, Green &amp;amp; Co. 1943. 112pp.
::The visit of the famed English composer Sir Arnold (1883-1953) to Russia in the late spring and summer of 1910 was in pursuit of the daughter of a wealthy Ukrainian landowner, Liubov’ (Liuba) Nikolaevna Korolenko, who sadly did not return his love. Arriving in Petersburg from Lausanne in April, he travelled on to Moscow and then to the Korolenko estate near Lubny in Ukraine (pp. 63-79).

 
======K209======
'''Goodrich, Joseph King, '''''Russia in Europe and Asia. ''Chicago: A.C. McClurg &amp;amp; Co., 1912. x+302pp.
::After several years teaching in the Imperial Government College, Kyoto, the American Goodrich (1850-1921), who seems to have visited Vladivostok for the first time in 1899, travelled from Japan in July 1910 and described in detail his fifteen-day journey on the Trans-Siberian to Moscow (pp. 112-29). His attempt at a comprehensive account of Russia is primarily based on secondary literature and discussions Goodrich had with Russians and those involved in Russian affairs while he was based in Japan.

 
======K210======
'''Hertz, Carl,''' ''A modern mystery merchant: the trials, tricks and travels of Carl Hertz, the famous American illusionist''. London: Hutchinson &amp;amp; Co., 1924. 319pp.
::Hertz (1859-1924), born in San Francisco to a Russian father and Polish mother, writes of his visit to Russia in the summer of 1910. The magician and his wife (stage name Emilie D’Alton) performed for two months at the ''Iar'' restaurant in Moscow, before visiting St Petersburg, where he apparently gave a command performance before the tsar and tsaritsa (pp. 227-31).

 
======K211======
'''Washington, Booker Taliaferro, '''''The man farthest down: a record of observation and study in Europe.'' With the collaboration of Robert E. Park. London: T. Fisher Unwin, 1912. 390pp.
::In a chapter entitled ‘A Russian border village’ the African-American political leader and author (1856-1915) on a European tour describes his brief visit in August 1910 to a village called Barany on the Russian side of the Austrian Poland/Russian Poland border and compares the status of the Russian peasant with that of the Mississippi Afro-American farmer (pp. 276-95).

 
======K212======
'''Crawford, Laura MacPherson, '''''Dear family: the travel letters and reminiscences of Laura MacPherson Crawford. ''Edited by Ruth Saunders. Claremont, California: privately printed, 1946. xx+360pp.
::Canadian society wife and later Red Cross worker Mrs Crawford describes in a letter home her and her husband’s visit to St Petersburg and Moscow, as well as to the Ponafidin estate near Ostashkov (see [[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J40|J40]]) in May 1910 (pp. 107-16).

 
======K213======
'''Eddy, Sherwood, '''''A pilgrimage of ideas; or the re-education of Sherwood Eddy.'' New York: Farrar &amp;amp; Reinhart Inc., 1934. xiv+336pp.
::Intellectual autobiography of the American Protestant missionary and prolific author (1871-1963), who visited Russia some ten times between 1910 and 1930 and had published an analytical work entitled ''The challenge of Russia'' in 1931. In the final chapter he briefly discusses his two visits to imperial Russia in 1910 and 1912, before turning to his experiences during the Soviet period. In 1912 he attended student meetings in Kiev, Moscow and St Petersburg, noting an epidemic of suicide (pp. 313-23).

 
======K214======
'''Wood, Ruth Kedzie, '''''Honeymooning in Russia. ''London: T. Fisher Unwin, 1911. vi+341pp.
::The honeymoon in Russia of an American couple, Mr and Mrs Philip D. Houghton, in the summer of 1910, related as much in dialogue as in descriptive prose by Joyce, Mrs Houghton, the alter ego, one assumes, of Ruth Wood (1880-1950), the author. It is nonetheless an informed and detailed tourist’s visit to the attractions of St Petersburg, Vologda, and Iaroslavl’, where they take the steamer down the Volga to Nizhnii Novgorod. They travel to Moscow and from there, through Ukraine to the Crimea and Odessa. They then journey to Kiev and exit to Warsaw. Contains Wood’s translations of poems by such as Krylov, Nikitin, Nekrasov, Lermontov, and Kozlov.

 
======K215======
'''Christie, Isabella [Ella] Robertson, '''''Through Khiva to golden Samarkand: the remarkable story of a woman’s adventurous journey alone through the deserts of Central Asia to the heart of Turkestan''. London: Seeley, Service &amp;amp; Co., 1925. 280pp.
::Renowned Scottish traveller and gardener, Christie (1861-1949), F.R.G.S., describes on the basis of her diaries her two expeditions as sole female traveller to Russian Turkestan in 1910 and 1912. Her first journey was from Constantinople to Andijan (pp. 1-217); and the second from St Petersburg to Khiva, which she was the first British woman ever to enter (pp. 218-63).

 
======K216======
'''McCaig, Archibald, '''''Wonders of grace in Russia.'' Riga: The Revival Press, 1926. 251pp.
::Dr McCaig, Principal of Spurgeon’s College in London, went to Russia for the first time in June 1910, accompanying the evangelical pastor William Fetler (author of ''The Stundist in Siberian exile and other poems'') whom he had initially met as a student at the college, to initiate the building of the missionary society’s tabernacle in St Petersburg. He also visited Novgorod, Schüsselburg and Moscow. He returned to Russia for the opening of the tabernacle in 1912 and paid two further visits in June 1912 and 1913 (pp. 11-185). His book is largely based on articles he wrote for various church periodicals during this period.

 
======K217======
'''Dobson, George, '''''St. Petersburg.'' London: Adam &amp;amp; Charles Black, 1910. xii+158pp.
::The first of three Russian guides commissioned by the Blacks, all enhanced by the paintings of the Belgian artist F. de Haenen (see [[#K246|K246]], [[#K261|K261]]). A lively contemporary account by the former long-time ''Times ''correspondent in St Petersburg (see [[#J78|J78]]).

 
======K218======
'''Garstin, Denis, '''''Friendly Russia''. With an introduction by H.G. Wells. London: T. Fisher Unwin, 1915. 248pp.
::Garstin (1890-1918) first went out to Russia as a tutor in about 1910, after leaving Cambridge, and living initially, it would seem, in the Crimea (pp. 15-184). The final section of his book (based largely on articles he contributed in 1913-14 to newspapers such as the ''Morning Post'' and the ''Daily News'') was entitled ‘The Russians in war’ (pp. 185-248). After service in WWI, Garstin returned to Russia with the British Propaganda unit (under Hugh Walpole) and lost his life in north Russia, fighting against the Bolsheviks.

 
======K219======
'''Price, Morgan Philips, '''''Siberia.'' London: Methuen &amp;amp; Co., 1912. xviii+308pp.
::Price (1885-1973), who was to write several books on Russia (see [[#K322|K322]], [[#K379|K379]]-[[#K381|81]]), provides a description of life in western and central Siberia, based on a trip undertaken during the spring and summer of 1910. In the first half of the book Price uses his stay in a number of locations to explore the social, religious and economic aspects of Siberia: Krasnoiarsk inspires an account of municipal life in a growing commercial town; the provincial town of Minusink allows him to explore a life less affected by “Western commercialism”; and the frontier village of Kushabar, on the north side of the Mongolian frontier on the Upper Enisei presents the life of the frontier peasant.

 
======K220======
'''Curtis, William Eleroy, '''''Turkestan: “the heart of Asia”.'' London: Hodder &amp;amp; Stoughton, 1911. 344pp.
::Curtis, who had first visited Russia in the reign of Alexander III (see [[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J68|J68]]), spent the spring and early summer of 1910 in Turkestan, writing a series of detailed descriptions of the region that were to appear first in the ''Chicago Record-Herald''. The photographs were taken by John T. McCutcheon during his earlier visit to Turkestan in the summer of 1906.

 
======K221======
'''Curtis, William Eleroy, '''''Around the Black Sea, Asia Minor, Armenia, Caucasus, Circassia, Daghestan, the Crimea, Roumania. ''London: Hodder &amp;amp; Stoughton, 1911. 456pp.
::Essentially a sequel both in time and manner to his previous book and similarly composed from letters sent to a Chicago paper. Curtis spent the rest of the summer and the autumn of 1910, visiting countries adjacent to the Black Sea, including Georgia (pp. 85-128), Baku and Daghestan (pp. 214-51), Odessa and the Crimea (pp. 252-347).

 
======K222======
'''Simpson, Eugene E., '''''Eugene E. Simpson’s'' ''Travels in Russia, 1910 and 1912. ''Taylorville, Ill.: for the author, 1916. 126pp.
::Based on articles Simpson (1871-1929) sent to the New York ''Musical Courier'' during visits down the Volga and to the Crimea in the two summers. Simpson was an enthusiast of Russian music, classical and folk, and an admirer of Tchaikovsky, whose home at Klin he visited.

 
======K223======
'''Lied, Jonas, '''''Return to happiness. ''London: Macmillan, 1943. xii+318pp. [American edition entitled: ''Prospector in Siberia: the autobiography of Jonas Lied.'' New York: Oxford University Press, 1945.]
::The Norwegian entrepreneur (1881-1969), inspired by the example of Captain Wiggins, paid his first of many visits to Russia in May 1910 to investigate trading possibilities in Siberia on behalf of a London firm. In 1913 he sailed to the Kara Sea with Nansen. Thereafter he made frequent trips both to St Petersburg and Siberia, meeting important Russian businessmen, political figures, and the tsar. The October revolution brought his highly successful shipping and trading business to an end, but he continued to visit and live in the Soviet Union until 1931 (pp. 52-293).

 
======K224======
'''Lied, Jonas, '''''Siberian Arctic: the story of the Siberian Company.'' London: Methuen, 1960. 217pp.
::Lied’s history of the Siberian Company, which he helped found, repeats much of what appears in his autobiography but expectedly provides less detail about his own actions (pp. 50-111).

 
======K225======
'''Buchanan, George, '''''My mission to Russia and other diplomatic memories.'' London: Cassell &amp;amp; Co., 1923. 2 vols.
::After relating his career as a diplomat from 1876, Sir George (1854-1924) describes in detail his long years as British ambassador in Petersburg from September 1910 to January 1918 (vol. I, pp. 91-253; II, pp. 1-248).

 
======K226======
'''Buchanan, Meriel, '''''Diplomacy and foreign courts.'' With an introduction by Sir Bernard Pares. London: Hutchinson &amp;amp; Co., 1928. 288pp.
::A devoted daughter’s parallel account of her father’s postings and a vigorous defence of his character and actions during WWI and the revolutionary events of 1917. Meriel (1886-1959), from 1925 Mrs Knowling, also provides descriptions of the social life of high society in St Petersburg and her own experiences of the effects of war and revolution (pp. 133-243).

 
======K227======
'''Buchanan, Meriel, '''''Ambassador’s daughter. ''With a foreword by Sir Robert Bruce Lockhart. London: Cassell &amp;amp; Co., 1958. ix+239pp.
::In the last of the books she devoted to Russia and published in the year before her death, Meriel returns to her defence of her father’s conduct, reworking much already familiar material (pp. 89-194).

 
======K228======
'''Buchanan, Meriel, '''''Recollections of imperial Russia.'' London: Hutchinson &amp;amp; Co., 1923. vii+277pp.
::Essentially a re-hash of previous books which soon goes beyond personal memories of her arrival and early years in Petersburg to unoriginal histories of eighteenth-century rulers and spun-out descriptions of places with minimal personal input.

 
======K229======
'''Buchanan, Meriel, '''''The dissolution of an empire.'' London: John Murray, 1932. 304pp.
::Meriel herself acknowledges that she might be “accused by some people of repeating myself to a monotonous degree”, and this book, tracing her family’s stay in Russia from 1910 to 1918 with added material for the following years, was not to be the last example.

 
======K230======
'''Buchanan, Meriel,''' ''Victorian gallery''. London: Cassell &amp;amp; Co., 1956. x+219pp.
::Includes chapters devoted to people, Russian and British, whom she knew in Russia, such as Princess Zinaida Iusupova and Sir Henry Wilson (pp. 28-67, 103-95).

 
======K231======
'''Baring, Maurice, '''''The Russian people. ''London: Methuen &amp;amp; Co., 1911. xix+366pp.
::A history and geography of Russia, based on the reading of many secondary sources, but also a summation of Baring’s years of studying the Russian people in situ and constantly illuminated by his own anecdotes and observations. His intention was “to supply the average reader with an introduction to the course of Russian affairs”.

 
======K232======
'''Graham, Stephen, '''''A vagabond in the Caucasus. With some notes of his experiences among the Russians. ''London: John Lane, 1911. vii+311pp.
::Graham’s long years “tramping” throughout Russia (and later America) began with his travels through the Caucasus in 1910. After spending a long winter in Kharkov, he remained in Moscow until Easter (to p. 116). The rest of the book concerns his time in the Caucasus as he moved from Vladikavkaz through the Gorge of Dariel to reach, months later, Tiflis. Amusingly, his book finishes with “a chapter for prospective tourists” (pp. 301-08).

 
======K233======
'''Graham, Stephen, '''''A tramp’s sketches. ''London: Macmillan, 1912. xiii+339pp.
::“Not so much a book about Russia as about the tramp”, it is a paean to the joys of unfettered wanderings through the south of Russia, by the Black Sea, into Georgia, in the Crimea, and accompanying peasant pilgrims to Jerusalem (see [[#K247|K247]]).

 
======K234======
'''Graham, Stephen, '''''Undiscovered Russia.'' London: John Lane, 1912. xvi+337pp.
::The particular area of Russia Graham was “discovering” for his readers was the north to which he had travelled from the Caucasus by way of Moscow in 1911. From Archangel he made long expeditions along the courses of the rivers Pinega and Dvina. When he finally left Archangel he returned to Moscow via Vologda and Kostroma, finishing his tramping at Vetluga, before making a detour to Rostov.

 
======K235======
'''Donner, Kai [Karl] Reinhold, '''''Among the Samoyed in Siberia''. Translated by Rinchart Kyler. Edited by Genevieve A. Highland. New Haven: Human Relations Area Files, 1954. xx+176pp.
::The Finnish linguist, ethnographer, and pioneer of Finno-Ugrian studies (1888-1935) describes two trips he made to Siberia, the first from August 1911 to June 1913 and the second in June-October 1914, illustrated by scores of his own photographs. He first travelled along the upper reaches of the Ob and the Enisei, describing the life of the Samoed peoples with whom he lived; and on his second journey he reached the northern slopes of the Saian mountains of south-eastern Siberia. The original account, ''Bland Samojeder i Sibirien åren 1911-1913, 1914'' was first published in 1915.

 
======K236======
'''Digby, George Bassett,''' ''Tigers, gold, and witch-doctors''. London: John Lane, (The Bodley Head), 1928. 341pp.
::Digby (1888-1962), F.R.G.S., offered an account of “things that I found out in the course of my wanderings in Siberia”, including not only those in his title, but also bears, wolves, mammoths, and much else. Well-read in earlier accounts of Siberia, he offers a lively narrative of several undated journeys in the years before WWI.

 
======K237======
'''Wright, Richardson Little, and Digby, George Bassett, '''''Through Siberia: an empire in the making. ''London: Hurst and Blackett, 1913. viii+260pp.
::American journalist Wright (1887-1961) and British prospector Digby decided to travel “with the Russians” by third-class slow train from Moscow to Siberia in the spring of 1911. They went on to Kharbin and into Japanese Manchuria. In many ways a familiar route, but with some unusual encounters and interesting detail.

 
======K238======
'''Herbert, Agnes,''' ''Casuals in the Caucasus: the diary of a sporting holiday''. London: John Lane, 1912. xii+331pp.
::Accompanied by her cousins Cecily Windus and Kenneth Baird (of the Petersburg Bairds), Miss Herbert sailed from Gibraltar to Batumi to “shoot a little, climb a little” during the summer of 1911 in the Caucasus. A frothy, gossipy travelogue from an author who had already shot her way around Somaliland and Alaska (pp. 27-331).

 
======K239======
'''Phelps, William Lyon, '''''Autobiography with letters''. London: Oxford University Press, 1939. xxiii+986pp.
::Phelps (1865-1943), Yale professor and author of ''Essays on Russian novelists'' (1917), paid his first and only visit to Russia with his wife in September 1911. They visited St Petersburg and Moscow, but only Nevskii Prospekt seems to have impressed him (pp. 522-28).

 
======K240======
'''Perry-Ayscough, Henry George Charles, and Otter-Barry, Robert Bruère,''' ''With the Russians in Mongolia''. With a preface by Sir Claude Macdonald. London: John Lane, The Bodley Head, 1914. xxiv+344pp.
::Perry-Ayscough of the Chinese Postal Service, F.R.G.S., and Captain Otter-Barry (b. 1879) of the Royal Sussex regiment, F.R.G.S., travelled through Mongolia at different times, the latter, in 1911, just before the Chinese Revolution, the former, in 1913, when Mongolia was already under Russian protection. Collaborating on the general chapters devoted to Mongolian history and Russo-Mongolian relations, the authors provide independent accounts of their travels through Mongolia and parts of Siberia. Only the final pages of Otter-Barry’s contribution concern Siberia as he passes through the border town of Kiakhta to Verkhne-Udinsk to meet his wife from the Trans-Siberian on 12 July 1911 (pp. 175-84). In February 1913 Perry-Ayscough travelled from China through Manchuria and went by train to Verkhne-Udinsk on his way to the Mongolian capital (pp. 189-94). In April, leaving for England, he crossed into Siberia at Kosh Agach, proceeded to Biisk, and from there by steamer to Novonikolaevsk to catch the Trans-Siberian (pp. 241-94).

 
======K241======
'''Shaft, Arthur, '''''My Russian and English connections''. Broadstone: for the author, 1993. xiii+154pp.
::Shaft (b. 1911), the youngest son of Anglo-French parents, both of whom had also been born in Russia, remembers his happy early years in Moscow and on an estate near Rzhev and then the increasingly difficult times under Soviet rule until the family was allowed to depart for England via Finland in the spring of 1920 (pp. 1-61).

 
======K242======
'''Lockhart, Robert Hamilton Bruce, '''''Memoirs of a secret agent, being an account of the author’s early life in many lands and of his official mission in 1918. ''London and New York: Putnam, 1931. xii+355pp.
::Lockhart (1887-1970) arrived in Moscow in January 1912 as British vice-consul, but was to serve throughout WWI as acting consul-general. His narrative of his life, or lives, “Russian and unofficial” and “official and mainly English”, over the next five years is anecdotal, often amusing, and informative. In early September 1917 he was recalled to London (pp. 53-192). The second half of the book is devoted to his more “famous” exploits after his return to Soviet Moscow in January 1918, his arrest in September for involvement in the alleged anti-Bolshevik “Lockhart plot”, and expulsion in early October.

 
======K243======
'''Lockhart, Robert Hamilton Bruce, '''''My Europe. ''London: Putnam, 1952. x+273pp.
::In the opening two chapters, ‘Moscow before the wars’ and ‘Prelude to revolution’, Lockhart recalls his Moscow years from 1912 to 1917 (pp. 3-28).

 
======K244======
'''Lockhart, Robert Hamilton Bruce,''''' Giants cast long shadows. ''London: Putnam, 1960. 253pp.
::In his collection of essays devoted to distinguished people in all walks of life is ‘Missionaries of sport’, in which Lockhart recalls Lancashire-born Clem Charnock, credited with introducing soccer into Russia in 1887 and members of the Charnock clan with whom he played for the “Morozovtsy” in Moscow in 1912 (pp. 172-80).

 
======K244a======
'''Lockhart, Robert Hamilton Bruce''', ''The diaries of Sir Robert Bruce Lockhart''. Edited by Kenneth Young. London: Macmillan, 1973-80. 2 vols.
::Lockhart’s brief diary entries from January 1915 to his departure in September 1917 with gap between March 1916 and March 1917 (vol. I, pp. 21-9. On Moscow and a vsit to Kiev.  Glimpses of such as Gor’kii and Kerenskii (to whom Lockhart was to devote a chapter entitled ‘Russian optimist’ in his book ''Friends, foes, and foreigners'' (1957), pp. 107-18).


======K245======
'''Young, Ernest, '''''From Russia to Siam, with a voyage down the Danube: sketches of travel in many lands. ''London: Max Goschen, 1914. xii+328pp.
::Young (1869-1952), travel writer, published a book on Finland in 1912. It was during his sojourn that he sailed from Sortovala (then in the Grand Duchy) to visit the Russian Orthodox monastery on the island of Walamo/Valaam in Lake Ladoga. He spent five days in a guest cell, interviewing the monks and observing their way of life (pp. 3-35).

 
======K246======
'''Grove, Henry Montgomery, '''''Moscow.'' London: Adam and Charles Black, 1912. viii+142pp.
::The companion to Dobson’s book on St Petersburg (see [[#K217|K217]]), Grove’s follows the same mixture of historical and contemporary commentary to accompany de Haenen’s paintings. (see also [[#K261|K261]]). Grove was the long-serving British consul-general in Moscow.

 
======K247======
'''Graham, Stephen, '''''With the Russian pilgrims to Jerusalem.'' London: Macmillan and Co., 1913. x+306 pp.
::It was in Constantinople in 1912 that Graham joined the boat containing some 500 Russian peasant pilgrims bound for Jaffa and then travelling on to Jerusalem but it was as if he found himself “in a populous Russian village on a market day”. Thus not strictly a travelogue through Russia, it merits inclusion for its vivid evocation of Russian peasants and their stories. He returned to Odessa after Easter to begin more solitary tramping.

 
======K248======
'''Graham, Stephen, '''''Changing Russia.'' London: John Lane, 1913. ix+309pp.
::“The journal of a tramp” through southern parts of Russia in 1912, mainly to Batumi via such resorts as Sochi and Sukhumi, and, later, through the Crimea. Written, allegedly, “with an eye to the ways and thoughts of the Intelligentsia” during a period of rapid change.

 
======K249======
'''Cripps, Frederick Heyworth, '''''Life’s a gamble. ''With a foreword by Lord Burnham. London: Odhams Press, 1957. 208pp.
::The hon. Fred Cripps, later 3rd Lord Parmoor (1885-1977), went to Russia in about 1912 as a merchant banker and he remained until the summer of 1914, when he returned to England to join the army. Although his office was in Peterburg, he travelled to other parts of Russia, including the Urals. Among his close friends was Shaliapin (pp. 81-97). In 1919 he went to Soviet Moscow and over the next five or six years was engaged in an astonishing variety of business and leisure activities, including organizing his own ballet company (pp. 127-55).

 
======K250======
'''Lee, Helena Crumett, '''''[https://archive.org/details/acrosssiberiaal00leegoog Across Siberian alone: an American woman’s adventures]''. London: John Lane, Bodley Press, 1914. 220pp.
::After attending her daughter’s wedding in Shanghai, Mrs Lee sails to Dalnii in September 1912 to begin her long lone train journey to Moscow, where her account ends. Boarding the Trans-Siberian Railway (or the Chinese Eastern Railway extension) at Chang Chung, she travels to Irkutsk, where she breaks her journey and dines with exiles and discusses the Siberian prison system. She then makes a detour to Tomsk and visits the university. Her declared aim was to spread knowledge in America of Siberia (pp. 40-220).

======K251======
'''Fraser, Eugenie, '''''A house by the Dvina: a Russian childhood. ''Edinburgh: Mainstream Publishing Co., 1984. ii+281pp.
::Daughter of a Russian father and a Scottish mother, Eugenie, née Sholts (1905-2002) was brought up in Archangel. She relates her family story from 1912 to 1920, when she and her mother escaped to Scotland. She and her husband visited the family home in 1972, described in ''The Dvina remains'' (1996).

 
======K252======
'''Bury, Herbert, '''''Russian life to-day.'' London, A.R. Mowbray &amp;amp; Co., 1915. vii+270pp.
::Bury (1853-1933), Anglican bishop of Northern and Central Europe, travelled extensively throughout Russia in 1912-14, visiting Anglican congregations in western Russia and beyond the Urals as far as Petropavlovsk, and into the Kirghiz steppe. Offers his book as an “impressionistic description of Russian life” and includes a very respectful audience with the tsar at Tsarskoe selo.

 
======K253======
'''Wood, Ruth Kedzie, '''''The tourist’s Russia. ''London: Andrew Melrose, 1912. viii+253pp.
::The first of several contributions by Wood (see [[#K214|K214]]) to the ‘Tourist’s’ series (Spain, California, etc.), adapted for both American and British publics. It is her substitute for an English-language Baedeker, the lack of which she regretted in her earlier book but which was to appear in 1914 (see [[#K271|K271]]).

 
======K254======
'''Steveni, William Barnes, '''''Things seen in Russia''. Seeley, Service &amp;amp; Co., 1913. 260pp.
::A contribution to the popular small-format ‘Things seen’ series that includes the statutory, but delightful, fifty illustrations, written by a long-time British resident of St Petersburg (see [[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J111|J111]], [[#K266|K266]]), who concentrates mainly on the capital, Moscow and Kiev.


======K254a======
'''Butler, Frank Hedges''', ''Through Lapland with skis &amp; reindeer with some account of ancient Lapland and the Murman coast''. London: T. Fisher Unwin, 1917. xii+386pp.
::Butler (1855-1928) F.R.G.S., was already well-known for his many roles as intrepid balloonist, treasurer of the Royal Automobile Club, founder of the Royal Aero Club, and travel writer, when he undertook his journey to Lapland in March 1913. Accompanied by his Lapp interpreter Johann Thürri, Butler travelled through Norway and Lapland and entered Russian Lapland en route for the monastery at Pechenga, the history of which is described in detail (pp. 141- 91).


======K255======
'''Forse, Edward John George,''' ''From Warsaw to Moscow: from the travel diaries of Edward J.G. Forse''. Southbourne: for the author. xvipp.
::Rev. Forse (1877-1942), F.R.G.S., vicar of Southbourne, Bournemouth and author of several books of travel and art history, describes in some detail his train journey, begun in Warsaw on 3 September 1913, to the old Russian capital. He explores the streets and famous, mainly religious, buildings of Moscow, which he leaves on 8 September, at which point his narrative comes to an abrupt end.

 
======K256======
'''Ransome, Arthur, '''''The autobiography of Arthur Ransome. ''Edited, with prologue and epilogue, by Rupert Hart-Davis. London: Jonathan Cape, 1976. 368pp.
::Author and journalist Ransome (1884-1967) first went to Russia in June-September 1913 to study the language and folklore and returned in May-August of the following year to write a guide to St Petersburg (pp. 159-70). On 30 December 1914 he was back in newly-named Petrograd and thereafter in and out of Russia (and later the Baltic states) until 1924, reporting war and revolution, writing, fishing, and wooing Trotskii’s secretary (pp. 159-319).

 
======K257======
'''Le Blond, Elizabeth, '''''Day in, day out''. [With a foreword by E.F. Benson.] London: John Lane, 1928. 264pp.
::The three-times married Mrs Le Blond, née Hawkins-Whitsted (1861-1934), the greatest lady mountaineer of the age, not to mention her prowess as cyclist and car driver, accompanied her husband on a tour that took them to China and Korea in 1912 and then by the Trans-Siberian railway to Moscow in June 1913. Her husband obliged to return to England, she remained sightseeing in Moscow and then St Petersburg (pp. 165-77).

 
======K258======
'''Wheeler, William Webb, '''''The other side of the earth. ''St. Joseph, Missouri: privately printed, 1913. 208pp.
::American merchant and author of various travel accounts, Wheeler (1845-1925) passed from Manchuria into Russia in June 1913 and travelled east on the Trans-Siberian. He proceeded to St Petersburg and then travelled to Moscow, noting the usual sights (pp. 169-201).

 
======K259======
'''Shelley, Gerard, '''''The blue steppes: adventures among Russians. ''London: John Hamilton, 1925. 268pp.
::Shelley (b. 1892) produced two versions of his memoirs of his years in Russia from 1913, when he was invited by Count and Countess Torlov to stay on their estate near Kharkov, to 1920, when he escaped from Petrograd to Finland. He spent time in Petrograd and Moscow and also visited the Crimea before the revolution. A linguist who quickly acquired Russian and worked for some time as an interpreter for the Russians during WWI, Shelley also translated the Russian poets, including Pushkin, Lermontov, and Blok.

 
======K260======
'''Shelley, Gerard, '''''The speckled domes: episodes of an Englishman’s life in Russia. ''Duckworth, 1925. 256pp.
::A less sprightly version of Shelley’s adventures, but also different in other respects, not least in the manner of his ultimate escape, here disguised as a woman! In both accounts there is much on his acquaintance with Rasputin, whom he met for the first time in April 1915 and whom he defended.

 
======K261======
'''Stewart, Hugh, '''''Provincial Russia. ''London: Adam and Charles Black, 1913. viii+173pp.
::The final volume in a series that was re-issued in the same year as a single volume under the title ''Russia'', displaying to the full the talent of the painter de Haenen (see [[#K217|K217]], [[#K246|K246]]). Stewart (1884-1934) who had travelled widely through Russia from about 1906, provided a succinct account of the various regions.

 
======K262======
'''Johnson, William Eugene, '''''The liquor problem in Russia. ''Westerville, Ohio: American Issue Publishing Co., 1915. 230pp.
::The American social reformer and prohibition campaigner Johnson (1862-1945) travelled to Russia in 1913 to do “some muckraking in connection with the vodka monopoly”, but found a welcome change in government attitudes towards alcohol production.

 
======K263======
'''Bruce, Henry James, '''''Silken Dalliance''. London: Constable, 1946. viii+183pp.
::After service in Vienna and Berlin Bruce (1880-1951) was posted as Head of Chancery to the British embassy in Petersburg at the end of August 1913 and remained there for five momentous years, before he and his wife, the ballerina Tamara Karsavina, whom he married in the Russian capital in 1915, finally left from Murmansk in July 1918. His memoirs are a curious mixture of personal adventures and observation and potted eighteenth-century Russian history (pp. 135-75).

 
======K264======
'''Bruce, Henry James, '''''Thirty-dozen moons. ''London: Constable and Co., 1949. 189pp.
::In his second book of memoirs Bruce describes his courting of Diagilev’s prima ballerina Karsavina from autumn 1913 to the following summer (pp. 1-10).

 
======K265======
'''Vecchi, Joseph, '''''“The tavern is my drum”: my autobiography. ''Preface by Negley Farson. London: Odhams Press, 1948. 224pp.
::The Italian restaurateur (d. 1961) moved from ''Claridge''’s in London to the ''Kaiserhof'' in Berlin, and in September 1913, on to the newly-opened hotel ''Astoria'' in St Petersburg, where he managed the French restaurant until the hotel was requisitioned in April 1916. He then worked at the ''Felicien'' and the ''Bear'', before going to Kiev’s ''Grand Hotel''. He returned to Petrograd in March 1917 and a final venture, ''The Little Palace''. At the end of 1917 he left his “beloved” Russia via Murmansk, eventually settling in London (pp. 29-152).

 
======K266======
'''Steveni, William Barnes, '''''The Russian army from within.'' London: Hodder &amp;amp; Stoughton, 1914. 184pp.
::Steveni (see [[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J112|J112]], [[#K254|K254]]) travelled from the capital to the Caucasus in 1913 to report on the state of the Russian army at the behest of London newspapers.

 
======K267======
'''Mears, John Henry, and Collyer, Charles B.D., '''''Racing the moon (and winning): being the story of the swiftest journey ever made, a circumnavigation of the globe by airplane and steamship in 23 days, 15 hours, 21 minutes and 3 seconds by two men and a dog''. New York: Rae D. Henkle Co., 1928. 320pp.
::It is not the journey of 1928, when Broadway producer Mears regained the world record, but his original journey in 1913 to gain the record for the first time that is relevant. Mears (1878-1956) left New York on 2 July 1913 and returned thirty-five days, twenty-one hours, thirty-five minutes, eighteen and four-fifths seconds later. In the course of that journey, by steamship and train, he passed, twelve days on, through St Petersburg and took the Trans-Siberian in Moscow to Omsk, then went into China (pp. 245-85).

 
======K268======
'''Nansen, Fridtjof Wedel-Jarlsberg, '''''Through Siberia, the land of the future. ''Translated from the Norwegian by Arthur G. Chater. London: William Heinemann, 1914. xvi+478pp.
::The Norwegian scientist and explorer Nansen (1861-1930), who was to win the Nobel Peace Prize in 1922, set out from Norway in August 1913 with, among others, Jonas Lied (see [[#K223|K223]]) to attempt “to open up a regular trade connexion with the interior of Siberia, via the Kara Sea and the mouth of the Yenisei”. They arrived back in Petrograd after penetrating as far as the Manchurian border, at the very end of October.

 
======K269======
'''Dickinson, Duncan, '''''Through Spain: the record of journey from St. Petersburg to Tangier, by way of Paris, Madrid, Cordova, Seville and Cadiz; and thence to Gibraltar, Ronda and Granada.'' London: Methuen &amp;amp; Co., 1914. xxiv+197pp.
::The author, it would seem, was born or raised in Russia (“home”) and thus in the summer of 1913 begins his long train journey to Spain at the Warsaw station in Petersburg, briefly describing the scenery on his way to the border (pp. 1-4).

 
======K270======
'''Bryce, James, '''''Memories of travel. ''London: Macmillan, 1923. 300pp.
::Posthumously published collection of travel sketches includes Lord Bryce’s recollections, written in 1922, of a journey on the Trans-Siberian in August-September 1913 with an excursion via Tomsk to the Altai mountains (pp. 254-95). Bryce had previously visited Russia in 1876 (see [[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander II (1855-1881)#I137|I137]]).

 
======K271======
'''Baedeker, Karl, '''''Russia with Teheran, Port Arthur, and Peking: handbook for travellers.'' London: George Allen &amp;amp; Unwin, 1914. lxiv+590.
::Published on the eve of WWI, the travellers’ indispensable ''vade mecum'', prepared with meticulous detail and accuracy by the “Baedeker” editor and his associates, “who have repeatedly explored the country with a view to procuring the latest possible information”. It was so soon to become a historical document.

 
======K272======
'''Keller, Otto, '''''St. Petersburg and its environs, Finland, Moscow, Kiev, Odessa, Warsaw, Riga, a tour on the Volga, the Crimea and the Caucasus, with plans of St. Petersburg and of the environs of St. Petersburg, railway map of Russia, sketches of the Hermitage and the museum of Alexander III''. London: Siegle, Hill &amp;amp; Co., 1914. ii+168pp.
::Keller (1838-1927) had been in resident in St Petersburg for ten years at the time of writing and the guide book is based on his personal experiences and specifically slanted for the English visitor.

 
======K273======
'''Williams, Harold Whitmore, '''''Russia of the Russians''. London: Sir Isaac Pitman &amp;amp; Sons, 1914. x+430pp.
::Although specifically written for the series ‘Countries and peoples’ and suppressing the personal element, it reflects the deep knowledge and love of pre-WWI Russia of the astonishing New-Zealand linguist and esteemed newspape''r'' correspondent. Hugely informative about all aspects of contemporary Russia, particularly the arts, it is especially notable for the chapter (pp. 389-424) on St Petersburg, where Williams (1876-1928) arrived for the first time in December 1904 and left finally in March 1918.

 
======K274======
'''Baring, Maurice, '''''The Mainsprings of Russia. ''London: Thomas Nelson and Sons, 1914. xi+328pp.
::An attempt to provide “a single idea of the more important factors in Russian life” based on Baring’s personal observations and travels over many years. The book is dedicated to H.G. Wells, recalling the time they spent together in St Petersburg in 1914.

 
======K275======
'''Byford, Charles Thomas''', ''The soul of Russia.'' London: Kingsgate Press, 1914. 396pp.
::Based on his extensive travels through Russia, including the Baltic States and the Crimea during presumably the early years of the century, Byford’s study aimed to present “a concise view of the spiritual and religious forces at work” in contemporary Russia, with an overarching theme of the growth in religious liberty in the Russian Empire. Each chapter, devoted to the Orthodox church and to sects such as the Dukhobors and Molokans, is prefaced by a list of secondary sources he has consulted.

 
======K276======
'''Graham, Stephen, '''''The way of Martha and the way of Mary. ''London: Macmillan, 1915. xii+291pp.
::In his quest for the essence of Eastern Christianity Graham travelled from Paris to Kiev in January 1914 and then to Moscow, where he completed his book in September of the following year. He visited the Convent of Martha and Mary in Moscow to see Nesterov’s painting of the saints, who embodied for him the paths of faith and service. In the third and final section he describes his journey in May 1915 to Egypt to visit monasteries and shrines and thence “to make a journey to Russia the way Christianity came to her”.

 
======K277======
'''Graham, Stephen, '''''The death of yesterday. ''London: Ernest Benn, 1930. iii+179pp.
::In the essay ‘At the Moscow Art Theatre: 1914’, Graham recalls his visits to see performances of ''Hamlet'', Chekhov’s ''The cherry orchard'' and Andreev’s ''Anathema'' (pp. 129-46).

 
======K278======
'''Keeling, H.V.,''' ''Bolshevism: Mr. Keeling’s five years in Russia''. Edited by E.H. Haywood. London: Hodder &amp;amp; Stoughton, 1919. 212pp.
::Keeling was sent to Russia in February 1914 to assist in the setting up of a patent photo-litho process in St Petersburg and to train Russian workers. He stayed on for five years, working as a jobbing mechanic in the capital and other towns, before escaping via Finland. The editor contributed both preface and final chapter entitled ‘The theory of Bolshevism’ (pp. 199-212).

 
======K279======
'''Bartlett, Robert A.,''''' The last voyage of the ‘Karluk’, flagship of Vilhjalmar Stefannson’s Canadian Arctic expedition of 1913-1916. ''As related by her master Robert A. Bartlett, and here set down by Ralph T. Hale. Boston: Small, Maynard and Co., 1916. vi+329pp.
::The Canadian captain “Bob” Bartlett (1875-1946) recounts the ill-fated last voyage of the ''Karluk'', which was crushed by ice and sank on 11 January 1914. In their overland trek to safety, Bartlett and others reached Wrangell Island on 12 March 1914, but only he and one companion then proceeded to cross over to the north-east Siberian mainland, which they reached on 4 April and encountered a settlement of Chukchis. After many further exploits and surprising meetings, he finally stepped on American soil on 28 May (pp. 161-281).

 
======K280======
'''Bartlett, Robert A., '''''The log of Bob Bartlett: the true story of forty years of seafaring and exploration. ''New York and London: G.P. Putnam’s Sons, 1928. xii+252pp.
::Includes a succinct account of the ''Karluk'' expedition (pp. 254-79).

 
======K281======
'''Moore, Benjamin Burges, '''''From Moscow to the Persian Gulf, being the journal of a disenchanted traveller in Turkestan and Persia''. New York and London: G.P. Putnam’s Sons, 1915. xx+450pp.
::The journal of an American traveller, reflecting his non-definitive “unfavourable opinion of Persia and her people”. He begins his journey from Moscow on 8 February 1914, en route for Samarkand and Bokhara in Russian Turkestan, before he crosses into Persia on 20 February (pp. 3-70).

 
======K282======
'''Dawe, Rosamond E., '''''A memoir of an English governess in Russia, 1914-1917. ''Chichester: Bishop Otter College, 1973. vi+26pp. [Revised edition, Woking: Unwin Brothers, 1976. x+45pp.]
::The eighteen-year-old Rosamond (1896-1990) left Norwich in May 1914 to teach English to the three eldest Naumov daughters on the family estate near Samara on the Volga. Two years later, she moved to the Tolstoi family at Tsarskoe selo and, finally, held a position with the Miklashevskii family, taking her to Kislovodsk before returning to Petrograd, where she witnessed the aftermath of the February revolution. She returned to England in June 1917 via Scandinavia.

 
======K283======
'''Wardell, John Wilford, '''''In the Kirghiz Steppes''. London: Gallery Press, 1961. 190pp.
::Wardell, a draughtsman and engineer with the London firm of Walter Perkins, was sent on a three-year contract to work for the British-owned Spasskii Copper Mine Ltd in southern Siberia. He left England on 16 May 1914 but it was the end of September 1919 before he and his wife Lily (who had joined him in July 1914) and other British, caught by war and revolution, were able to leave from Vladivostok for China and home. Unusual and fascinating account of life and work among the Kazaks.

 
======K284======
'''Czaplicka, Marya Antonina,''' ''My Siberian year''. London: Mills &amp;amp; Boon, 1916. 306pp.
::The Polish-born Oxford anthropologist (1884-1921), already author of ''Aboriginal Siberia'' (1914), based on printed sources, spent a year of fieldwork between May 1914 and the spring of the following year in the northern tundra by the Enisei, accompanied by the American anthropologist Hubert Hall.

 
======K285======
'''Anderson, Herbert Foster, '''''Borderline Russia. ''London: Cresset Press, 1942. 238pp.
::Recently graduated, Anderson (b. 1890) accepted a position as manager of an estate in Tambov ''guberniia'', where he remained from June 1914 to the summer of the following year, when he sought to return to England on hearing of the declaration of war (pp. 1-29).

 
======K286======
'''Levings, Grace M., '''''Travel sketches of Norway, Sweden, Russia, Austria, Belgium and Holland.'' Boston: Richard G. Badger; Toronto: Copp Clark Co., 1916. 168pp.
::A European tour by an American couple – Mrs Levings refers to her husband throughout as “Doctor” – that took them from Stockholm by boat to Petrograd. Routine tourist notes of the sights of Petrograd and Moscow, before they move on to Vienna (pp. 64-103). No dates and no mention of the war, but use of Petrograd, if not afterthought, suggests possibly summer of 1914.

 
======K287======
'''Boultbee, Rosamond, '''''Pilgrimages and personalities''. London: Hutchinson &amp;amp; Co., 1924. 328pp.
::The Canadian journalist (1878-1957) paid a first brief visit to Russia in early 1914 to visit friends in Kiev (pp. 64-66). She returned for a second time in July 1915, spending two months in Petrograd, before leaving in September for Kiev. After a lengthy stay in Kiev, she moved to Odessa, where she was to remain for three months, before departing for Romania via Kishinev in the spring of 1916. In April she returned to Odessa and then visited Moscow, which delighted her and where she stayed until July (pp. 96-148, 176-89).

 
======K288======
'''Graham, Stephen, '''''Through Russian Central Asia. ''London, New York: Cassell and Co., 1916. xii+279pp.
::In the early summer of 1914 Graham left Vladikavkaz by train for Bokhara and Tashkent. A little beyond Tashkent he began his travels on foot, by cart, and horse and eventually crossed into Siberia and reached Semipalatinsk, the place of Dostoevskii’s exile. His articles to the ''Times ''chronicled his progress at the time, but he delayed publication of the book until after its successor ([[(title of the correspondent chapter)#K289|K289]]).

 
======K289======
'''Graham, Stephen, '''''Russia and the world. A study of the war and a statement of the world-problems that now confront Russia and Great Britain. ''London: Cassell and Co., 1915. xi+259pp. [Revised and enlarged edition, 1917. 301pp.]
::Graham was in the Altai mountains by the Mongolian frontier when news of the outbreak of WWI reached him in July 1914. He travelled to Moscow in September (pp. 3-32). The rest of the book is devoted to general essays and memorable meetings, before he made his first ever visit to St Petersburg/Petrograd on his way home to England (pp. 243-46).

 
======K290======
'''Haviland, Maud Doria, '''''A summer on the Yenisei (1914)''. London: Edward Arnold, 1915. xii+328pp.
::The ornithologist Miss Haviland, inspired by the writings of Seebohm, joined the expedition to the Enisei, organized by Marya Czaplicka and Hubert Hall. She and the artist Dora Curtis reached Golchika on the Enesei on 29 June 1914 and remained there for two months, observing and registering birds. Leaving their other two companions, they returned to a Britain at war, via Norway, on 9 October.

 
======K291======
'''Nicholas, Prince of Greece,''' ''Political memoirs 1914-1917: pages from my diary''. London: Hutchinson &amp;amp; Co., 1928. 319pp.
::In July 1914 Prince Nicholas (b. 1872) and his family travelled from Athens to pay their yearly visit to his mother-in-law Grand Duchess Vladimir and, two days before the declaration of war, reached Tsarskoe selo, where they met the tsar (pp. 18-22). In July 1916 he was sent to Russia on an unsuccessful mission from the Greek government to explain Greek neutrality in the war. He was received by the tsar at Mogilev, before he proceeded to Petrograd, where he had meetings with Russian ministers and foreign ambassadors. He left in October (pp. 133-80).

 
======K292======
'''Lethbridge, Alan Bourchier, '''''The new Russia: from the White Sea to the Siberian steppe. ''London: Mills and Boon, 1915. xvi+314pp.
::Lethbridge had been in Russia several times, including Siberia in 1907, before he resolved to undertake his northern journey in 1914, inspired by a reading of Kliuchevskii “to whet the appetite for a first-hand experience of that wonderful North that is so bound up with the creation of the modern Russian Empire”. He and his wife Marjorie followed a route that took them from Archangel to Solovets and via Velikii Ustiug and Viatka to Perm and across the Urals to Ekaterinburg. They went as far as Tiumen and Omsk, before returning to Petrograd by train, but, because of the war, were obliged to return to England from Archangel.

 
======K293======
'''Lethbridge, Marjorie Colt and Lethbridge''', '''Alan Bourchier,''' ''The soul of the Russian''. London: John Lane, 1916. xii+238pp.
::A collection of twenty-eight sketches, ten written by Marjorie (b. 1882), who also published semi-fictional tales under the title ''Russian chaps'' (1916), and eighteen by Alan, on a wide variety of subjects, historical, cultural, geographical, and social, appearing originally in London newspapers and journals in 1914-15.

 
======K294======
'''Merry, Walter Mansell, '''''Two months in Russia July-September, 1914. ''Oxford: B.H. Blackwell, 1916. iv+202pp.
::Invited to St Petersburg to be the temporary chaplain to the British community, Rev. Merry, vicar of St Michael’s, Oxford, arrived in the Russian capital on 13 July 1914 and left with considerably more difficulty on 3 September for Sweden. Offers selections from his journal without additions other than the division into three parts: before the war; the beginning of the war, when he undertook a journey to Odessa in the vain hope of leaving via the Black Sea; and wartime Petrograd and ultimate departure (pp. 8-172).

 
======K295======
'''Scudder, Jared Waterbury, '''''Russia in the summer of 1914, with discussion of her pressing problems.'' Boston: Richard G. Badger, 1920. 193pp.
::American theologian and missionary, best known for his Latin textbooks, Scudder (1863-1934) arrived at Cronstadt from Stockholm on 23 July 1914. After a few days of sightseeing in St Petersburg, he was in Moscow when war was declared, witnessed anti-German riots, and hurried back to the capital, eventually managing to leave for Finland on 11 August.

 
======K296======
'''Gaunt, Mary, '''''A broken journey; wanderings from the Hoang-Ho to the island of Saghalien and the upper reaches of the Amur River''. London: Werner Laurie, 1919. 295pp.
::Tourist travelling from China to Russia, partly via train, partly via steamer along the Amur River, found herself in the Russian far east in July 1914, just as war is declared. Describes her long, arduous return journey to St Petersburg and then the difficulty in getting from Russia to Finland (pp. 157-268).

 
======K297======
'''Paléologue, Maurice, '''''An ambassador’s memoirs. ''Translated from the French by F.A. Holt. London: Hutchinson &amp;amp; Co., 1924-25. 3 vols.
::France’s last ambassador to imperial Russia, Paléologue (1859-1944) kept meticulous diaries of his four-year sojourn in St Petersburg, beginning with the entry for 20 July 1914, marking the visit of President Poincaré, and ending with 17 May 1917, when he was already in Finland.

 
======K298======
'''Kroeger, Theodor,''''' The forgotten village: four years in Siberia''. London: Hutchinson &amp;amp; Co., 1936. 320pp.
::Russian-born and educated, but a German national, Kroeger (1891-1958) recalls twenty years after the events his experiences as a POW during WWI. He had attempted to flee to Germany following the declaration of war in August 1914, but was arrested on suspicion of being a spy and sent first to Schlüsselberg, then to a camp near Baikal. Charges against him were dropped in March 1916, but he had married and continued to live in Siberia until his eventual departure for Germany after the death of his wife in late 1919.

 
======K299======
'''Fortescue, Granville Roland, '''''Russia, the Balkans and the Dardanelles. ''London: Andrew Melrose, 1915. 285pp.
::Fortescue (1875-1952), American soldier and military attaché during the Russo-Japanese war, was the special correspondent of the ''Daily Telegraph ''with the Russian army in Poland in 1914-15, before illness forced him to leave for England. He considered “the campaigns I had witnessed there will rank among the greatest military events in history” and believed “the Russian infantryman one of the finest soldiers in the world” (pp. 15-139).

 
======K300======
'''Morse, John, '''''An Englishman in the Russian ranks.'' London: Duckworth &amp;amp; Co., 1915. vi+337pp.
::When WWI began, Morse, an English businessman, was in Germany and to avoid internment he crossed over into Russian Poland on 2 August 1914. Intent on returning home, in the event he stayed and fought for nine months with the Russian army, until he was captured by the Germans. He was to escape and make his way to the Russian lines. He eventually reached Riga, which he left on 20 May 1915 for Sweden and England.

 
======K301======
'''Fraser, John Foster, '''''[https://archive.org/details/russiaoftoday00frasiala Russia of today]. ''London: Cassell and Co., 1915. viii+289pp.
::Fraser (see also [[#K29|K29]], [[#K88|K88]], [[#K90|K90]], [[#K161|K161]]) had first visited Russia in 1896 and his ‘Russia of today’ was the Petrograd and Moscow with its “happy British colony” that he visited in 1914 at the beginning of WWI. He ends with guarded optimism for the changed Russia that will emerge after the war!

======K302======
'''West, Julius, '''''Soldiers of the tsar and other sketches and studies of the Russia of to-day''. London: The Iris Publishing Co., 1915. xvi+167pp.
::West (1891-1918), born in Russia but leaving when two months old with his journalist father Semen Rappoport, returned during the first months of WWI. He offers an attractive collection of sketches based on “long chats with Russians of all classes”, alongside articles on Petrograd, Moscow and Warsaw in wartime, and on the vogue for translations from Russian literature (several of which – from Andreev and Chekhov – he himself made).

 
======K303======
'''Brändström, Elsa, '''''Among prisoners of war in Russia and Siberia. ''Translated from the German by C. Mabel Rickmers. With a preface by Nathan Söderblom. London: Hutchinson &amp;amp; Co., 1929. 284pp.
::Daughter of the Swedish ambassador to Russia and living in St Petersburg since 1908, Brändström (1888-1948) describes her activities and experiences as an official Swedish Red Cross delegate from winter 1914 until summer 1920, during which time she travelled to all the concentration centres for prisoners of war in European Russia and in Siberia as far as Vladivostok, her work bringing her in touch with an estimated 700,000 POWs. The German original was entitled ''Unter Kriegsgefangenen in Rußland und Sibirien, 1914–1920'' (Leipzig, 1927).

 
======K304======
'''Gibson, William J., '''''Wild career: my crowded years of adventure in Russia and the Near East. ''London: George G. Harrap, 1935. 288pp.
::Born in Canada, but brought up in St Petersburg, Gibson volunteered for the Russian army in the summer of 1914. He subsequently worked for the Russian secret service in Central Asia. He was in Petrograd, working as a newspaper correspondent, during the February Revolution and witnessed Lenin’s arrival at the Finland Station. After a spell as a Soviet commissar, he eventually left Petrograd at the end of 1918 (pp. 1-200).

 
======K305======
'''Buchanan, Meriel, '''''Petrograd the city of trouble, 1914-1918. ''[With a foreword by Hugh Walpole.] London: W. Collins &amp;amp; Sons, 1918. 262pp.
::The first published of Meriel’s books on Russia, it describes in detail her experiences of life in Petrograd from the declaration of war through to the rise and succession to power of the Bolsheviks.

 
======K306======
'''Bauermeister, Alexander (‘Agricola’), '''''Spies break through: memoirs of a German secret service officer. ''Translated [from the German] and introduced by Hector C. Bywater''. ''London: Constable and Co., 1934. 185pp.
::The leading German spymaster on the Eastern Front in WWI, Lt. Bauermeister (1899-1940), was born in St Petersburg, which he left in 1914 and was based in Königsberg, decoding Russian communiqués. He assumed a prominent role in the Russo-German armistice negotiations in November 1917.

 
======K307======
'''Dietrich, Johann, '''''Tovarish; the odyssey of a Siberian exile.'' Narrated by Paul Cölestin Ettighoffer. Translated from the German by M.H. Jerome. London: Hutchinson &amp;amp; Co., 1935. 288pp.
::The account of an escape from Siberia by the German telepathist and hypnotist Johann Dietrich (b. 1885), as told to the novelist Ettighoffer (1896-1975). Dietrich, in St Petersburg on business just as WWI began, sought to flee Russian territory, but was arrested at Orenburg and exiled in early 1915. In 1917 he escaped to Irkutsk, where he developed his telepathic skills, and left Russia via Vladivostok in the autumn.

 
======K308======
'''Arbenina, Stella, '''''Through terror to freedom: the dramatic story of an Englishwoman’s life and adventures in Russia before, during &amp;amp; after the Revolution.'' London: Hutchinson &amp;amp; Co., 1930. 288pp.
::Née Whishaw, member of a British family that had been in Russia since the eighteenth century, Stella (1885-1976) was the wife of Baron Pavel Meyendorf at the time of the October revolution. She relates in somewhat chaotic fashion her early life, her passion for acting, and her experiences during and after the revolution, before they escaped initially to Revel (pp. 9-273). It was in Berlin in 1921 that she assumed the stage name of Arbenina, which she retained in England, where she arrived in June 1923.

 
======K309======
'''Pares, Bernard, '''''[https://archive.org/details/daybydaywithruss00pareuoft Day by day with the Russian army, 1914-1915]. ''London: Constable &amp;amp; Co., 1915. xi+287pp.
::Pares (see [[#K58|K58]], [[#K59|K59]], [[#K123|K123]], [[#K124|K124]]) left England in August 1914, spent six weeks in newly-named Petrograd, and begins his day-by-day account on 8 October from Vilna and ends on 19 June 1915, when he left the front. The book finishes with the diary of an Austrian officer serving in Galicia, March-May 1915 (pp. 261-82).

======K310======
'''Hanbury-Williams, John, '''''The Emperor Nicholas II as I knew him. ''London: Arthur L. Humphreys, 1922. xii+271pp.
::Major-General Sir John (1859-1946) was chief of the British Military Mission in Russia between August 1914 and April 1917. He was attached to the G.H.Q. of the Russian armies at Mogilev and had a unique opportunity to observe and converse with the tsar. His book consists principally of diary entries, followed by sketches of the emperor, the tsarevich, Grand Duke Nikolai Nikolaevich and General Alekseev (pp. 217-64).

 
======K311======
'''Knox, Alfred William Fortescue,''''' With the Russian army 1914-1917, being chiefly extracts from the diary of a military attaché''. London: Hutchinson &amp;amp; Co., 1921. 2 vols.
::Military attaché at the Petersburg embassy from 1911 and a fluent Russian speaker, Major-General Sir Alfred (1870-1964), later a Conservative politician, was appointed liaison officer to the Russian army in 1914-17 and kept the detailed diaries which form the substance of these volumes, augmented by additional later comment and analysis. The first volume describes warfare on the eastern front, particularly in Poland, between September 1914 and September 1915; the second continues with an account of the fighting during 1916, particularly the Brusilov Offensive. The later chapters describe Knox’s observations of growing political unrest within the Russian army and an eye-witness account of the February Revolution, subsequent rapid decline of order within the army, the failed Kerenskii offensive and the October Revolution. He left Russia on 8 January 1918.

 
======K312======
'''Blair, Dorian, '''''Russian hazard: the adventures of a British secret agent in Russia.'' Edited (?) by C.H. Dand. London: Robert Hale &amp;amp; Co., 1937. 288pp.
::Allegedly born in St Petersburg c.1893 to Scoto-Russian parents, Blair returned to Russia in August 1914 to embark on a succession of increasingly implausible undercover adventures that involved burning the body of Rasputin, plotting to kidnap the tsar, and later, Trotskii and Lenin, at the instigation of Kerenskii. (pp. 13-147). The “scarlet pimpernel”, as he styles himself, was captured by the Cheka on 31 December 1917, but survived to be involved in even more unlikely exploits before escaping to England in 1920.

 
======K313======
'''Washburn, Stanley, '''''Field notes from the Russian front. ''London: Andrew Melrose, 1915. 291pp.
::Washburn, who had covered the Russo-Japanese war (see [[#K154|K154]]), returned to Russia in 1914 as the special war correspondent of ''The Times'' with the Russian armies. The dispatches, which were largely published in ''The'' ''Times'' and American newspapers, begin with his report from Petrograd on 10 September 1914 and continue from the Polish front, from where his final report is datelined 15 January 1915. This became the first volume of a trilogy of dispatches (see [[#K340|K340]], [[#K363|K363]]). The book is also notable for the photographs by the'' Daily Mirror''’s George Mewes, the only “official” English photographer with the Russian armies.

 
======K314======
'''Britnieva, Mary, '''''One woman’s story. ''London: Arthur Barker, 1934. 287pp.
::Born to Anglo-Russian parents in Russia, Mary (maiden name unknown) begins her “story” on 29 September 1914, the day she, a new Red Cross nurse, was to leave with her field hospital for the eastern front. She recounts her experiences in East Prussia, on the Warsaw front, and in Warsaw itself up to the beginning of the great retreat in July 1915. During a period of leave in April 1916 she visited her mother’s estate at Chistopol in Kazan province (pp. 9-64). The rest of the book is devoted to her life from the beginning of 1918, when she married Aleksandr Britnev, the head doctor, her departure for England in 1922, her subsequent return visits, and final farewell in 1930.

 
======K315======
'''Walpole, Hugh, '''''Hugh Walpole: a biography. ''By Rupert Hart-Davis. London: Macmillan &amp;amp; Co., 1952. xiv+503pp.
::The prolific and once-popular novelist (1884-1941) arrived in Russia at the end of September 1914 as a correspondent for the ''Daily Mail'' and ''Saturday Review. ''He also found material and inspiration for his two Russia-centred novels ''The dark forest'' (1916) and ''The secret city'' (1919). He joined a Russian Red Cross unit in the Carpathians, before leaving for England in October 1915. He returned in February 1916 as head of a new British propaganda unit in Petrograd. He left finally for home on 8 November 1917, the morning after the start of the October Revolution. Excerpts from his journal and his letters, especially to Henry James (pp. 123-64). The text of the long memorandum on the February Revolution that he composed at the request of the British ambassador is on pp. 449-69.

 
======K316======
'''Marye, George Thomas,''' ''Nearing the end in imperial Russia''. London: Selwyn &amp;amp; Blount, 1929. 479pp.
::Lawyer and banker Marye (1849-1933) arrived in Petrograd on 24 October 1914 as the American ambassador to Russia. He remained until mid-March 1916. Although his title was obviously influenced by later events, Marye stresses that he was publishing his “notes and jottings” with their “first impressions of events” just as they were written.

 
======K317======
'''Thurstan, Violetta,''' ''Field hospital and flying column, being the journal of an English nursing sister in Belgium &amp;amp; Russia.'' London and New York: G.P. Putnam’s Sons, 1915. viii+184pp.
::Red Cross nurse Thurstan (1879-1978), after service in Belgium, volunteered for the Russian Red Cross. She left Copenhagen on 24 October 1914 for Petrograd via Lapland and Finland and was sent to Warsaw and the eastern front. Wounded by shrapnel and ill with pleurisy, she convalesced in Petrograd, where she finished her journal of an eventful 1914 (pp. 106-78).

 
======K318======
'''Roberts, Carl Eric Bechhofer, '''''Russia at the cross roads''. With an introduction by A.H. Murray. London: Kegan Paul, Trench, Trübner &amp;amp; Co, 1916. viii+201pp.
::The work arose from a year-long stay in Russia from late 1914 by Roberts (1894-1949), styling himself at that period Bechhofer, and offers in ten chapters his thoughts on the Russian character and society, developments in literature and ideas, and musings on Russia’s future.

 
======K319======
'''Roberts, Carl Eric Bechhofer, '''''A wanderer’s log: being some memories of travel in India, the Far East, Russia, the Mediterranean &amp;amp; elsewhere''. London: Mills &amp;amp; Boon, 1922. 246pp.
::In late 1914 Bechhofer, wanting to learn Russian, took a post as a tutor with a Ukrainian family, before leaving it to go to Kiev, and then to Batumi. Back in Petrograd, he recalls his visit to the literary cabaret, ‘The Stray Dog’, and his encounter with Rasputin (pp. 127-54). A further chapter describes his experiences with Denikin’s army around Moscow in 1919 and a final trip to Moscow and around the Volga as a newspaper correspondent in the autumn of 1921, the subjects of subsequent books (pp. 155-82).

 
======K320======
'''Farson, Negley, '''''The way of a transgressor. ''London: Victor Gollancz, 1935. 640pp.
::In his lively autobiography, the American adventurer (1890-1960) recounts his first visit to Russia in the winter of 1914 to sell munitions to the Russian military authorities. He also visited Archangel, Moscow and the Crimea until illness forced him to return to America (pp. 126-226). He returned in 1916 to a Petrograd inexorably moving towards revolution and describes in detail events of “the Kerensky revolution” and its aftermath before he left to join the American air force (pp. 252-316). In 1928-29 he was in Soviet Russia with his wife for an extensive tour (pp. 542-81).

 
======K321======
'''Krist, Gustav, '''''Prisoner in the forbidden land.'' Translated from the German by E[mily] O[verend] Lorimer. London: Faber &amp;amp; Faber, 1938. 344pp.
::“Gurk” Krist (1894-1937), an Austrian POW, captured by the Russians on the eastern front in November 1914, describes his long years of captivity in Turkestan, first at Katta-Kurgan, near Samarkand, from which he escaped into Persia, but was re-captured and remained in camps into the Soviet period. He was finally repatriated in late 1921. German original entitled ''Pascholl plenny!'' (Vienna, 1936).

 
======K322======
'''Price, Morgan Philips, '''''War and revolution in Asiatic Russia.'' London: Allen &amp;amp; Unwin, 1918. 296pp.
::Price (see [[#K219|K219]], [[#K379|K379]]-[[#K381|81]]) returned to Russia in November 1914 as special correspondent for the ''Manchester Guardian.'' Frustrated in his attempt to report from the eastern front, he made his way to the less controlled Caucasus, where he spent much of 1915 and all of 1916. His book, written in Tiflis and completed in Petrograd in 1916-17, provides an overview of the Caucasus campaign, followed by an account of Price’s activities as journalist and relief worker in the region, and finishes with his analysis of Russian involvement in Central Asia and the impact of the February revolution.

 
======K323======
'''Cantacuzène, Julia, '''''My life here and there''. New York: Charles Scribner’s Sons, 1921. 322pp.
::Princess Cantacuzène, née Grant, also styled Countess Speranskaia (1876-1975), the granddaughter of U.S. president Ulysses Grant, married the Russian diplomat Prince Mikhail Cantacuzène (Kantakuzen) in 1899 and moved to Russia, where she was to remain until 1917. In this, the last of her three books to be published, she recalls her first years in Russia.

 
======K324======
'''Cantacuzène, Julia, '''''Revolutionary days: recollections of Romanoffs and Bolsheviki 1914-1917''. London: Chapman &amp;amp; Hall, 1920. vi+411pp. [See ''Revolutionary days, including passages from My life here and there 1876-1917''. Edited by Terence Emmons. Chicago: R.R. Donnelly &amp;amp; Sons, 1999. lx+442pp.]
::Princess Cantacuzène in the first of her three books to be published traces her family’s fortunes from the beginning of WWI, in Petrograd, Kiev and the Crimea, to their escape to Finland in 1917.

 
======K325======
'''Cantacuzène, Julia, '''''Russian people: revolutionary recollections.'' New York: Charles Scribner’s Sons, 1920. 358pp.
::Chapters on Kolchak and Denikin as well as vignettes of Russian life, first published in the ''Saturday Evening Post.''

 
======K326======
'''Urch, Reginald Oliver Gilling, '''''“We generally shoot Englishmen”: an English schoolmaster’s five years of mild adventure in Moscow (1915-1920). ''London: Allen &amp;amp; Unwin, 1936. 300pp.
::“Five years in Russia of a rather ordinary English family not connected with any official missions, consulates, or services, but sharing the lot of average families then living in Russia.” The Urches, husband, wife, and two children had apparently been for some time in Riga before being forced by war events to move to Moscow in the autumn of 1915. There Urch began to teach at the re-established Riga Polytechnic as lecturer in commerce and his wife established an English kindergarten until the Bolsheviks won the battle for Moscow (pp. 19-93). Thereafter it is a tale of Urch’s vicissitudes under the Soviets, including imprisonment in the Butyrskii prison.

 
======K327======
'''Price, Hereward Thimbleby,''' ''[https://archive.org/details/bochebolshevikex00pricrich Boche and Bolshevik: experiences of an Englishman in the German army and in Russian prisons].'' London: John Murray, 1919. viii+247pp.
::Son of a missionary, Madagascar-born, Oxford-educated, Price (1880-1964), later professor of English at University of Michigan, was drafted into the German army while lecturing at Bonn in 1915. Captured by the Russians on the eastern front, he was marched to a POW camp near Stretensk in Siberia. Released following the February Revolution, he moved to Irkutsk, working as a tutor in a Russian family until he escaped in 1918 with the help of the British consul (pp. 94-243). His book consists of a series of articles he contributed to the ''China Illustrated Weekly'' between November 1918 and February 1919.

======K328======
'''Fyfe, Henry Hamilton, '''''My seven selves. ''London: George Allen &amp;amp; Unwin, 1935. 320pp.
::The renowned Scottish newspaper editor and war correspondent (1869-1951) was sent from the Western front to Russia in 1915, and was eventually allowed to the Galician front the following year. In August 1916 he was ordered to Bucharest, from where he returned in December, reaching Petrograd on the 30th, the day after the murder of Rasputin, which he was the first British journalist to report (pp. 191-202, 210-13). Many of his (censored) articles from Russia appeared in such publications as the ''War Illustrated'', but, sadly, his war articles and “a vast quantity of matter that could not be printed” have never been collected.

 
======K329======
'''Pierce, Ruth,''''' Trapped in “Black Russia”'': ''letters June-November 1915''. Boston and New York: Houghton Mifflin Company, 1918. 150pp.
::A series of letters that the American traveller Mrs Pierce sent to her parents from Kiev, where she stayed between 30 June and early November 1915. For six weeks in August-September she was under house arrest for alleged espionage. She witnessed the transportation of Galician Jews through Kiev to Siberia, visited a Jewish detention camp, and recorded scenes in the city as the Germans approached after the fall of Warsaw.

 
======K330======
'''Cresson, William Penn, '''''The Cossacks: their history and country. ''New York: Brentano’s, 1919. x+239pp.
::One-time captain in the American Expeditionary Force and formerly secretary at the American embassy in Petrograd, Cresson (1873-1932) attempts to produce a “comprehensive study of Cossack life and history”, based in part on his travels through Cossack regions between 1915 and 1917 (see particularly pp. 196-239).

 
======K331======
'''Pollock, John, '''''War and revolution in Russia: sketches and studies''. London: Constable &amp;amp; Co., 1918. xviii+280pp.
::Sir John (1878-1963), 4th Baronet of Haddon and a former Fellow of Trinity College, Cambridge, went to Poland in 1915 as a representative of the Great Britain to Poland Committee, set up to aid refugees during WWI. He subsequently became an International Commissioner with the Russian Red Cross. He was in Petrograd during both revolutions and visited Kiev, Saratov, Voronezh, and Ekaterinodar. He also acted as correspondent for the ''Manchester Guardian'' and other English newspapers and his book, completed in Russia in September 1917, largely comprises articles he sent to them.

 
======K332======
'''Pollock, John, '''''Time’s chariot. ''London: John Murray, 1950. xii+280pp.
::In his memoirs Sir John recalls succinctly (pp. 213-35) the four years he spent in Russia from March 1915 to May 1919, the “red” months of which he described in his ''Bolshevik adventure ''(1919).

 
======K333======
'''Sykes, Ella Constance, and Sykes, Percy,''''' Through deserts and oases of Central Asia''. London: Macmillan, 1920. xii+340pp.
::In March 1915 Miss Sykes (d. 1939) accompanied her brother, Brigadier-General Sir Percy (1867-1945), to Kashgar in Chinese Turkestan, where he was to deputise for the British consul-general. They were obliged to travel via Scandinavia to Petrograd and then they took the train to Tashkent and proceeded by carriage to their destination (pp. 7-35). They later set out on a tour to the Russian Pamirs and the “roof of the world”, crossing into Russian territory on 18 June and returning in mid-July (pp. 129-47). Miss Sykes wrote all the initial chapters of travel and adventure which form part I; her brother contributed the (non-Russian) material on Chinese Turkestan in part II.

 
======K334======
'''Dwinger, Edwin Erich''','' The army behind barbed wire: a Siberian diary.'' Translated by Ian F. D. Morrow. London: George Allen &amp;amp; Unwin Ltd, 1930. 341pp.
::German soldier, nationalist, and prolific author Dwinger (1898-1981) relates experiences as POW in Siberia between 1915 and 1918 and his enduring relationships and friendships with fellow POWs. Taken prisoner at Windau in Latvia, Dwinger spent much of 1915 recuperating in a Moscow hospital, before being sent in 1916 to Siberia, imprisoned in various camps until his eventual release and departure from Russia in late 1918.

 
======K335======
'''Liddell, Robert Scotland, '''''On the Russian front. ''London: Simpkin, Marshall, Hamilton, Kent, 1916. x+273pp.
::Liddell (1885-1972) arrived in Petrograd in the spring of 1915 and soon moved to Warsaw, where he served as a member of the Group of Polish Red Cross Volunteers with the Russian army. He also contributed articles to the ''Sphere'' as its special correspondent, writing “nearly every line to the accompaniment of guns”. He describes the Russian retreat through Poland in May-August, evincing great admiration for the ordinary Russian soldier. Preface dated March 1916 “with the active Russian army”.

======K336======
'''Liddell, Robert Scotland, '''''Actions and reactions in Russia. ''London: Chapman and Hall, 1917. viii+227pp.
::“Russia to-day is not the Russia of two years ago. Russia has changed miraculously”, Liddell wrote in his sequel to ''On the Russian front''. He had in the interim been to Romania and to the Caucasian front and became, he claimed, the only British subject in command of a Russian army unit. His narrative ranges widely over Russia, from Odessa and the Crimea to Georgia and Minsk.

 
======K337======
'''Liddell, Robert Scotland, '''''“Sestra” (Sister): sketches from the Russian front. ''London: Hodder and Stoughton, 1917. viii+244pp.
::The final contribution to an impressive trilogy. Fourteen sketches, several of which, including the title sketch, have as their heroines nurses who figured prominently in his earlier accounts.

 
======K338======
'''Steveni, William Barnes,''' ''Petrograd past and present''. London: Grant Richards, 1915. viii+319pp.
::Steveni (see [[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J111|J111]], [[#K254|K254]], [[#K266|K266]]), who arrived as a boy of sixteen at the end of the reign of Alexander II and lived seven years in Cronstadt before moving to the capital, produced one of the best, if little-known, books on the Russian capital with a particular emphasis on British presence and influence and a happy mixture of history, anecdote, and personal observation.

 
======K339======
'''Templeton, Isabel Molison, '''''The old lady in room 2''. Bearsted, Kent: for the author, 1976. ii+184pp.
::Mrs Templeton, née Young (1886-1976) sailed out to Archangel in January 1915 to join her husband, a Scottish engineer working for the Maikop Pipeline &amp;amp; Transport Co. in Ekaterinodar in the Kuban, where they were to live until April 1917, when worsening conditions forced them to leave, although her husband was subsequently detained in Russia until May 1918 (pp. 1-9, 47-97).

 
======K340======
'''Washburn, Stanley, '''''The Russian campaign. April to August, 1915, being the second volume of “Field notes from the Russian front.” ''London: Andrew Melrose, 1916. 348pp.
::Washburn (see [[#K154|K154]], [[#K313|K313]], [[#K362|K362]]) details fighting between the Russian and Austro-German armies in the early summer of 1915 and the decline in Russian fortunes. The dispatches alternate between the Warsaw and Galician fronts and include chapters on Eugene Hurd (an American doctor working for the Russian Red Cross), the German gas attacks, meetings with the Russian generals Ivanov and Brusilov.

 
======K341======
'''Kohn, Hans, '''''Living in a world revolution: my encounters with history''. New York: Simon and Schuster, 1964. xxii+211pp.
::The Czech Jewish philosopher and historian (1891-1971) recalls the years he spent as a POW in Russia during and after WWI. Taken prisoner on 21 March 1915 during the Carpathian campaign, he was to remain in Russia until 12 January 1920. Initially marched off to Lemberg, he was then sent to a camp in Samarkand, from which he escaped in February 1916. Recaptured a month later, he was moved to camps in Siberia, where he learnt Russian and came to admire his captors, before being freed in 1918 and starting his slow exit from Russia (pp. 88-99).

 
======K342======
'''McCormick, Robert Rutherford, '''''With the Russian army, being the experiences of a national guardsman. ''London: Macmillan, 1915. xvi+306pp.
::Son of a former ambassador to Russia and a major in the First Cavalry of the Illinois National Guards, McCormick (1880-1955) arrived in Petrograd in April 1915 as foreign correspondent of ''Chicago Tribune'', interviewing the tsar and foreign minister Sazonov, before leaving for Warsaw and the eastern front.

 
======K343======
'''Balch, Emily G., '''''Women at The Hague: the International Congress of Women and its results. By three delegates to the Congress from the United States [Jane Addams, Emily G. Balch, Alice Hamilton]''. New York: Macmillan Co., 1915. 171pp.
::Wellesley professor Balch (1867-1961) was a member of a delegation from the Congress assigned to Scandinavia and Russia. She arrived in Petrograd on 10 June 1915, interviewing during her two-week stay the Minister of Foreign Affairs Sazonov (pp. 103-04).

 
======K344======
'''Simpson, James Young, '''''The self-discovery of Russia. ''London: Constable and Company, 1916. viii+227pp.
::Young, who had first visited Russia in 1896 ([[(title of the correspondent chapter)#K27|K27]]) and was by now professor of natural science at his ''alma mater'' Edinburgh, offers his views on a number of topics, including the prohibition of vodka, conditions on the Galician front, and religion, based on his observations and conversations with Russians in the summer of 1915.

 
======K345======
'''Coxwell, Charles Fillingham, '''''Through Russia in war-time. ''London: T.F. Unwin, 1917. 311pp.
::Thwarted by the sinking of the ''Lusitania'' from sailing from New York to London, Coxwell (b. 1856) was redirected to Archangel in June 1915 and decided to seize the opportunity to tour Russia, visiting many towns and provinces in the south west Russia over the following eleven weeks. Returning to Petrograd in mid-August 1915, Coxwell, in later years a prolific translator from Russian literature, made an excursion into Lapland on his journey back to England.

 
======K346======
'''[Stopford, Albert Henry], '''''The Russian diary of an Englishman: Petrograd, 1915-1917. ''London: William Heinemann, 1919. xiv+228pp.
::A member of the Irish aristocracy, Stopford had previously visited Petrograd in March 1914, but his book, comprising extracts from his diary and letters, covers the period from 18 July 1915 to 26 September 1917. His exact role and the nature of his “affairs” are unclear, although he was uncommonly well connected with the Russian elite, including the emperor, but particularly with the Grand Duchess Vladimir, and with the British embassy. He returned briefly to England in October 1916, but he travelled fairly extensively in Russia, visiting Mogilev, Moscow, the Crimea, and the Caucasus.


======K346a======
'''Martin, Alexander Gustav,''' ''Mother country, fatherland: the story of a British-born German soldier.'' London: Macmillan, 1936. xi+390pp.
::Born in England to Anglo-German parents, Martin (1874 -1946) had been a cavalry officer in the Prussian army since 1890 when WWI began. After service in France, he was transferred to the  Galician front and on 30 August 1915 was captured by the Russians. After eighteen months in captivity, mostly in a camp at Krasnoiarsk in Siberia, Martin was exchanged on medical grounds with a Russian officer and following a lengthy stay in Petrograd, was eventually repatriated in March 1917 (pp. 190-294).


======K347======
'''Grow, Malcolm Cummings, '''''Surgeon Grow: an American in the Russian fighting. ''New York: Frederick A. Stokes Co., 1918. xvi+304pp.
::Grow (1887-1960) was a lieutenant-colonel in the Imperial Russian Army Medical Corps during WWI and finished his career as the first surgeon-general of the U.S. air force. Arriving in Petrograd in September 1915, he was soon sent to the front. In his book he describes his activities at the front during two periods: September 1915 to Easter 1916 and June 1916 to March 1917. He met the tsar at a staff dinner and he was awarded the cross of St George. He was in Petrograd in mid-1917, noting the increasing poverty and unrest, and left Russia after the October revolution to join the American army.

 
======K348======
'''Thurstan, Violetta, '''''The people who run, being the tragedy of the refugees in Russia.'' London and New York: G. P. Putnam’s Sons, 1916. x+175p.
::In her second book Thurstan (see [[#K317|K317]]) describes the demographics, conditions, and first-hand experiences of Polish, Baltic, Rumanian, and Russian refugees fleeing the eastern front during the summer and autumn of 1915. Arriving in Petrograd from Newcastle in December 1915, she spent Christmas with refugee children in Gatchina and Petrograd before travelling to Moscow. She went on to Kiev and Kazan to observe, and report in glowing terms, the government response to the refugee problem.

 
======K349======
'''Gorer, Geoffrey, and Rickman, John, '''''The people of Great Russia: a psychological study. ''London: Cresset Press, 1949. iv+236pp.
::An attempt to understand the people of Russia in terms of their “principal motives” and “typical behaviour”. It is Rickman, a country doctor with the Friends’ War Victims Relief Unit between 1916 and 1918, who provided the on-the-spot experience of life in Russian villages in his ‘Russian Camera Obscura. Ten Sketches of Russian Peasant Life (1916-1918)’ (pp. 23-89).

 
======K350======
'''Bury, Herbert, '''''Here and there in the war area''. London: A.R. Mowbray &amp;amp; Co., 1916. xii+325pp.
::Bishop Bury (see [[#K252|K252]]) paid “a particularly inspiring visit to Russia” in the first months of 1916, arriving from Scandinavia. He was mainly in St Petersburg and in Moscow, travelling there with Sir George Buchanan, who was to receive the freedom of the city, and everywhere records Russian enthusiasm for Britain (pp. 238-325). Bury was later to make and describe visits to Soviet Russia in the 1920s in his ''Russia from within ''(1927).

 
======K351======
'''Hoare, Samuel John Gurney, '''''The fourth seal: the end of a Russian chapter.'' London: William Heinemann, 1930. iii+377pp.
::Sir Samuel, Viscount Templeton (1880-1959), having learnt Russian, was sent to Petrograd by British intelligence in March 1916 to work with the Russian general staff. He eventually became head of the British military mission and remained, together with his wife Lady Maud Lygon, in Russia until March 1917 when his services were required in Rome. Interesting pen-portraits of many prominent Russian and British figures in the Russian capital (pp. 34-359).

 
======K352======
'''Graham, Stephen, '''''Russia in 1916. ''London: Cassell &amp;amp; Co., 1917. vii+179pp.
::Graham’s last book on pre-Revolutionary Russia is essentially a series of essays, reflecting his journey to Ekaterina and Archangel and on to Moscow, followed by travels into central Russia down as far as the Caucasus and return to England in October 1916 via Petrograd. A reprise of his old themes, offered as “my little book of the hour” to keep in touch with our allies.

 
======K353======
'''Francis, David Rowland, '''''Russia from the American embassy, April, 1916 – November, 1918''. New York: Charles Scribner, 1922. xiii+349pp.
::Appointed American ambassador to Russia by President Woodrow Wilson, the democrat politician Francis (1850-1927) was in Petrograd throughout the revolutionary period and provides a chronological account of events, based on his letters, diary, and official papers.

 
======K354======
'''Francis, David Rowland''', ''Dollars and diplomacy: ambassador David Rowland Francis and the fall of tsarism, 1916-1917.'' Edited by Jamie H. Cockfield. Durham, N.C.: Duke University Press, 1981. x+149pp.
::Eighty-one of Francis’s letters to friends and family from April 1916 to March 1917. See also the microfilmed ''Russia in transition: the diplomatic papers of David. R. Francis, U.S. ambassador to Russia, 1916-1918''. Edited by Robert Chadwell Williams and Robert Lester. Frederick, Maryland: University Publications of America, 1986.

 
======K355======
'''Ruhl, Arthur,''''' White nights and other Russian impressions: With illustrations from photographs''. New York: Charles Scribner’s Sons, 1917. x+248pp.
::American journalist and travel writer (b. 1876) spent the summer of 1916 in Russia, visiting Petrograd and Kiev, then the front near Minsk, before travelling down the Volga to Astrakhan. An interesting chapter is devoted to his attending a performance of Chekhov’s ''Three Sisters'' at the Moscow Art Theatre that gave him insight into the Russian character.

 
======K356======
'''Barber, Margaret H., '''''A British nurse in Bolshevik Russia. ''London: A.C. Fifield, 1920. 64pp.
::Daughter of an Anglican clergyman, Barber came to Russia as a Red Cross nurse during WWI and lived and worked in hospitals in a number of Russian cities from Petrograd to Astrakhan between April 1916 and December 1919.

 
======K357======
'''Power, Rhoda, '''''Under Cossack and Bolshevik. ''London: Methuen &amp;amp; Co., 1919. iv+279pp.
::Power (1890-1957), later known as a broadcaster and children’s author, sailed from Newcastle for Petrograd via Scandinavia in 1916 to work as governess to the daughter of a Russian businessman in Rostov-on-Don. She describes life there and a trip in autumn 1917 to Odessa, where hostile attitudes towards her employers induced them to flee, leaving Rhoda behind. She witnessed fighting between the Red Guards and Cossack forces, the subsequent Cossack victory and life under their rule during the winter of 1917-18, the following Bolshevik victory in the spring of 1918, and their subsequent retreat in the face of advancing White Army. Power finally flees to Murmansk and leaves for England on a refugee boat.

 
======K358======
'''Child, Richard Washburn, '''''Potential Russia. ''London: T. Fisher Unwin, 1916. vi+221pp.
::Massachusetts lawyer and journalist, later U.S. ambassador to Italy and apologist of fascism, Child (1881-1935) was sent to Russia early in 1916 by ''Collier’s Weekly'', in which and in other journals he first published many of the sketches gathered together for his book. He sought to assess the effect of the war on the Russian people and the economy and he ended by calling for greater American investment in “an empire of contradictions” but of great potential.

 
======K359======
'''Beable, William Henry, '''''Commercial Russia.'' London: Constable, 1918. 263pp.
::Beable organized and led the Anglo-Russian Trade Commission, visiting Russia between April and October 1916 and during the spring of 1917. He travelled widely throughout western Russia, seeking to demonstrate the potential opportunities available to English manufacturers in Russia.

 
======K360======
'''Stanford Doreen, '''''Sun and snow: a Siberian adventure.'' London: Longmans, 1963. 158pp.
::In May 1916 the twenty-year-old Doreen left England to join her parents in Siberia, where her father, a mining engineer, had worked for the previous eight years. Met in Petrograd by her parents, she travelled with them by train to Krasnoiarsk, by steamer along the Enisei, then by ''tarantas'' to their final destination of Ulen and its copper mine. A year later, they were forced to move and her father found employment until June 1919 at a gold mine in Olkhovskii beyond Minusinsk. They were eventually able to escape from Vladivostok in May 1920.

 
======K361======
'''Heald, Edward Thornton, '''''Witness to revolution: letters from Russia 1916-1919''. Edited by James B. Gidney. Kent, Ohio: Kent State University Press, 1972. xx+367pp.
::Informal family letters and diary entries written by Heald (1885-1967), who arrived in Petrograd in late September 1916 as field secretary of the American YMCA for its prisoner of war relief programme. He remained in Petrograd until July 1917, when he was assigned to the Russian army in Minsk, which the German advance forced him to leave for Kiev in September. He witnessed the February revolution in Petrograd and the October in Kiev, where he described the battles between Ukrainian nationalists and Bolsheviks for control of the city. He was in Siberia during the first few months of the Russian Civil War and was in Vladivostok when the American Expeditionary Force landed.

 
======K362======
'''Washburn, Stanley, '''''The Russian offensive. Being the third volume of “Field notes from the Russian front,” embracing the period from June 15 to September 1, 1916.'' London: Constable, 1917. 193pp.
::In his final volume (see [[#K313|K313]], [[#K340|K340]]) Washburn covers the successful Russian offensive that culminated in the taking of the town of Brody during the summer of 1916.

 
======K363======
'''Boleslavski, Richard, and Woodward, Helen, '''''Way of the Lancer. ''Indianapolis: Bobbs-Merrill, 1932. 316pp.
::Autobiographical account of Boleslavski’s (1887-1937) experience fighting with a Polish volunteer lancer regiment within the Russian army. The account describes his experiences of life on the eastern front from autumn 1916 onwards, and charts the breakdown of discipline within the Russian army following the February Revolution. Following Nicholas II’s abdication, Boleslavski’s regiment withdraws from the Russian army and attempts to make its way back to Poland.

 
======K364======
'''Inglis, Elsie Maud, '''''Dr Elsie Inglis''. By Lady Frances Balfour: London: Hodder and Stoughton, 1918. x+253pp.
::The famed Scottish suffragette and doctor (1864-1917), after serving in Serbia during the first years of WWI, left with her seventy-six-strong nursing unit of the Scottish Women’s Hospitals for Russia in September 1916. From Archangel they travelled via Moscow south to Odessa, where they were to remain until the following October. They left Archangel on the return journey on 18 November 1917; but Dr Inglis died on 27 November, the day after the ship reached Newcastle. Letters to her family and friends (pp. 197-233).

 
======K365======
'''Inglis, Elsie Maud, '''''Between the lines: letters and diaries from Elsie Inglis’s Russian unit''. Arranged and edited by Audrey Fawcett Cahill. Edinburgh: Pentland Press, 1999. x+372pp.
::The “choral narrative” the editor promised in her earlier book.

 
======K366======
'''Fawcett, Margaret, '''''The First World War papers of Margaret Fawcett: letters and diaries from Russia and Roumania 1916-1917. ''Edited and with an introduction by Audrey Fawcett Cahill. Pietermaritzburg: Wyllie Desktop Publishing, 1993. viii+144pp.
::The specific Russian element in the two diaries and a letter-book of Margaret Fawcett (b. 1892), an orderly in Dr Inglis’s unit, is the initial journey from Archangel to Odessa in September 1916 and the return journey a year later (pp. 28-32, 38-40, 52-55, 62-67, 78-80, 104-08, 121-22).

 
======K367======
'''Colquhoun, James, '''''Adventures in red Russia from the Black Sea to the White Sea. ''London: John Murray, 1926. viii+193pp. [Printed for private circulation.]
::Chairman of the Caucasus Copper Company and with previous visits to Russia, the Scottish engineer (b. c.1858) arrived in Petrograd on 13 October 1916 en route for Tiflis. His final destination was Borchka near the Turkish border, where he was to supervise the reconstruction of the metallurgical plant, damaged by Turkish forces. He was subsequently caught up in the revolutionary events of 1917 and continuing incursions by the Turks. He eventually made his escape through Georgia and reached Tsaritsyn, whence he went by steamer to Nizhnii Novgorod. He made his way to Murmansk and sailed for England on 16 June 1918.

 
======K368======
'''Austin, Walter, '''''A war zone gadabout: being an authentic account of four trips to the fighting nations during 1914, ’15, ’16''. Boston: R.H. Hinkley Co., 1917. 161pp.
::A “mere gadabout tourist” and correspondent for the Massachusetts weekly ''Dedham Transcript'', Austin arrived in Petrograd on 11 November 1916, leaving three weeks later on 1 December after a round trip to Moscow. He attended a meeting of the Duma and heard Sturmer and Miliukov speak (pp. 112-54).

 
======K369======
'''Dosch-Fleurot, Arno Walter, '''''Through war to revolution, being the experiences of a newspaper correspondent in war and revolution, 1914-1920. ''London: John Lane, 1931. xii+242pp.
::Dosch-Fleurot (1879-1951), correspondent of the New York ''World,'' was sent from the western front to Petrograd in November 1916 and was soon caught up by the revolutionary events of 1917. He was to remain until the end of 1918, when he escaped via Finland (pp. 97-215).

 
======K370======
'''Harper, Florence MacLeod, '''''Runaway Russia. ''New York: Century Co., 1918. ix+321pp.
::Harper left Vancouver in December 1916 to spend nine months in Russia as staff war correspondent of ''Leslie’s Weekly'', reaching Petrograd via the Trans-Siberian from Kharbin. Working with the magazine’s photographer Donald Thompson (see [[#K372|K372]]), she was witness to the street violence during the February Revolution. In April she travelled to the eastern front to work as a surgical nurse at a Red Cross field hospital. Back in the capital in June 1917, she met members of the Women’s Battalion of Death and visited Cronstadt, before departing for Finland in early September.

 
======K371======
'''Thompson, Donald C., '''''Donald Thompson in Russia.'' New York: Century Co., 1918. xix+353pp.
::American photographer Donald Thompson (b. 1895), who had previously been in Russia in March-May 1915, returned to work with Florence Harper for ''Leslie’s Weekly'', arriving via the Trans-Siberian in Petrograd on 17 February 1917. In a series of letters to his wife and in photographs, he captured the events through the spring and summer of 1917 not only in the capital but also at the Galician front and in Moscow, before leaving in August 1917 via the Manchurian border.

 
======K372======
'''Thompson, Donald C. and Harper, Florence Macleod, '''''From Czar to Kaiser: the betrayal of Russia''. Garden City, New York: Doubleday, Page &amp;amp; Co., 1918. viii+200pp.
::The work is a collection of extraordinary photographs taken by Thompson and arranged thematically: before the revolution, during the February revolution in Petrograd, the May parades and labour riots in Petrograd, hospital conditions on the eastern front, the women’s battalion, the July riots in Petrograd, from the front line and riots by the Bolsheviks during the autumn. Harper provides brief descriptions of each photograph.

 
======K373======
'''Petersson, C.E.W., '''''How to do business with Russia: hints and advice to business men dealing with Russia.'' With notes and additional chapters by W. Barnes Steveni and a foreword by Charles E. Musgrave. London: Sir Isaac Pitman &amp;amp; Sons., 1917. xviii+202pp.
::Designed as a sort of businessman’s ''Baedeker'' to encourage trade with Russia and written by an experienced merchant operating in Riga and St Petersburg, it fell foul of revolutionary events immediately on publication. The preface by the secretary of the London chamber of commerce is dated February 1917 and the preface written by long-standing Petersburg resident Steveni is dated April 1917, acknowledging that the February revolution would modify “mostly for the best” conditions – but October rendered it an historical document with fascinating information about what was.

 
======K374======
'''Souiny-Seydlitz, Leonie Ida Philipovna, '''''Russia of yesterday and to-morrow. ''New York: The Century Co., 1917. 382pp.
::Following her marriage to Baron Seidlits, the Czech-born author (b. 1865) moved to Russia, where she lived for many years until emigrating to the USA in 1914. The most interesting chapters in her attempt to give a wide-reaching survey are her comparison of Russia and America (pp. 220-55) and ‘Russian art, dramatic literature and music’, where she discusses, among other topics, the Moscow Arts Theatre and Stanislaslavskii (pp. 256-85). Published in June 1917, the book finishes with guarded optimism after the events of February.

 
======K375======
'''Wilson, Henry Hughes, '''''Field-Marshall Sir Henry Wilson: his life and diaries''. By Major-General Sir C.E. Callwell. With a preface by Marshal Foch. London: Cassell and Co., 1927. 2 vols.
::Wilson (1864-1922), director of British military operations since 1910, after visiting Paris and Berlin, paid a brief first visit to St Petersburg, Moscow, and Kiev in September 1912 (vol. I, p. 117). In January 1917 he headed the joint allied mission to Russia (a party of some fifty British, French and Italian representatives) that sailed on the ''Kildonan Castle ''for the White Sea. He arrived in Petrograd on 29 January, proceeded to the front at Pskov on 8 February, and journeyed on to Moscow via Riga and Minsk. After further talks in Petrograd, he sailed from Russia on 25 February (vol. I, pp. 312-22).

 
======K376======
'''De Windt, Harry, '''''[https://archive.org/details/russiaasiknowit00dewi Russia as I know it].'' London: Chapman and Hall, 1917. xii+232pp.
::De Windt’s final summing-up of his experiences of Russia, where he covered some 50,000 miles and spent some four years between 1887 and the time of writing (preface dated April 1917). Includes more on European Russia than previously (including Petrograd, which he disliked) but also covers Finland, Ukraine, the Crimea, the Caucasus, Siberia and central Asia. (See [[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)#J53|J53]] for full listing of other entries.)

======K377======
'''Hall, Bert, '''''One man’s war: the story of the Lafayette escadrille. ''Edited by John Jacob Niles. London: John Hamilton, 1929. 352pp.
::Hall, a “seasoned Soldier of Fortune”, was a member of an American volunteer squadron within the French air service that arrived in Russia on 12 January 1917 to aid the Russian air service. He witnessed the major events of the revolutions before escaping via the Trans-Siberian to China at the end of the year (pp. 226-76).

======K378======
'''Houghteling, James Lawrence, Jr., '''''A diary of the Russian revolution.'' New York: Dodd, Mead &amp;amp; Co., 1918. xxii+195pp.
::Houghteling (1883-1937), an attaché in the American embassy in Petrograd from 19 January 1917, provides a diary of events leading up to and during the February revolution in the capital and in Moscow. He left Petrograd for Siberia on 3 April.

 
======K379======
'''Price, Morgan Philips, '''''My reminiscences of the Russian revolution. ''London: George Allen &amp;amp; Unwin, 1921. 402pp.
::Price’s third book (see [[#K219|K219]], [[#K322|K322]], [[#K380|K380]]-[[#K381|81]]) was dedicated to those in Britain who, like himself, “defended the Soviet republic of Russia against the onslaughts of the international bondholders”. He offers “a consecutive account”, relying on his own experiences and diaries for the first one and a half years of the revolution and devoting only the last two chapters to developments in the period after he left Russia in 1919.

 
======K380======
'''Price, Morgan Philips, '''''My three revolutions. ''London: George Allen &amp;amp; Unwin, 1969. 310pp.
::The three revolutions were the Russian, the German, and the British, and of these the Russian had a major impact “in the most critical period of my life” and “greatly influenced my critical thinking for a time”. Writing in his eighties, Price reviews all his visits to Russia between 1908 and 1917 (pp. 21-94).

 
======K381======
'''Price, Morgan Philips, '''''Dispatches from the revolution: Russia 1916-18.'' Edited by Tania Rose. Foreword by Eric Hobsbawn. London: Pluto Press, 1997. xiv+181pp.
::A skilfully edited “selection of Price’s unpublished memoranda, letters to his family, and some of his published articles [from the ''Manchester Guardian''] with a bearing on the revolutions which reflect not only the events as they unfolded but also his own reactions to them” represents a significant addition to Price’s four other published books on Russia. The letters and articles were written from Tiflis, Kutais, Rostov-on-Don, Samara, Petrograd, and Moscow (pp. 18-154).

 
======K382======
'''Brennan, Hugh, '''''Sidelights on Russia. ''London: David Nutt, 1918. 112pp.
::Lecturer in Russian at the University of Glasgow, Brennan refers to an earlier visit to the south of Russia c.1908. It is, however, the events of 1917 (pre-October), when he was in Petrograd, that are the centre of attention as he assesses the British – and British colony’s – position in the light of revolutionary events, stressing the need to study the language and seize business opportunities in the context of persisting hopes for the emergence of “a new, great, and democratic Russia”.

 
======K383======
'''Jones, Stinton, '''''Russia in revolution, being the experiences of an Englishman in Petrograd during the upheaval. ''London: Herbert Jenkins, 1917. xvi+279pp.
::The British engineer arrived in Moscow for the first time in November 1905, but remained for twelve years, married a Russian, travelled extensively throughout Russia, and viewed the February revolution from his office on Nevskii Prospect and on the streets. He provides a graphic account of the five days from 10 to 14 March.

 
======K384======
'''Pollock, John, '''''The Bolshevik adventure.'' London: Constable &amp;amp; Co., 1919. 276pp.
::The second instalment of Pollock’s adventures in Russia (see [[#K331|K331]], [[#K332|K332]]), here specifically the period of the February and October revolutions and ending with his escape.

 
======K385======
'''Rivet, Charles,''''' The last of the Romanovs''. Translated, with an introduction by Hardress O’Grady. London: Constable and Company, 1918. 246pp.
::Rivet (b. 1881), who had been in Russia since 1901, firstly as a university teacher and then as Petrograd correspondent of the Paris ''Le'' ''Temps, ''provides a sympathetic analysis of the February revolution, presented in three parts ‘Unknown Russia’, ‘The Revolution’ and ‘France and Germany’.

 
======K386======
'''Maugham, William Somerset, '''''A writer’s notebook. ''London: William Heinemann, 1949. xvi+349pp.
::Maugham (1874-1965) was in Petrograd between the February and October revolutions, operating as a British “secret agent”, such as he was later mockingly to portray in his novel ''Ashenden'' (1928). Under the heading ‘1917’ his notebook contains his jottings on Russian literature and Dostoevskii in particular, on the Russian character and such personalities as Kerenskii and Savinkov, and on Nevskii Prospekt and the Alexander Nevskii lavra (pp. 139-79).

 
======K387======
'''De Robien, Louis, '''''The diary of a diplomat in Russia, 1917-1918''. Translated from the French by Camilla Sykes. London: Michael Joseph, 1969. 319pp.
::Comte Louis (1888-1958) was attached to the French embassy in St Petersburg from 1914 but it was only at the beginning of March 1917 that he began to record daily events in the capital. His diary is a fascinating and opinionated record of events and personalities not only in the capital but also, from March 1918, in Helsingfors and Vologda, to where the embassy was relocated, and from Archangel, whence he and his wife left in December 1919 for Paris.

 
======K388======
'''Anet, Claude [pseudonym of Schopher, Jean], '''''Through the Russian revolution: notes of an eye-witness, from 12th March-30th May. ''London: Hutchinson, 1917. 253pp. illus.
::Schopher (see [[#K141|K141]]) returned to Russia as the correspondent of the ''Petit Parisien'' and sent off to Paris vivid daily accounts of the events he witnessed in Petrograd over a twelve-week period, beginning in fact on 7 March 1917. This is a translation of the first of the four volumes of the French original, covering a longer period.

   
http://dx.doi.org/10.11647/OBP.0042.11


←[[In_the_Lands_of_the_Romanovs:_An_Annotated_Bibliography_of_First-hand_English-language_Accounts_of_the_Russian_Empire_(1613-1917)|[Back to contents]]]

{{CourseCat}}

[[Category:19th century in Russia]]</text>
      <sha1>36qk71lz3wkz7ou1nge538mj60ohxby</sha1>
    </revision>
  </page>
  <page>
    <title>Haskell programming in plain view</title>
    <ns>0</ns>
    <id>203942</id>
    <revision>
      <id>2818543</id>
      <parentid>2817963</parentid>
      <timestamp>2026-07-19T18:09:11Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Lambda Calculus */</comment>
      <origin>2818543</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6273" sha1="0kj02yhjxt1v7cb4r8qw4chrp7k80dz" xml:space="preserve">
==Introduction==
* Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]])
* Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]])
* Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]])
* Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]])
* Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]])
&lt;/br&gt;

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

&lt;/br&gt;

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

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


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

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

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

==Lambda Calculus==
* Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]])
* Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]])
* Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]])
* Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]])
* Encoding Datatypes 
:- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]])
:- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]])
:- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]])
:- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]])
:- Recursions ([[Media:LCal.9A.Recursion.20260707.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]])
&lt;/br&gt;
&lt;/br&gt;

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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

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


[[Category:Haskell|programming in plain view]]</text>
      <sha1>0kj02yhjxt1v7cb4r8qw4chrp7k80dz</sha1>
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    <revision>
      <id>2818545</id>
      <parentid>2818543</parentid>
      <timestamp>2026-07-19T18:10:35Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Lambda Calculus */</comment>
      <origin>2818545</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
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==Introduction==
* Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]])
* Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]])
* Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]])
* Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]])
* Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]])
&lt;/br&gt;

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

&lt;/br&gt;

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

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


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

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

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

==Lambda Calculus==
* Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]])
* Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]])
* Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]])
* Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]])
* Encoding Datatypes 
:- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]])
:- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]])
:- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]])
:- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]])
:- Recursions ([[Media:LCal.9A.Recursion.20260713.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]])
&lt;/br&gt;
&lt;/br&gt;

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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

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


[[Category:Haskell|programming in plain view]]</text>
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    </revision>
    <revision>
      <id>2818547</id>
      <parentid>2818545</parentid>
      <timestamp>2026-07-19T18:11:24Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Lambda Calculus */</comment>
      <origin>2818547</origin>
      <model>wikitext</model>
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==Introduction==
* Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]])
* Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]])
* Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]])
* Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]])
* Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]])
&lt;/br&gt;

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

&lt;/br&gt;

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

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


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

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

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

==Lambda Calculus==
* Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]])
* Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]])
* Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]])
* Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]])
* Encoding Datatypes 
:- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]])
:- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]])
:- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]])
:- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]])
:- Recursions ([[Media:LCal.9A.Recursion.20260714.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]])
&lt;/br&gt;
&lt;/br&gt;

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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

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


[[Category:Haskell|programming in plain view]]</text>
      <sha1>nwqbh097ywexbmsjbwfhusm3c39ueu5</sha1>
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    <revision>
      <id>2818573</id>
      <parentid>2818547</parentid>
      <timestamp>2026-07-19T19:57:13Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Lambda Calculus */</comment>
      <origin>2818573</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6273" sha1="g1defkldktnyo4w7sv2tfwrcq8zsu4a" xml:space="preserve">
==Introduction==
* Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]])
* Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]])
* Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]])
* Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]])
* Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]])
&lt;/br&gt;

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

&lt;/br&gt;

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

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


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

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

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

==Lambda Calculus==
* Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]])
* Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]])
* Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]])
* Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]])
* Encoding Datatypes 
:- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]])
:- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]])
:- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]])
:- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]])
:- Recursions ([[Media:LCal.9A.Recursion.20260720.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]])
&lt;/br&gt;
&lt;/br&gt;

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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

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


[[Category:Haskell|programming in plain view]]</text>
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    <title>Python programming in plain view</title>
    <ns>0</ns>
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      <id>2818597</id>
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      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
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      <text bytes="2696" sha1="7aqqgn5oyxebvtdcvtnihf6lcv39owx" 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.20260720.pdf |A]], [[Media:Python.Work2.Library.1B.20260720.pdf |B]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


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

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

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

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

=== Handling a Big Work ===

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

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

=== Handling Series of Data ===

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

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

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


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

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

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

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

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
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      <comment>Update for 2026</comment>
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      <text bytes="533" sha1="1agzbib7bcjgkb6i6sarj3qkfx6pw4k" xml:space="preserve">&lt;noinclude&gt;{{title|Timetable}}&lt;/noinclude&gt;
{{info|See [https://cloud.timeedit.net/au_canberra/web/public/ri1Q7.html Timetable 2026]}} 
# Enrol in 7124 ON-CAMPUS or ONLINE REALTIME
# Allocate to a tutorial group (either on-campus, virtual, or asynchronous/recorded)&lt;noinclude&gt;

==See also==
* [[Motivation and emotion/Drop-in|Drop-in]]
* [[Motivation and emotion/Lectures|Lectures]]
* [[Motivation and emotion/About/Schedule|Schedule]]
* [[Motivation and emotion/Tutorials|Tutorials]]

[[Category:Motivation and emotion]]
&lt;/noinclude&gt;</text>
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      <comment>Update for 2026</comment>
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{{info|See [https://cloud.timeedit.net/au_canberra/web/public/ri1lA204065Z0BQ81QQ2ZEA8t11Fnk868851lk7d9271769kAB0t5FY20501F461095033X51AB5913906gZ30Y0y1054gQ6ZQ23Y6377Y1323QX67627.phtml Timetable 2026]}} 
# Enrol in 7124 ON-CAMPUS or ONLINE REALTIME
# Allocate to a tutorial group (either on-campus, virtual, or asynchronous/recorded)&lt;noinclude&gt;

==See also==
* [[Motivation and emotion/Drop-in|Drop-in]]
* [[Motivation and emotion/Lectures|Lectures]]
* [[Motivation and emotion/About/Schedule|Schedule]]
* [[Motivation and emotion/Tutorials|Tutorials]]

[[Category:Motivation and emotion]]
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    <title>History of Topics in Special Relativity/Twin paradox</title>
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      <text bytes="108753" sha1="r8b60b8nxz7cg3zbt9x1c33foxd8cag" xml:space="preserve">{| style=&quot;width:20%; font-size:13px;&quot; align=right

|{{../Other Topics (header)}}

|}

==Early history of the twin paradox==

{{Lorentzbox|Text={{center|Date of article creation: 9 November 2023; Last major revision: 2 March 2026}}}}

a) When was the [[:w:twin paradox]] applied to life forms and human beings?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref group=S name=pes /&gt;&lt;ref group=S name=during /&gt; report that {{slink||Einstein 1911-HU}} discussed the aging of living organisms, and that {{slink||Langevin 1911-HU}} and {{slink||Wiechert 1911-HU}} explicitly discussed the aging of human beings. 

:*More details in sections {{slink||Human beings in 1911|Twins from 1911 to 1920}}, including newspaper articles from 1911 written by {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}} that clearly show that Einstein was the first to explicitly discuss the aging of human beings as well.

b) Who was the first to formulate the principle of maximal proper time along straight worldlines, upon which differential aging in the standard twin paradox is based?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref group=S name=during /&gt; mention Langevin (1911), Laue (1911). 

:*More details in section {{slink||Maximal proper time}} with the contributions of Langevin (1911), Wiechert (1911), Study (1911), Laue (1911-13).

c) Who was the first to formulate [[w:Triangle inequality#Reversal in Minkowski space|inverse triangle inequality]] in Minkowski space, which represents the simplest version of the twin paradox?

:*See details in section {{slink||Triangle inequality}} with the contributions of Robb (1914-20), Eddington (1922), Rogers (1922).

d) Who was the first to show that any influence of proper acceleration on clocks can be neglected in the computation of the twin paradox from the viewpoint of the stay-at-home twin?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref group=S name=pes /&gt; mention Einstein (1911), Laue (1913). 

:*More details in section {{slink||Negligibility of proper acceleration}} with the contributions of Einstein (1911), Wiechert (1911), Laue (1913), Lorentz (1913).

e) Who was the first to introduce the three clock/brother example that completely removes acceleration from the clock/twin paradox?

:*Historical accounts&lt;ref group=S name=debs /&gt;&lt;ref group=S name=alizzi /&gt; date it back to Lange (1927) and Lord Halsbury (1957).

:*More details in section {{slink||Relay (three brothers) experiment}} with the contributions of Grünbaum (1911) and Wiechert (1920-22).

f) Who was the first to use acceleration as an asymmetry indicator?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref name=cuvaj group=S /&gt;&lt;ref group=S name=pes /&gt; mention Langevin (1911), Einstein (1918). 

:*More details in section {{slink||Acceleration as asymmetry indicator}} with the contributions of Langevin (1911), Sommerfeld (1913), Lorentz (1913), Einstein (1914-20).

g) Who was the first to use different frame distribution as asymmetry indicator as an asymmetry indicator?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref group=S name=pes /&gt; mention Laue (1911-13). 

:*More details in section {{slink||Frame distribution as asymmetry indicator}} with the contributions of Laue (1911-13), Bloch (1918).

h) Who was the first to describe the perspective of the traveler?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref group=S name=beng /&gt; mention Langevin (1911), Lorentz (1914), Einstein (1918). 

:*More details in section {{slink||Perspective of the traveler}} with the contributions of Langevin (1911), Lorentz (1913-14), Einstein (1918), Thirring (1921).

i) Who was the first to describe a round-trip experiment in curved spacetime?

:*See section {{slink||Curved spacetime}} with the contribution of Becquerel (1922).

j) Who was the first to denote the round-trip experiment as paradoxical?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref group=S name=during /&gt; point to Laue (1911).

:*See section {{slink||Paradoxical?}} for details.

k) Who was the first to misunderstand the twin paradox?

:*See section {{slink||Misunderstandings}} with the contributions of Berg (1910), Wiechert (1911), Campbell (1911/12), Gruner (1912).

l) What were Einstein's contributions?

:*See section {{slink||Einstein's contributions}}.

==Human beings in 1911==

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

![[w:Albert Einstein|Einstein]] 

|-

|{{anchor|Einstein 1905}}In 1905&lt;ref name=einstein05 /&gt; he showed that a clock moving on a round-trip away from A and back along a polygonal or curved path, is retarded with respect to a clock stationary at A by approximately &lt;math&gt;\tfrac{1}{2}t(v/V)^{2}&lt;/math&gt; at reunion. For example, a clock on the equator is retarded with respect to a clock on the pole. He described this consequence as being &quot;peculiar&quot; (German: eigentümlich).

{{anchor|Einstein 1911-HU}}In a lecture given on January 1911&lt;ref name=einstein11a /&gt; (published in November), he extended this &quot;funny&quot; (German: drollig) experiment to living organisms:

{| 

! width=55% | Einstein wrote

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Wenn wir z. B. einen lebenden Organismus in eine Schachtel hineinbrächten und ihn dieselbe Hin- und Herbewegung ausführen lassen wie vorher die Uhr, so könnte man es erreichen, dass dieser Organismus nach einem beliebig langen Fluge beliebig wenig geändert wieder an seinen ursprünglichen Ort zurückkehrt, während ganz entsprechend beschaffene Organismen, welche an den ursprünglichen Orten ruhend geblieben sind, bereits längst neuen Generationen Platz gemacht haben. Für den bewegten Organismus war die lange Zeit der Reise nur ein Augenblick, falls die Bewegung annähernd mit Lichtgeschwindigkeit erfolgte!

| style=&quot;padding: 0px 20px 0px 20px;&quot; |For example, if we put a living organism in a box and make it undergo the same back and forth movement as the clock before, we could achieve that this organism returns to its original location with arbitrary little change after a flight of arbitrary length, whereas completely identical organisms that remained at rest in the original location have long since made room for new generations. To the moving organism, the long journey was only a moment if the movement happened close to the speed of light!

|}

{{Lorentzbox|Text=Two participants of that lecture, {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}}, report that Einstein also talked about the aging of ''human beings''.}}

|-

!{{anchor|Lämmel 1911-HU}}[[w:Rudolf Lämmel|Lämmel]]

|-

|He attended Einstein's 1911 lecture and gave a popular report about it in the Swiss newspaper &quot;[[w:Neue Zürcher Zeitung|Neue Zürcher Zeitung]]&quot; published on 28 April 1911,&lt;ref name=lammel /&gt; including additional details. Regarding the round-trip clock experiment he wrote:

{| 

! width=50% | Lämmel wrote

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Bewegt sich eine Uhr mit Lichtgeschwindigkeit längs einer Geraden, auf der gerichtete Uhren stehen, so scheint die bewegte Uhr, beurteilt vom Standpunkt der ruhenden aus, im oben stizzierten Sinn, stillzustehen. Kehrt die Uhr, nach einem Ruck, mit Lichtgeschwindigkeit wieder zurück zur Zentral-Uhr, so ist, nach Einstein, für den Beobachter bei der Zentral-Uhr die Sache so, als ob ein mit der bewegten Uhr mitgeführter Beobachter (samt dessen Uhr) nicht gealtert hätte. Hinge also des letzteren Alter von den Angaben des ruhenden Beobachters ab, so könnte der von einer großen Reise ins Weltall zurückkehrende Beobachter bei der Zentral-Uhr spätere Generationen antreffen – er selber hätte nicht gealtert. Welche Bedeutung diese ''ad absurdum'' geführte Gedankenspielerei etwa hat, läßt sich heute nicht absehen – vielleicht, ja wahrscheinlich ist sie ohne jeden Einfluß auf die tatsächlichen Verhältnisse. Aber man sieht dabei immerhin, daß die Physik imstande ist, die kühnsten Träume der Phantasie noch – zu überbieten.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Let a clock be moving at speed of light along a line on which regulated clocks are standing, then the moving clock's hand appears to be standing still (in the sense described above) as judged from the standpoint of the resting one. If the clock, after one jolt, comes back with light speed to the central clock, then according to Einstein the matter presents itself to the observer at the central clock, as if the observer comoving with the clock (together with his clock itself) hasn't been grown older. Thus if the age of the latter would depend on the indications of the resting observer, the observer returning from a great journey into space could meet later generations at the central-clock – he himself hasn't been grown older. The importance of this play of thought led ''ad absurdum'' cannot be seen today – maybe, or even probably, it is without any influence on the actual situations. Though at least one can see that physics is able to – surpass – even the boldest dreams and fantasies.

|}

Lämmel in December 1920 (published 1921)&lt;ref name=lammel2 /&gt; again alluded to Einstein's lectures in Zürich (possibly the one from 1911, and maybe also later ones), describing a discussion between himself and Einstein. After Einstein concluded that the travelers who came back after their journey will probably meet their former contemporaries as old men while they themselves could have been away for only a few years, Lämmel objected that this conclusion is only drawn with respect to rods and clocks, but not with respect to living beings. Einstein responded though, that all processes in the blood, in the nerves etc. are eventually periodical oscillations, i.e. motions. Yet to any such motion the relativity principle applies, thus the conclusion regarding the unevenly rapid aging it permissive.

{{Lorentzbox|Text=While the official publication of Einstein's January lecture ({{slink||Einstein 1911-HU}}) mentions the aging of organisms, Lämmel recalls the reference to the aging of a human space traveler (&quot;observer returning from a great journey into space&quot;). This means that Einstein was the first to use human beings in the clock/twin paradox on January 16 which was first published by Lämmel on April 28, 1911. In comparison, {{slink||Langevin 1911-HU}} used space travelers in a lecture on April 10 with publication in July, and {{slink||Wiechert 1911-HU}} used space travelers in lectures held between March 25 and May 23 with publication in July/September. It seems very unlikely that before April 28, Lämmel became somehow aware of the content of Langevin's or Wiechert's lectures held a few weeks earlier, in order to use them in his description of Einstein's lecture.}}

|-

!{{anchor|Langevin 1911-HU}}[[w:Paul Langevin|Langevin]]

|-

|On 10 April 1911, published July 1911,&lt;ref name=langevin1 /&gt; he held a now famous lecture popularizing the clock/twin paradox which he derived from the proper time integral as described in {{slink||Langevin 1911-PT}}. He demonstrated that a moving radioactive sample of radium is less evolved and less aged and therefore more active at return then the ones that remained in the laboratory. He also used light signals and the Doppler effect to visualize the effect. The most famous part concerned his description of the aging of human space travelers:

{| 

! width=50% | Langevin wrote

! [[:s:Translation:The Evolution of Space and Time|English Wikisource translation]]

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Cette remarque fournit le moyen, à celui d’entre nous qui voudrait y consacrer deux années de sa vie, de savoir ce que sera la Terre dans deux cents ans, d’explorer l’avenir de la Terre en faisant dans la vie de celle-ci un saut en avant qui pour elle durera deux siècles et pour lui durera deux ans, mais ceci sans espoir de retour, sans possibilité de venir nous informer du résultat de son voyage puisque toute tentative du même genre ne pourrait que le transporter de plus en plus avant.

Il suffirait pour cela que notre voyageur consente à s’enfermer dans un projectile que la Terre lancerait avec une vitesse suffisamment voisine de celle de la lumière, quoique inférieure, ce qui est physiquement possible, en s’arrangeant pour qu’une rencontre, avec une étoile par exemple, se produise au bout d’une année de la vie du voyageur et le renvoie vers la Terre avec la même vitesse. Revenu à la Terre ayant vieilli de deux ans, il sortira de son arche et trouvera notre globe vieilli de deux cents ans si sa vitesse est restée dans l’intervalle inférieure d’un vingt-millième seulement à la vitesse de la lumière. Les faits expérimentaux les plus sûrement établis de la physique nous permettent d’affirmer qu’il en serait bien ainsi.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |This remark provides the means for any among us who wants to devote two years of his life, to find out what the Earth will be in two hundred years, and to explore the future of the Earth, by making in his life a jump ahead that will last two centuries for Earth and for him it will last two years, but without hope of return, without possibility of coming to inform us of the result of his voyage, since any attempt of the same kind could only transport him increasingly further.

For this it is sufficient that our traveler consents to be locked in a projectile that would be launched from Earth with a velocity sufficiently close to that of light but lower, which is physically possible, while arranging an encounter with, for example, a star that happens after one year of the traveler's life, and which sends him back to Earth with the same velocity. Returned to Earth he has aged two years, then he leaves his ark and finds our world two hundred years older, if his velocity remained in the range of only one twenty-thousandth less than the velocity of light. The most established experimental facts of physics allow us to assert that this would actually be so.

|}

{{Lorentzbox|Text=Reading his lecture in full, one finds the word &quot;paradoxical&quot; only in relation to the constancy of light speed, not on relation to the round-trip clock experiment.}}

|-

!{{anchor|Wiechert 1911-HU}}[[w:Emil Wiechert|Wiechert]]

|-

|In lectures on 25 March and 23 May 1911, submitted July and published September 1911,&lt;ref name=wiechert11 /&gt; he described the round-trip clock experiment with two equal clocks regulated to the same rate and brought to the same pointer position, or by introducing the same chemical process two times, or by introducing ''two life forms that began their life at the same time''. At the end of his paper he applied this to human travelers:

{|

! width=50% | Wiechert wrote

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Nehmen wir aber wieder eine Relativgeschwindigkeit an, die bis auf 3 Proz. der Lichtgeschwindigkeit nahekommt, dann wird das Verhältnis der empfundenen Zeitlängen wie 4:1. Das Bild mag etwas weiter noch ausgemalt werden. Denken wir uns, daß ein Beobachter durch den Raum unseres Sternhimmels mit dieser Geschwindigkeit in einer Kreisbahn mit einem Radius von 16 Lichtjahren fährt, dann wird er nach unserer Zeitrechnung nach je 100 Jahren wieder an unserem Sonnensystem vorüberkommen. In seinem Gefährt wird dabei die Zentrifugalkraft so auf ihn einwirken, daß sie gemäß den Relativitätsgesetzen der Einwirkung der Schwerkraft auf uns Erdenbewohner gleich erscheint. Es sind also die wirkenden Kräfte nur so groß, daß der Phantasie die Möglichkeit geboten wird, den Reisenden als menschliches Wesen zu denken. Da hier dauernd &lt;math&gt;\sqrt{1-v^{2}/c^{2}}&lt;/math&gt; ist, fließt die Eigenzeit für den Reisenden viermal langsamer dahin, als für die Bewohner der Gestirne. Wenn er also nach 100 unserer Jahre wieder zu unserem Sonnensystem zurückkehrt, wird er sich selbst nur um 25 Jahre gealtert fühlen. Erreicht er nach der Entwicklung seines Körpers und nach seiner Zeitempfindung ein Alter von 75 Jahren, so entspricht dies doch einer dreimaligen Wiederkehr zu unserem Sonnensystem, also 300 unserer Erdenjahre.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Yet if we again assume a relative velocity approximating the speed of light by 3 percent, then the ratio of the experienced duration of time becomes 4:1. This image can be further extended. Let's imagine that an observer travels with that velocity on a circular path at a radius of 16 light years through the space of our galaxy, then according to our time calculation he passes by our solar system every 100 years. In his vehicle the centrifugal force will act on him in such a way, that in accordance with the relativity laws it will appear to be equal to the force of gravity acting upon the inhabitants of Earth. Thus the acting forces are only thus big, in order to give our fantasy the possibility to imagine the traveler as a human being. Since we have &lt;math&gt;\sqrt{1-v^{2}/c^{2}}&lt;/math&gt; throughout, proper time flows four times slower for the traveler than for the inhabitants of the stars. Thus when he comes back to our solar system after 100 of our years, he will feel to have aged only by about 25 years. If he reaches an age of 75 years according to the development of his body and his own time experience, then this corresponds to a threefold return to our solar system, i.e. 300 of our Earth years.

|}

{{Lorentzbox|Text=a) Wiechert (1915)&lt;ref name=wiechert15 /&gt; later provided a short historical survey of the clock/twin paradox. He referred to the fact that already {{slink||Einstein 1905}} considered the case of two clocks (&quot;Einstein's clock experiment&quot;), and even though [[w:Hermann Minkowski|Minkowski]] himself didn't consider the case, his proper time formula provides the result in a straight forward manner. The latter was done by himself in lectures on 25 March and 23 May 1911, as well as by Langevin published in July 1911. Wiechert pointed out that he himself and Langevin used &quot;humorist&quot; examples in order to clarify the situation: While Wiechert argued that one has to make a journey in order to stay young, Langevin argued that one has to romp about in a laboratory in order to stay young. Both of them used human beings, arguing that their physical and mental life should have been influenced in the same way as any other process in nature.

b) The dates given by Wiechert (1915) are not complete. The correct ones are:

*Langevin's lecture on 10 April 1911, published in July.

*Wiechert's lectures on 25 March and 23 May 1911, submitted on July 26, published in September.

*He was still unaware of Einstein's lecture from January 1911, published in November 1911.}}

|-

!{{anchor|Müller 1911-HU}}[[w:Fritz Müller-Partenkirchen|Müller]]

|-

|The freelance writer and law student Fritz Müller (who was later known as [[w:Fritz Müller-Partenkirchen|Müller-Partenkirchen]]) attended Einstein's lecture and wrote a popular report about it in the German newspaper &quot;[[w:Berliner Tageblatt|Berliner Tageblatt]]&quot; on 16th and 23rd October 1911,&lt;ref name=muller /&gt; in which he gave further details (compare with {{slink||Lämmel 1911-HU}}). Regarding the clock/twin paradox he wrote:

{| 

! width=50% | Müller wrote

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Zwei gleichgehende Uhren sollen je einen Beobachter haben und nebeneinander ruhen. Nun soll die eine mit ihrem Beobachter plötzlich mit Lichtgeschwindigkeit in den Weltenraum hinausreisen. Vorher haben die beiden vereinbart, sich alle Sekunden mit einem Lichtsignal die Zeit zu telegraphieren. [...] In unserem Grenzfall, wo die Reise mit Lichtgeschwindigkeit vor sich geht, müßte der ruhende Beobachter erklären, jene andere Uhr käme in der Zeit überhaupt nicht voran. Die Zeit stünde dort still. Tatsächlich kommen die Einsteinschen Gleichungen zu diesem Resultat. Für den mit der Uhr reisenden Beobachter, sagt Einstein, gelte dasselbe. Das heißt, im Urteil des Zurückbleibenden würde jener niemals alt. „Und wenn er auf einer gebrochenen Reiselinie wieder an seinen Ausgangspunkt zurückkehrte?&quot; fragt man den Vortragenden in der Diskussion. – „So bliebe er in unserem Urteil so jung wie bei der Ausreise,&quot; erwidert Einstein mit vollem Ernst, „selbst wenn wir Zurückgebliebenen inzwischen Männer mit weißen Bärten geworden sind – die Gleichungen liefern für jede Richtung der Bewegung, auch für eine gebrochene Bewegung, unerschütterlich die selben Resultate.&quot; – Wir sehen einander an. Das klingt märchenhaft. Märchenhaft? Gewiß, die alten Märchen vom Mönch von Heisterbach, vom Rip van Winkle, von Urashima Taro steigen auf. Merkwürdig, wie die Volksphantasie bei den Deutschen, bei den Amerikanern, bei den Japanern in der gleichen Richtung gearbeitet hat – alle drei Märchen erzählen ja von Leuten, deren Leben still steht, viele hundert Jahre lang, während die andern altern. So fanden sie bei ihrer Rückkehr ein anderes Land und eine andere Generation.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Two synchronous clocks at rest next to each other, shall each be accompanied by an observer. Now one of them, together with its observer, suddenly travels into space at the speed of light. Previously, both have arranged that every second they telegraph their time to each other using light signals. [...] In our limiting case where the journey happens at light speed, the resting observer would have to declare that the other clock would not proceed in time at all. Time would stand still at this place. Einstein's equations indeed produce this result. As to the observer traveling with the clock, says Einstein, the same is true. That means in the judgment of the remaining one, the other one would never become old. Then the lecturer [i.e. Einstein] was asked in the discussion: &quot;And if he comes back to his starting point on a curved travel path?&quot;, to which Einstein replied in full earnest: &quot;Then in our judgment he would remain as young as he was at departure, even if we remaining ones became men with white beards in the meantime, the equations unshakably give the same result in every direction of motion, also for curved motion&quot;. We look at each other. That sounds fabulous. Fabulous? Of course, the old fairy tales of [[w:Heisterbach Abbey|w:The monk of Heisterbach]] or [[w:Rip Van Winkle]] or [[w:Urashima Tarō]] come forward. Strange, how the folk fantasy of the Germans, the Americans, the Japanese worked in the same direction, all three fairy tales indeed tell about people whose life stands still, many hundred years long, while the other ones grow old. Thus they found another country and another generation when they returned.

|}

{{Lorentzbox|Text=Müller's account confirms {{slink||Lämmel 1911-HU}} that Einstein indeed mentioned human beings, but his description also suggests that Einstein was the first to use mutually sent light signals. However, as this was published in October, it cannot be excluded that Müller's description of light signals was influenced by {{slink||Langevin 1911-HU}}, published in July, in which light signals were used as well.}}

|}

==Twins from 1911 to 1920==

We now provide a list of authors who employed ''twins'', i.e. ''two'' life forms or humans that initially were of ''same age'' when the round-trip began:

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Date !! Description

|-

|[[w:Emil Wiechert|Wiechert]]&lt;ref name=wiechert11 /&gt;

|1911

|Two life forms that begin their life at the ''same time'' (German: &quot;Zwei Lebewesen [..] die ihr Leben gleichzeitig beginnen&quot;), of which the moving one returns retarded in its progression with respect to the stationary one.

|-

|[[w:Paul Gruner|Gruner]]&lt;ref name=gruner /&gt;

|1912

|Two persons of ''same age'' (French: &quot;deux personnes du même âge&quot;), of which the moving one returns less developed than stationary one.

|-

|[[w:Max von Laue|Laue]]&lt;ref name=laue3 /&gt;

|1913

|The moving life form returns younger than its ''former agemates'' (German: &quot;ehemaligen Altersgenossen&quot;).

|-

|[[w:Hermann Weyl|Weyl]]&lt;ref name=weyl /&gt;

|Easter 1918

|

{| 

! width=50% | German original

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Von zwei Zwillingsbrüdern, die sich in einem Weltpunkt A trennen, bleibe der eine in der Heimat (d. h. ruhe dauernd in einem tauglichen Bezugsraum), der andere aber unternehme Reisen, bei denen er Geschwindigkeiten (relativ zur »Heimat«) entwickelt, die der Lichtgeschwindigkeit nahekommen; dann wird sich der Reisende, wenn er dereinst in die Heimat zurückkehrt, als merklich jünger herausstellen denn der Seßhafte.

|Suppose we have two twin-brothers who take leave from one another at a world-point A, and suppose one remains at home (that is, permanently at rest in an allowable reference-space), whilst the other sets out on voyages, during which he moves with velocities (relative to “home”) that approximate to that of light. When the wanderer returns home in later years he will appear appreciably younger than the one who stayed at home. 

|}

{{Lorentzbox|Text=Weyl was the first to ''explicitly use twins'' in relation to the round-trip experiment. The fourth edition (1920) of that book was translated from German into English and French in 1922.}}

|-

|[[w:Albert Einstein|Einstein]]&lt;ref name=einstein20 /&gt;

|1920/21

|{{Anchor|Einstein 1921-TW}}

{| 

! width=50% | German original

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; | Trifft A wieder bei B ein, so kann es sich ereignen, daß der beharrende Zwilling inzwischen 60 Erdjahre alt geworden ist, während der zurückkehrende nur 15 Jahre zählt, oder sich gar noch im Säuglingsstadium befindet. [..] Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls -Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. 

|If A then returns to B, it may happen that the twin who stayed at home is now sixty years old, whereas the wanderer is only fifteen years of age, or is perhaps only an infant still. [..] In the case of these two twins, Einstein declared, we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures. 

|}

{{Lorentzbox|Text=This was based on an interview of Einstein by Moszkowski. While the expression &quot;clock paradox&quot; was used since 1911/12 (see section {{slink||Paradoxical?}}), this seems to be the first time that it was rebranded as &quot;twin paradox&quot;. The copyright mark indicates 1920, while the title page indicates 1921. The translation from German into English also appeared in 1921.}}

|}

==Maximal proper time==

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Early examples

|-

|[[w:Paul Langevin|Langevin]]

1911

|{{anchor|Langevin 1911-PT}}In April 1911 (published July),&lt;ref name=langevin1 /&gt; he described the round-trip experiment without formulas using two portions of matter present at two events happening at the same place. The ''integration of proper time'' along the entire wordlines shows that the portion of matter that starts a closed cycle by receding and finally coming back, will have a ''smaller proper time'' than the one that stayed behind.

In October 1911 (published 1912),&lt;ref name=langevin2 /&gt; Langevin again showed that the portion of matter that described a closed cycle will have a ''smaller proper time'' &lt;math&gt;R&lt;/math&gt; than the one that stayed in an inertial frame, which is defined by the equation:

:&lt;math&gt;\begin{matrix}V^{2}\left(t-t_{0}\right)^{2}=d^{2}-R\\
\left[d^{2}=\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right]
\end{matrix}&lt;/math&gt; 

|-

|[[w:Emil Wiechert|Wiechert]]&lt;ref name=wiechert11 /&gt;

Lectures March-May 1911

submitted July

published September

|{{anchor|Wiechert 1911-PT}}Let two equal processes be observed in two equal material systems colocated in two moments (1) and (2), and let there velocities have been changed in arbitrarily different ways in the meantime. It follows that the ratio of advancement of those processes is given by the two intervals &lt;math&gt;\Delta\tau &lt;/math&gt; of their respective ''proper times''. He concluded that any round-trip clock experiment can be easily comprehended from that theorem by computation. The corresponding integral is:

:&lt;math&gt;\Delta\tau=\int_{1}^{2}d\tau=\int_{1}^{2}dt\sqrt{1-\frac{\mathfrak{v}^{2}}{c^{2}}}&lt;/math&gt; 

|-

|[[w:Eduard Study|Study]]&lt;ref name=study /&gt;

June 1911

|Minkowski's concept of worldlines implies that the straight path between two points of the same worldline is the ''longest'' among all paths between those points, if the path length on a worldline is defined by the related proper time.

{{Lorentzbox|Text=Study's book was purely mathematical without mentioning clocks or the round-trip experiment, alluding to his result only in a footnote.}}

|-

|[[w:Max von Laue|Laue]]

1911-13

|{{anchor|Laue 1911/12-PT}}In December 1911 (published 1912),&lt;ref name=laue1 /&gt; Laue showed without formulas that the round-trip experiment is represented by a curved worldline, which at worldpoint A decomposes into a row of curves, after which all of them will be re-united at worldpoint B to a single line. Of all curves connecting the points A and B having time-like direction throughout, the straight connection has the ''longest proper time.''

{{anchor|Laue 1912/13-PT}}In December 1912 (published 1913) in the second edition of this relativity book,&lt;ref name=laue1 /&gt; Laue described the proper time integral between events 1 and 2 of a slowly accelerated clock covering a broken line and a stationary clock covering a straight worldline. Of all worldlines covering 1 and 2, the straight line has the ''longest proper time''. Therefore the traveling clock in the round-trip experiment is retarded at reunion, because its curved worldline corresponds to a shorter proper time. This result he presented in terms of the following inequality, of which the right-hand side refers to the straight curve of the stationary clock, while all others possible curves are represented on left-hand side:

:&lt;math&gt;\tfrac{1}{c}\int_{1}^{2}\sqrt{du^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)}&lt;\tfrac{1}{c}\int_{1}^{2}du&lt;/math&gt; 

{{Lorentzbox|Text={{anchor|Sommerfeld 1913-PT}}Similar treatments can be found in the textbooks of [[w:Arnold Sommerfeld|Sommerfeld]] (1913),&lt;ref name=sommerfeld /&gt; [[w:Hermann Weyl|Weyl]] (1918),&lt;ref name=weyl /&gt; [[w:Wolfgang Pauli|Pauli]] (1921),&lt;ref name=pauli /&gt; [[w:August Kopff|Kopff]] (1921),&lt;ref name=kopff /&gt; [[w:Jean Becquerel|Becquerel]] (1922).&lt;ref name=becqu1 /&gt;}}

|}

==Triangle inequality==

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Early examples

|-

|valign=top|[[w:Alfred Robb|Robb]]

1914-1920

|{{anchor|Robb 1914-TR}}In 1914&lt;ref name=robb1 /&gt; he showed that there are three types of triangles formed by intervals in Minkowski space, depending on whether one deals with &quot;separation lines&quot; (spacelike intervals), &quot;optical lines&quot; (lightlike intervals), or &quot;inertia lines&quot; (timelike intervals representing the path of nonaccelerated particles defined by &lt;math&gt;{\scriptstyle \left(x_{1}-x_{0}\right)^{2}+\left(y_{1}-y_{0}\right)^{2}+\left(z_{1}-z_{0}\right)^{2}-c^{2}\left(t_{1}-t_{0}\right)^{2}&lt;0}&lt;/math&gt;). As to a triangle formed by inertia lines, he showed that the sum of a certain two sides is ''less'' than that of the third one. 

{{Lorentzbox|Text=So the triangle inequality derived from time-like intervals in Minkowski space is ''[[w:Triangle inequality#Reversal in Minkowski space|inverse]]'' to the inequality in Euclidean space. This inverse inequality directly represents the most simple variant of the twin paradox: the traveler follows two sides of the time-triangle, while the stay-at-home observer follows the third side indicating maximal proper time.}}

[[File:RobbTriangle.svg|right|150px]]
In 1920&lt;ref name=robb2 /&gt; Robb gave a numerical example of the triangle ABC with time-like intervals (&quot;inertia lines&quot;) defined by coordinates

:&lt;math&gt;\begin{matrix} &amp; x &amp; y &amp; z &amp; t\\
A\  &amp; 0 &amp; 0 &amp; 0 &amp; 0\\
B\  &amp; 0 &amp; 0 &amp; 0 &amp; 10\\
C\  &amp; 4 &amp; 0 &amp; 0 &amp; 5
\end{matrix}&lt;/math&gt; 

which he plugged into 

:&lt;math&gt;\bar{s}^{2}=\left(t_{1}-t_{0}\right)^{2}-\left(x_{1}-x_{0}\right)^{2}-\left(y_{1}-y_{0}\right)^{2}-\left(z_{1}-z_{0}\right)^{2}&lt;/math&gt; 

from which he obtained the sides AB=10, AC=3, CB=3 and the inequality &lt;math&gt;AC+CB&lt;AB&lt;/math&gt;.

|-

|[[w:Arthur Eddington|Eddington]]&lt;ref name=edding2 /&gt;

1922

|He distinguished between the &quot;space-triangle&quot; for spacelike intervals, and the &quot;time-triangle&quot; for time-like intervals. The latter is measured with a clock from A to B and from B to C, with the sum of those readings ''is always less'' than the reading of a clock measuring directly from A to C. In the ordinary space-triangle any two sides are together greater than the third side; in the time-triangle two sides are together ''less'' than the third side. 

|-

|Rogers&lt;ref name=rogers /&gt;

1922

|He showed that the &quot;pure time-triangle&quot; C, A, B (in their proper time order) satisfies the relation &lt;math&gt;\cosh C=\tfrac{\alpha^{2}+\beta^{2}-\gamma^{2}}{2\alpha\beta}&lt;/math&gt;, where &lt;math&gt;\cosh C&lt;/math&gt; denotes the unit-scalar product of the vectors CA, CB, and &lt;math&gt;\alpha,\beta,\gamma &lt;/math&gt; the real and positive intervals BC, CA, AB. Since &lt;math&gt;\alpha&gt;\beta &lt;/math&gt; and &lt;math&gt;\cosh C&gt;1&lt;/math&gt;, it follows that &lt;math&gt;\alpha&gt;\beta+\gamma &lt;/math&gt;. That is, &quot;the greatest side of pure time-triangle is greater than the sum of the other two sides&quot;. It follows at once that the stationary value of the proper time integral is an &quot;absolute maximum&quot;.

|}

==Negligibility of proper acceleration==

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Early examples

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|valign=top |[[w:Albert Einstein|Einstein]]

1905-1918


|In 1905,&lt;ref name=einstein05 /&gt; Einstein used velocity time dilation &lt;math&gt;\tau=t\sqrt{1-\left(\frac{v}{V}\right)^{2}}&lt;/math&gt; to derive the retardation of a clock performing a round-trip with constant speed &lt;math&gt;v&lt;/math&gt; along a polygonal path or a continuously curved line, without mentioning any influence of acceleration at turnaround.

{{anchor|Einstein 1911-VA}} In 1911 (published 1912),&lt;ref name=einstein3 /&gt; Einstein said that special relativity doesn't say anything about what happened to the clock's pointer position during the acceleration that changes the clock's direction along the round-trip, yet the influence of this change must be getting smaller the longer the clock ''is moving uniformly'', i.e. the longer one chooses the dimensions of the path.

{{anchor|Einstein 1912-VA}}In an unpublished manuscript on special relativity from 1912,&lt;ref name=einst12manu /&gt; he pointed out that any influence of acceleration during the round-trip experiment, can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygonal path. 

{{anchor|Einstein 1914a-VA}}In a letter from April 1914,&lt;ref name=einstpetz /&gt; Einstein showed that any ''finite'' acceleration at turnaround during the round-trip experiment can only influence the clock in a ''finite'' way, thus it can be neglected by minimizing the time of acceleration with respect to the time of uniform translation. So it ''must be concluded'' that the clock is retarded at reunion after traveling on a polygonal path.

{{anchor|Einstein 1914b-VA}}During a conversation in May 1914,&lt;ref name=rowe group=S /&gt; Einstein is reported to have replied that the accelerations during the round-trip are &quot;irrelevant for the amount of the time difference&quot;. (Compare with {{slink||Einstein 1914b-AC}}) 

{{anchor|Einstein 1918-VA}}In his famous &quot;Dialog about Objections against the Theory of Relativity&quot; from 1918,&lt;ref name=einstein18 /&gt; Einstein pointed out that any effect of velocity changes at turnaround must be limited, thus the traveling clock must be retarded at reunion due to time dilation if one makes the path AB and back along the round-trip long enough. (Compare with {{slink||Einstein 1918-AC}})

|-

|[[w:Emil Wiechert|Wiechert]]&lt;ref name=wiechert11 /&gt;

1911

|{{Anchor|Wiechert 1911-VA}}[[File:WiechertTwin.svg|110px|right]] He demonstrated that differential aging along the round-trip cannot be caused during the passage from one velocity to another (i.e. acceleration) at turnaround, because the same result also follows when ''both'' A and B experience the ''same velocity changes'' with respect to another frame, only with the difference that B has relative velocities &lt;math&gt;+u&lt;/math&gt;  and &lt;math&gt;-u&lt;/math&gt; for a long time, while A is brought after a short time from relative velocity &lt;math&gt;+u&lt;/math&gt; to relative rest at which it remains a long time, and then it is brought to relative velocity &lt;math&gt;-u&lt;/math&gt;  for a short time.

{{Lorentzbox|Text=He was probably the first to use an example in which both accelerate with same magnitude.}}

|-

|[[w:Max von Laue|Laue]]&lt;ref name=laue3 /&gt;

1913

|{{anchor|Laue 1913-VA}}He showed that the problem of the influence of acceleration at turnaround in the round-trip experiment, can be eliminated by ''arbitrarily'' enlarging the time in inertial motion.

{{Lorentzbox|Text=This is the same argument as given in {{slink||Einstein 1911-VA}}. The Einstein-Laue argument was also used by others such as [[w:Hans Thirring|Thirring]] (1921)&lt;ref name=thirring /&gt; or [[w:Max Born|Born]] (1921).&lt;ref name=born /&gt;}}

|-

|[[w:Hendrik Lorentz|Lorentz]]&lt;ref name=lorentz1 /&gt;

1913

|He pointed out that any effect of acceleration on the traveling clock at turnaround, can be separated from the time dilation effect since only the latter depends on the distance traversed along the round-trip.

{{Lorentzbox|Text=Similarly, [[w:Wolfgang Pauli|Pauli]] (1921) stated that the arising infinitesimal accelerations at turnaround are certainly independent of the total travel time and ''therefore easy to eliminate''.&lt;ref name=pauli /&gt;}}
|}

==Relay (three brothers) experiment==

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Early examples

|-

|[[w:de:Fritz Grünbaum (Physiker)|Grünbaum]]&lt;ref name=gbaum /&gt;

1911

|He discussed a one-way time dilation experiment in which the first clock is set into motion from the origin and then moving to the second clock. He argued that one can avoid the problem of acceleration experienced by the first clock when set into motion, by replacing it with a ''third'' clock that is already in motion with constant velocity and is synchronized at the origin with the first clock.

{{Lorentzbox|Text=While Grünbaum didn't discuss round-trip experiments, his introduction of a third clock in order to avoid acceleration is the basis of the three-brother experiment.}}

|-

|valign=top|[[w:Emil Wiechert|Wiechert]]

1920-1922

|In 1920 (published 1921),&lt;ref name=wiechert20 /&gt; Wiechert explained how to completely remove acceleration from the round-trip experiment: Bodies A, B, C move undisturbed and non-accelerated in different directions. A and B pass each other at time (1), B and C pass each other at a later time (2), and C and A finally pass each other at an even later time (3). So in this setup, the condition of C is the continuation of the condition of B. On any of the three bodies one can count the oscillations of light of a certain spectral-line, in which case relativity predicts that the ''combined sum of all oscillations'' on B+C is smaller than the number of oscillations on A alone. Wiechert also held that one can replace the light oscillations by the life functions of human-like beings which live on A, B and C. For instance, while the inhabitants of B+C only had time for one meal, there were arbitrarily many generations on A who follow after each other by death and birth.

[[File:Wiechert1922a.png|180px|right]]
In 1921 (published 1922),&lt;ref name=wiechert21 /&gt; Wiechert extended his previous acceleration-free round-trip experiment to an arbitrary number of non-accelerated bodies &lt;math&gt;B_{1}&lt;/math&gt;, &lt;math&gt;B_{2}&lt;/math&gt;, ..., which constitutes a &quot;relay&quot; (German: Stafette) starting from body A and back again. The first B passes A and moves away, and after some time the last B comes back to A. Since any B body continues the fate of the previous one, all bodies &lt;math&gt;B_{1}&lt;/math&gt;, &lt;math&gt;B_{2}&lt;/math&gt;, ..., combined have emitted fewer oscillations than A alone during the relay race. Wiechert pointed out that instead of light oscillations one can also choose the aging of life forms.

{{Lorentzbox|Text=Such relay experiments were later independently rediscovered in English language papers&lt;ref name=debs group=S /&gt; such as by Lange (1927)&lt;ref group=S name=lange /&gt; in which the brothers synchronize their times when they pass each other (“three brother experiment”).}}
|}

==Acceleration as asymmetry indicator==

While it was known that any direct influence of [[w:proper acceleration]] on clocks can be neglected in the computation of the inertial frame of the stay-at-home twin (see previous section {{slink||Negligibility of proper acceleration}}), the very fact that only one of them is accelerating is still useful as an asymmetry argument in order to show that there is no contradiction to the relativity principle.

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Early examples

|-

|[[w:Paul Langevin|Langevin]]&lt;ref name=langevin1 /&gt;

1911

|{{Anchor|Langevin 1911-AC}}He derived differential aging in the round-trip experiment using the proper time integral along worldlines (see {{slink||Langevin 1911-PT}}) and used acceleration as an asymmetry indicator: The result of the round-trip experiment is &quot;another example of the absolute character of acceleration&quot; in which the &quot;asymmetry occurred because only the traveler, in the middle of his journey, has undergone an acceleration that changes the direction of his velocity&quot;. 

|-

|[[w:Arnold Sommerfeld|Sommerfeld]]&lt;ref name=sommerfeld /&gt;

1913

|After he showed (see {{slink||Sommerfeld 1913-PT}}) that retardation of time in the round-trip experiment derived from the proper time integral rests on the assumption that the clock's rate ''only depends on its momentary velocity'' (now called &quot;clock hypothesis&quot;), he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle since one of the clocks has to be accelerated in order to come back, thus the retardation in the round-trip experiment does not demonstrate &quot;motion&quot;, but &quot;accelerated motion&quot;.

|-

|[[w:Hendrik Lorentz|Lorentz]]

1913&lt;ref name=lorentz1 /&gt;

|After he derived differential aging in the round-trip experiment from velocity time dilation and pointed out the negligibility of proper acceleration for the computation, he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle, since one of them changes velocity and accelerates; the relativity principle does not require symmetry between inertial and non-inertial observers.

|-

|valign=top|[[w:Albert Einstein|Einstein]]

1914-1920

|{{anchor|Einstein 1914b-AC}} During a conversation in 1914,&lt;ref name=rowe group=S /&gt; Einstein is reported to have said that moving clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are ''irrelevant'' for the ''amount'' of the time difference, their ''presence'' nevertheless cause B to fall behind (&quot;accelerated motions are absolute&quot;).

{{anchor|Einstein 1918-AC}}In his famous &quot;Dialog about Objections against the Theory of Relativity&quot; from 1918&lt;ref name=einstein18 /&gt;, Einstein pointed out the negligibility of velocity changes from the viewpoint of an inertial frame (see {{slink||Einstein 1918-VA}}). Then he used ''acceleration as an asymmetry indicator'' in order to show, that there is no contradiction to the relativity principle, because relativity only predicts the equivalence of non-accelerated inertial frames: &quot;only K is such a frame while K' is temporarily accelerated, thus the retardation of U2 with respect to U1 cannot be used to construe a contradiction against the theory.&quot; 

{{anchor|Einstein 1920-AC}}Einstein is reported to have said in an interview from 1920:&lt;ref name=einstein20 /&gt;
{| 

! width=50% | German original

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls-Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. Dieser Grund für das Jüngerbleiben des A ergibt sich vom Gesichtspunkt der speziellen Relativitätstheorie aus der Tatsache, daß das betreffende Geschöpf — und nur dieses — Beschleunigungen erlitten hat.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |In the case of these two twins,&quot; Einstein declared, &quot;we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures . This ground, which counts for the comparative youth of A, is given, from the point of view of the special theory of relativity, by the fact that the creature in question, and only this creature, has been subject to accelerations.&quot;
|}

In a discussion from 1922,&lt;ref name=morand /&gt; Einstein is reported to have said that there is no contradiction in the round-trip experiment (in terms of a train leaving the station and returning later): The relativity principle is not applicable to this case, because the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two frames having velocities in opposite direction is not an inertial frame. There is no reciprocity between a frame that changes direction and one that doesn't.

|}

==Frame distribution as asymmetry indicator==

Because any direct influence of proper acceleration on the traveling clock at turnaround can be neglected (see {{slink||Negligibility of proper acceleration}}), the importance of {{slink||Acceleration as asymmetry indicator}} is limited to the mere fact that it reveals that only the traveler was in a non-inertial frame as only he changed his inertial frames, thus instead of emphasizing the occurrence of proper acceleration at turnaround, it's possible to describe the asymmetry more geometrically by emphasizing the different distribution of inertial frames of the twins along their worldlines.

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Early examples

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|valign=top|[[w:Max von Laue|Laue]]

1911-1913

|{{Anchor|Laue 1911/12-VA}} In 1911/12,&lt;ref name=laue1 /&gt; he pointed out that during the time of separation, that clock is most advanced which was at rest in an inertial frame all the time; namely there is ''always one, and only one inertial frame'', in which the locations of separation and re-encounter lie in the same geometric point. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1911/12-PT}}).

In 1912/13,&lt;ref name=laue2 /&gt; he argued that in the round-trip experiment, we indeed can decide, which one of the clocks was steadily at rest in one and the same reference system, and which one was in the meantime at rest in two or more such systems. Among them there is of course a real physical difference. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1912/13-PT}}).

In 1913&lt;ref name=laue3 /&gt; Laue pointed out:

{| 

! width=50% | German original

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; | Aber nach unseren Voraussetzungen ruht während der Zeit der Trennung die erste Uhr in ''einem'' berechtigten Bezugssystem, die zweite hingegen ruht zwar sowohl bei der Hin- wie bei der Rückbewegung in berechtigten Bezugssystemen, aber notwendig in ''zwei verschiedenen. Deshalb'' unterscheiden sich beider Schicksale physikalisch. Ließe man die zweite Uhr in der ihr anfangs erteilten Bewegung und schickte man ihr dafür die erste Uhr nach einiger Zeit mit größerer Geschwindigkeit nach, so würde beim Zusammentreffen die erste gegen die zweite zurückgeblieben sein; denn jetzt hat die erste während der Trennung in zwei verschiedenen Systemen geruht. (Footnote: Dem naheliegenden Einwand, daß wir über den Gang einer Uhr während eines Geschwindigkeits''wechsels'' nichts aussagen können, begegnet man am einfachsten mit dem Hinweis, daß man die Zeiten der gleichförmigen Bewegung ''beliebig'' groß gegen die der Beschleunigung machen kann.)

| style=&quot;padding: 0px 20px 0px 20px;&quot; | However, by our presuppositions, one clock is at rest in ''one'' valid reference system during the time of separation, while the second one is at rest in valid reference systems both during the forward- and the backward motion, but necessarily in ''two different ones. Therefore'' the two fates differ physically. If the second clock remains in the motion which was given to it at the start, and if after some time it is followed by the first clock with greater velocity, then the first one would be retarded with respect to the second one at the encounter; since now it was the first one that was at rest in two different systems during the separation. (Footnote: The objection which is near at hand, that we cannot say anything about the rate of a clock during a velocity ''change'', can be met most simply by the allusion, that we can render the times of uniform motion ''arbitrarily'' great with respect to acceleration..)
|}

|-

|[[w:Werner Bloch|Bloch]]&lt;ref name=bloch /&gt;

September 1918

|{{anchor|Bloch 1918-VA}} He represented the frames with three movable slots K, K' and K”, provided with hooks on which one can hang clocks at the origins of K and K'; while one clock always hangs on a hook of slot K, the other clock moved away with K' and after some time was transferred (neglecting any effect of acceleration) by a mechanical device to slot K” that moves in the other direction, by which it comes back; there is no contradiction to the relativity principle, as one clock rested in one inertial frame while the other one rested in two such frames.

|}

==Perspective of the traveler==

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Early examples

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|[[w:Paul Langevin|Langevin]]&lt;ref name=langevin1 /&gt;

1911

|{{anchor|Langevin 1911-LI}}[[Image:rstd4.gif|170px|right]] After deriving differential aging from the proper time integral in {{slink||Langevin 1911-PT}} and using human beings in {{slink||Langevin 1911-HU}}, he described the perspectives of both observers using light signals and the Doppler effect. When they separate they see each other live 200 times slower, while at return they see each other live 200 times faster. So ''from the explorer's viewpoint'', in the first year he sees the Earth perform the actions of two days, while in the second year he sees the Earth perform the actions of two centuries. The asymmetry can be seen by noticing, that the observer on Earth in 200 years sees the explorer performs the actions of 1 year. Then the explorer turns around, after which the observer on Earth in 2 days sees the projectile perform the actions of another year.

{{Lorentzbox|Text=Langevin used &lt;math&gt;v=c\left(1-\tfrac{1}{20000}\right)&lt;/math&gt;, producing Lorentz factor &lt;math&gt;\gamma\approx100&lt;/math&gt; and Doppler factor &lt;math&gt;\sqrt{\tfrac{c+v}{c-v}}\approx200&lt;/math&gt;.}}

|-

|[[w:Hendrik Lorentz|Lorentz]]

Lectures published in 1913&lt;ref name=lorentz1 /&gt;

Similar treatment in 1914&lt;ref name=lorentz3 /&gt;

|{{anchor|Lorentz 1913/14-LI}}Described the round-trip experiment in terms of inertial observer A (equipped with clock K) and traveling observer B (equipped with clock K'). In the frame of A, clock K' is retarded with respect to K at reunion due to time dilation. He then described the perspective of the traveling observer B by using two-way propagation of light from K' to K and back to K', leading to three periods defined by the moment of B's turnaround: In the first period the light signals return to K' before turnaround; in the second period the signals are emitted before turnaround and return after turnaround; in the third period emission and return of the signals are both happening after turnaround. Lorentz showed that K is time dilated by a factor of &lt;math&gt;\sqrt{1-v^{2}/c^{2}}&lt;/math&gt; with respect to K' in the first and third period, but in the second period K is ticking ''faster'' than K' by a factor of &lt;math&gt;\sqrt{\tfrac{c+v}{c-v}}&lt;/math&gt; which overcompensates the dilation in the other periods and explains, even from the perspective of B, why K' is retarded with respect to K at reunion.

{{Lorentzbox|Text=In a review of the German translation of Lorentz's book, Einstein (1914) didn't directly mention Lorentz's treatment of the twin paradox, but he wrote that nobody who is seriously interested in relativity should neglect to read that book.&lt;ref name=einstlor /&gt; [[w:Wolfgang Pauli|Pauli]] (1921) refers to Lorentz's book as one of three papers that analyze the twin paradox more closely.&lt;ref name=pauli /&gt;}}

|-

|valign=top| [[w:Albert Einstein|Einstein]]

1916-1920

|{{anchor|Einstein 1916-EP}}In a lecture from 1916,&lt;ref name=einstein16 /&gt; of which only an abstract was published, Einstein spoke about the &quot;clock paradox of special relativity from the standpoint of [[w:general relativity]].&quot;

{{anchor|Einstein 1918a-EP}}In a letter from September 1918,&lt;ref name=einadl /&gt; Einstein showed that general relativity makes the inertial frame K and and the accelerated frame K' of the clocks in the round-trip experiment &quot;equally justified&quot;, explaining the time difference in K' by combining the influence of velocity and gravitational potential on clocks.

{{anchor|Einstein 1918-EP}}In his famous &quot;Dialog about Objections against the Theory of Relativity&quot; from November 1918,&lt;ref name=einstein18 /&gt; aimed at clarifying misconceptions of the clock paradox, he explained that there is no paradox in special relativity because there is no symmetry between clock U1 at rest in inertial frame K and clock U2 at rest in accelerated frame K' (see {{slink||Einstein 1918-AC}}). Yet [[w:general relativity]] and the [[w:equivalence principle]] allow the treatment of this problem also from the standpoint of frame K', where clock U2 remains at rest all of the time while U1 makes the following movements: (1) It is accelerated by a homogeneous gravitational field in the negative direction, (2) it moves with constant velocity &lt;math&gt;-v&lt;/math&gt;, (3) it is accelerated in the positive direction until it turns around and comes by with constant velocity &lt;math&gt;+v&lt;/math&gt;, (4) it moves with velocity &lt;math&gt;+v&lt;/math&gt;, (5) it is accelerated in the negative direction until it stops. Clock U1 is retarded with respect to U2 in periods 2) and 4) due to velocity time dilation, but this retardation is overcompensated by the faster rate of U1 during period 3), because U1 is at a higher gravitational potential. He argued that the computation (which he didn't provide) shows that the advance of U1 in period 3) is double its retardation during periods 2) and 4). Einstein concluded that by this consideration &quot;the paradox is completely resolved&quot;. Using [[w:Mach's principle]], he pointed out that the gravitational field in K' might be induced by the masses of the universe that are accelerated in this frame.

{{anchor|Einstein 1918b-EP}}In a letter to Einstein from December 1918, [[w:Max Jakob|Jakob]] doubted the result that the advance in period 3) is double the retardation during periods 2) and 4). Einstein responded by letter,&lt;ref name=einstein18b /&gt; in which he used the gravitational time dilation factor &lt;math&gt;1+\Phi/c^{2}&lt;/math&gt; in K' in order to show that U1 at distance &lt;math&gt;l&lt;/math&gt; is advancing by &lt;math&gt;\Phi/c^{2}=2vl/c^{2}&lt;/math&gt; in period 3), which is indeed the double of approximated delay &lt;math&gt;vl/c^{2}&lt;/math&gt; caused by velocity time dilation during periods 2) and 4).

{{anchor|Einstein 1921-EP}}Einstein is reported to have said in an interview from 1920,&lt;ref name=einstein20 /&gt; that while acceleration explains the age difference between the stationary twin B and the traveling twin A in terms of special relativity (see {{slink||Einstein 1920-AC}}), the &quot;proper&quot; description in terms of general relativity is as follows:

{| 

! width=50% | German original

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; | Eine tiefere Erfassung des Grundes ist indeß nur auf dem Boden der „Allgemeinen Relativitätstheorie&quot; zu erlangen, die uns erkennen läßt, daß von A aus beurteilt ein Zentrifugalfeld existiert, von B aus betrachtet aber nicht; und dieses Feld hat einen Einfluß auf den relativen Ablauf und die Raschheit der Lebensvorgänge.

| style=&quot;padding: 0px 20px 0px 20px;&quot; | A proper grasp of the reason is furnished only when we adopt the general theory of relativity, which tell us that, from the point of view of A, a centrifugal field exists, whereas it is absent from the point of view of B. This field exerts an influence on the relative rate of happening of the events of life.&quot;
|}

{{Lorentzbox|Text=a) Einstein's explanation was quickly adopted in the textbooks of [[w:Werner Bloch|Bloch]] (1920),&lt;ref name=bloch2 /&gt; [[w:Wolfgang Pauli|Pauli]] (1921),&lt;ref name=pauli /&gt; [[w:August Kopff|Kopff]] (1921),&lt;ref name=kopff /&gt; [[w:Karl Bollert|Bollert]] (1921),&lt;ref name=bollert1 /&gt; [[w:Max Born|Born]] (1921),&lt;ref name=born /&gt; expressing the view that general relativity is &quot;necessary&quot; to provide the &quot;complete&quot; solution of the twin paradox.
b) From a modern standpoint, however, Einstein's explanation has nothing to do with general relativity, but is rather an application of accelerated frames and &quot;pseudo&quot;-gravitational fields to flat Minkowski space of ''special'' relativity.&lt;ref name=weiss group=S /&gt;}}

|-

|[[w:Hans Thirring|Thirring]]&lt;ref name=thirring /&gt;

April 1921

|{{anchor|Thirring 1921-DS}}[[Image:Twin Paradox Minkowski Diagram.svg|right|200px]]
He described the round-trip experiment by using two platforms K (clock A) and K' (clock B) each equipped with rows of clocks. He first demonstrated the symmetry of time dilation and the mutual relativity of simultaneity on the platforms and its effect on clock synchronization. The K clocks that B passes are all advanced because of &lt;math&gt;t'=\gamma\left(t-vx/c^{2}\right)&lt;/math&gt;, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation &lt;math&gt;t'-t'_{0}=\gamma\left(t+vx/c^{2}\right)&lt;/math&gt; leading to the effect of clock desynchronization, where &lt;math&gt;t'_{0}&lt;/math&gt; is a constant depending on which clock one uses as standard for the new synchronization. He graphically showed using Minkowski diagrams, that this simultaneity jump due to desynchronization amounts to double the velocity time dilation during the inertial phases, explaining why A is more advanced than B at reunion. 

{{Lorentzbox|Text=Using clock B as synchronization standard, Thirring's constant is given by &lt;math&gt;t'_{0}=2l\gamma v/c^{2}=2t\gamma v^{2}/c^{2}&lt;/math&gt; with &lt;math&gt;l=vt&lt;/math&gt; as position of turnaround. A similar explanation was subsequently given by Langevin (1922).&lt;ref name=morand /&gt;}}

|}

==Curved spacetime==

While the previous examples are defined in flat Minkowski spacetime and therefore can be fully discussed in terms of special relativity, [[general relativity]] is required when [[:w:spacetime curvature]] in the presence of mass and energy cannot be neglected any more.&lt;ref name=koks group=S /&gt; 

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Early examples

|-

|[[w:Jean Becquerel|Becquerel]]&lt;ref name=becqu1 /&gt;

1922

|After defining gravitational time dilation &lt;math&gt;d\tau=\sqrt{1-\tfrac{2GM}{c^{2}r}}dt&lt;/math&gt; in terms of the [[w:Schwarzschild metric]] around a material center, he discussed the following round-trip experiment: There are two identical clocks A and B placed next to each other, at a point very far from the material center, initially marking the same time &lt;math&gt;t&lt;/math&gt;. Let us transport clock A to a point where the field is more intense, at a distance &lt;math&gt;r&lt;/math&gt; from the center; this clock will measure time &lt;math&gt;\int d\tau&lt;/math&gt; which is shorter than &lt;math&gt;\int dt&lt;/math&gt;, thus it will run more slowly. If we bring clock A back to clock B, we will have to note that it is retarded with respect to B.

|}

==Paradoxical?==

{| class=wikitable style=&quot;background-color:white;&quot;

! width=50% | German original of [[w:Max von Laue|Laue]] (1911/12):&lt;ref name=laue1&gt;Laue introduces the word &quot;paradox&quot;, alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource&lt;/ref&gt;

! English translation

|-

|Unter all den paradox erscheinenden Folgerungen aus der Zeittransformation der Relativitätstheorie gibt es wohl keine, gegen welche sich der natürliche Menschenverstand bei jedem, der der Sache noch ungewohnt ist, so sehr sträubt, wie gegen die, daß die Zeitangabe einer Uhr von ihrem Bewegungszustand abhängen soll. Schon in seiner grundlegenden Arbeit hat Einstein diese Paradoxie auf die Spitze getrieben in einem Gedankenexperiment, welches neuerdings von Langevin in einem auch sonst sehr lesenswerten Vortrage besonders hübsch erläutert worden ist.

|Of all apparently paradox consequences that stem from the time-transformation of the theory of relativity, there is probably none against which the common sense of anyone who is still unfamiliar with the matter is more reluctant, than the one according to which the time indication of a clock shall be dependent on its state of motion. Already in his fundamental paper, Einstein has driven this paradox to the extreme by a thought experiment, recently explained in a very nice way by Langevin in a lecture that is also very readable in other respects.

|-

|colspan=2|{{Lorentzbox|Text=Laue was probably the first to denote the round-trip experiment as paradoxical (even though he pointed out that there are no real contradictions). Subsequently, [[:w:Paul Gruner|Gruner]] (1912)&lt;ref name=gruner /&gt; and others including Einstein (1918)&lt;ref name=einstein18 /&gt; explicitly used the expression &quot;clock paradox&quot; (French: Paradoxe des horloges, German: Uhrenparadoxon), whereas [[w:Rudolf Seeliger|Seeliger]] (1913)&lt;ref name=seel /&gt; spoke of the &quot;familiar Einstein-Langevinian paradox&quot; (German: &quot;bekannte Einstein-Langevinsche Paradoxon&quot;).}}

|}

==Misunderstandings==

{| class=wikitable style=&quot;background-color:white;&quot; 

! width=50% padding=10 | German original by [[w:Otto Berg (scientist)|Berg]] (1910):&lt;ref name=berg /&gt;

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Im Punkte &lt;math&gt;x = 0&lt;/math&gt; des Systems S befinde sich eine Uhr, eine andere im Punkte &lt;math&gt;x'=0&lt;/math&gt; von S'. Diese zweite bewege sich mit S' bis zum Punkte &lt;math&gt;x = a&lt;/math&gt;, kehre dort um und bewege sich nun mit der Geschwindigkeit &lt;math&gt;v&lt;/math&gt; zurück bis zum Punkte &lt;math&gt;x= 0&lt;/math&gt;. Welche Zeit müssen beide Uhren in dem Moment angeben, wo sie sich wieder treffen? Wir beantworten diese Frage zunächst vom Standpunkt des Beobachters in S. Die Uhr in &lt;math&gt;x' = 0&lt;/math&gt; hat sich mit der Geschwindigkeit &lt;math&gt;v&lt;/math&gt; bis zum Punkte &lt;math&gt;x = a&lt;/math&gt; bewegt; dazu brauchte sie die Zeit &lt;math&gt;\tau=\tfrac{a}{v}&lt;/math&gt;. Zum Rückweg ist dieselbe Zeit nötig. Nach der Zeit &lt;math&gt;2\tau=2\tfrac{a}{v}&lt;/math&gt; ist die Uhr also wieder im Punkte &lt;math&gt;x = 0&lt;/math&gt; angelangt. Wir stellen uns nun auf den Standpunkt des Beobachters in S'. Für diesen führt nach dem Relativitätsprinzip das System S genau dieselben Bewegungen aus wie das System S' für den Beobachter in S, nur in entgegengesetzter Richtung. Die Zeit bis zum Zusammentreffen beider Uhren ist also im System S' ebenfalls gegeben durch &lt;math&gt;2\tau=2\tfrac{a}{v}&lt;/math&gt;. Betrachtungen, die auf anschauliche Vorstellungen, wie Nachgehen von Uhren, gestützt sind, führen hier leicht zu Irrtümern, von denen auch die Fachlitteratur nicht frei ist.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |There is a clock at point &lt;math&gt;x=0&lt;/math&gt; of system S, and another one at point &lt;math&gt;x'=0&lt;/math&gt; of S'. The second one moves together with S' until point &lt;math&gt;x=a&lt;/math&gt;, turns around and now moves back with speed &lt;math&gt;v&lt;/math&gt; to point &lt;math&gt;x=0&lt;/math&gt;. Which time must both clocks indicate at the moment at which they encounter again? We answer this question at first from the standpoint of the observer in S. The clock at &lt;math&gt;x=0&lt;/math&gt; has been moving with speed &lt;math&gt;v&lt;/math&gt; until point &lt;math&gt;x=a&lt;/math&gt;, for which it required time &lt;math&gt;\tau=\tfrac{a}{v}&lt;/math&gt;. The same time is required for the way back. After time &lt;math&gt;2\tau=2\tfrac{a}{v}&lt;/math&gt; the clock has thus arrived again at point &lt;math&gt;x=0&lt;/math&gt;. Let's now take the standpoint of the observer in S'. In his view in accordance with the relativity principle, system S is conducting exactly the same motions as those of system S' with respect to the observer in S, only in opposite direction. Thus the time until the meeting of both clocks is given by &lt;math&gt;2\tau=2\tfrac{a}{v}&lt;/math&gt; in system S' as well. Considerations based on illustrative notions, such as the retardation of clocks, easily lead to mistakes at this place, of which also the professional literature isn't free.

|-

|colspan=2|{{Lorentzbox|Text=Berg was probably the first to turn the relativity principle against asymmetric aging in the round-trip experiment, claiming that both clocks must indicate the same time at reunion. See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem.}}

|-

! width=50% | German original by [[w:Emil Wiechert|Wiechert]] (1911)&lt;ref name=wiechert11 /&gt;

! English translation

|-

|colspan=2| Even though he correctly derived differential clock aging in the round-trip experiment, he claimed that effects like time dilation are &quot;apparent&quot; if one admits Einstein's &quot;unconditional&quot; relativity principle in which there is no aether and all &quot;strides&quot; (i.e. non-accelerated motions) are physically equivalent, but they are &quot;real&quot; if one admits the existence of an aether in the framework of a &quot;conditional&quot; relativity principle in which all strides are physically non-equivalent or anisotropic. This led him to the following interpretation of the clock paradox:

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |[...] so muß am Schluß des Versuches B in seinem Fortschritt gegenüber A im Verhältnis &lt;math&gt;1:\sqrt{1-u^{2}/c^{2}}&lt;/math&gt; zurückgeblieben sein. Und dieses Zurückbleiben ist unbedingt reell, denn die beiden Gebilde A und B können ja unter gleichen Umständen unmittelbar beieinander verglichen werden. Hier ist es ganz sicher ausgeschlossen, an einen Schein zu glauben, der durch unsere Auffassung der Zeit bewirkt wird. So ist denn also auch die Folgerung unabwendbar, daß für den Verlauf der Weltvorgänge die Schreitungen nicht gleichwertig sind, ''und damit sind wir von neuem zu einem Schluß gekommen, welcher der Unbedingtheit des Relativitätsprinzipes durchaus widerspricht.'' [...] Man kann den Versuch noch mannigfach variieren, z. B. so, daß A ebenso wie B zwei verschiedene Schreitungen, &lt;math&gt;+u&lt;/math&gt; und &lt;math&gt;-u&lt;/math&gt;, nacheinander inne hat. Wird dann zu A der Wert &lt;math&gt;u_{1}&lt;/math&gt;, zu B  der Wert &lt;math&gt;u_{2}&lt;/math&gt;, zugeordnet, so muß der Vergleich von A  und B am Schluß des Versuches ergeben, daß B  oder A in seinem Fortschritt zurückgeblieben erscheint, je nachdem die Schreitungen &lt;math&gt;+u_{1}&lt;/math&gt;, &lt;math&gt;-u_{1}&lt;/math&gt;, oder &lt;math&gt;+u_{2}&lt;/math&gt;, &lt;math&gt;-u_{2}&lt;/math&gt; weiter auseinanderliegen. ''Vielleicht ist gerade diese Formulierung des Satzes besonders geeignet, um die Ungleichwertigkeit der verschiedenen Schreitungen klar und deutlich zu zeigen.''

| style=&quot;padding: 0px 20px 0px 20px;&quot; | [...] thus B's progress must be retarded with respect to A's in the ratio &lt;math&gt;1:\sqrt{1-u^{2}/c^{2}}&lt;/math&gt; at the end of the experiment. And this retardation is definitely real, since both bodies A and B indeed can be immediately compared side by side under the same conditions. Here it is certainly excluded to believe that this is an appearance due to our conception of time. Thus the consequence is unavoidable too, that the strides are not equivalent in the course of the world processes, ''and therefore we again came to a conclusion that completely contradicts the unconditionality of the relativity principle.'' [...] One can vary this experiment in many ways, for instance, so that A in the same way as B successively undergoes two different strides &lt;math&gt;+u&lt;/math&gt; and &lt;math&gt;-u&lt;/math&gt;. If we apply the value &lt;math&gt;u_{1}&lt;/math&gt; to A and &lt;math&gt;u_{2}&lt;/math&gt; to B, then the comparison of A and B at the end of the experiment must give the result, that B or A is retarded in its progress depending on whether the strides &lt;math&gt;+u_{2},-u_{2}&lt;/math&gt; or &lt;math&gt;+u_{1},-u_{1}&lt;/math&gt; are further apart. ''Probably it is precisely this formulation of the theorem that is particularly suitable to demonstrate the non-equivalence of the different strides clearly and explicitly.''

|-

|colspan=2|{{Lorentzbox|Text=This interpretation was directly rebutted by Laue (1911/12) who demonstrated the geometrical meaning of differential aging in Minkowski space, see sections {{slink||Laue 1911/12-PT|Laue 1911/12-VA}}, showing that there is no need to assume non-equivalance or anisotropy of motions. Laue added, that as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.&lt;ref name=laue1 /&gt;}}

|-

! width=50% | German original by [[w:Norman Robert Campbell|Campbell]] (November 1911, published 1912)&lt;ref name=camp /&gt;

! English translation

|-

|colspan=2|After describing the round-trip experiment (as given by Wiechert) according to which the traveling clock B is retarded when it returns with respect to stationary clock A, he abandoned differential clock aging as follows: 

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Dieser Schluß ist nicht richtig. Die Beziehung zwischen &lt;math&gt;t&lt;/math&gt;, der Ablesung an der Uhr auf A seitens des Beobachters auf A und &lt;math&gt;t'&lt;/math&gt;, der Ablesung an der Uhr auf B seitens des Beobachters auf A, ist (unter der Annahme, daß zu Beginn des Versuchs &lt;math&gt;t=t'&lt;/math&gt; ist)
:&lt;math&gt;t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)&lt;/math&gt;. 
Der Unterschied zwischen &lt;math&gt;t'&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt; ist eine Funktion von &lt;math&gt;z&lt;/math&gt; und &lt;math&gt;v&lt;/math&gt; allein. Wenn man diesen Größen ihre früheren Werte wiedergibt, indem man die beiden Uhren wieder zur Koinzidenz bringt, während sie relativ zueinander ruhen, so geht der Unterschied zwischen &lt;math&gt;t'&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt; wieder auf null zurück, gleichviel, welche Werte &lt;math&gt;z&lt;/math&gt; und &lt;math&gt;v&lt;/math&gt; während der Zwischenzeit gehabt haben mögen. Wenn an irgendeinem Punkte der Bahn die Geschwindigkeit von B relativ zu A eine endliche plötzliche Änderung erfährt, so erfährt auch der Wert von &lt;math&gt;v&lt;/math&gt; eine endliche plötzliche Änderung.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |This conclusion is not correct. The relationship between &lt;math&gt;t&lt;/math&gt; as the reading on the clock on A by the observer on A, and &lt;math&gt;t'&lt;/math&gt; as the reading on the clock on B by the observer on A, is given by (assuming that &lt;math&gt;t=t'&lt;/math&gt; at the beginning of the experiment)
:&lt;math&gt;t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)&lt;/math&gt;. 
The difference between &lt;math&gt;t'&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt; is a function of &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;v&lt;/math&gt; alone. If these quantities are given their previous values by bringing the two clocks back to coincidence during which they are at rest relative to one another, the difference between &lt;math&gt;t'&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt; goes back to zero, no matter what values &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;v&lt;/math&gt; may have had in the meantime. If at any point on the path the speed of B experiences a finite sudden change relative to A, then the value of &lt;math&gt;t'&lt;/math&gt; also undergoes a finite sudden change.

|-

|colspan=2|{{Lorentzbox|Text=So Campbell claims that any time difference during the outbound path is wiped out during the inbound path. His mistake is obvious: Campbell is confusing coordinate differences stemming from the Lorentz transformation of ''events'' (which indeed depend on position and direction) with differences in ''clock aging'' derived from the proper time integral (which is ''accumulative'' and independent of position and direction.)}}

|-

! width=50% | French original by [[w:Paul Gruner|Gruner]] (March 1912):&lt;ref name=gruner /&gt;

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |[...] deux personnes du même âge, se séparant dans des systèmes de « marche » très différents et retournant après un laps de temps assez long, constateront une différence d'âge très sensible. [...] le principe de relativité exige toujours la ''réciprocité parfaite'' des phénomènes entre deux systèmes qui possèdent un mouvement relatif. Si, dans l'exemple cité, les deux personnes du même âge se séparent avec une vitesse relative pour se retrouver plus tard, la constatation d'une différence d'âge sera parfaitement mutuelle : A dira positivement que B est resté en arrière dans son développement, et B affirmera avec le même droit que c'est A qui ne s'est pas développé assez vite. Ainsi le principe absolu de la relativité montre ses conséquences les plus extrèmes et il est clair que l'introduction de l’éther n'est plus en état de résoudre cette contradiction irréductible et inconcevable.

| style=&quot;padding: 0px 20px 0px 20px;&quot; | [...] two people of same age, separating into very different systems of motion and returning after a quite long period of time, will notice a very significant age difference. [...] the principle of relativity always requires the ''perfect reciprocity'' of the phenomenons between two systems that possess relative motion. When, in the cited example, the two persons of same age are separated by some relative velocity only to meet again later, the finding of an age difference will be perfectly mutual: A will positively say that B stayed behind in its development, and B will assert with same right that it was A who has not developed fast enough. By that, the absolute relativity principle shows its most extreme consequences and it is clear, that the introduction of the aether is no longer able to resolve this irreducible and inconceivable contradiction.

|-

|colspan=2|{{Lorentzbox|Text=Gruner was probably the first to claim that combining the round-trip experiment with the symmetry of time dilation leads to the contradictory situation, that both must attribute younger age to one another at reunion. At the end of his paper, we also find the expression &quot;clock paradox&quot; (French: paradoxe des horloges). See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem.}}

|}

==Einstein's contributions==

1905:&lt;ref name=einstein05 /&gt; Introduction of the &quot;peculiar&quot; (German: eigentümlich) round-trip experiment with clocks, describing a polygonal path, an continuously curved path, and an experiment comparing a clock at the pole with one at the equator.

January 1911 (published November):&lt;ref name=einstein11a /&gt; In a lecture from January, Einstein extended the &quot;funny&quot; (German: drollig) round-trip experiment to living organisms. According to [[w:Rudolf Lämmel]] in April 1911&lt;ref name=lammel /&gt; and [[w:Fritz Müller-Partenkirchen|Fritz Müller]] in October 1911,&lt;ref name=muller /&gt; Einstein spoke of human beings as well.

January 1911 (published January 1912):&lt;ref name=einstein3 /&gt; During a discussion with Einstein directly after the previous lecture, [[w:Fritz Müller-Partenkirchen|Fritz Müller]] claimed that any time difference during the round-trip should vanish at reunion, in analogy to the fact that the Lorentz contraction of a moving rod vanishes when it is at rest again. Einstein showed that the analogy is incorrect: While the clock rates are indeed the same again when they are mutually at rest, the clocks do not indicate the same time at reunion because &quot;clocks are carriers of the time differential&quot;; he went on to show that any possible influence of acceleration during the turnaround can be made negligible by elongating the constant velocity periods. 

1912:&lt;ref name=einst12manu /&gt; In an unpublished manuscript on special relativity, Einstein showed that if system &lt;math&gt;\Sigma'&lt;/math&gt; makes a round-trip along a polygon, then its inner processes will be retarded with respect to resting system &lt;math&gt;\Sigma &lt;/math&gt; at reunion. He pointed out that any influence of acceleration can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygon. 

April 1914:&lt;ref name=einstpetz /&gt; [[w:Joseph Petzoldt]] criticized asymmetric clock aging in the round-trip experiment as a &quot;fallback into absolutist way of thinking&quot;, claiming that special relativity requires that any difference between the clocks vanishes when their relative velocity is zero again, even though he added that any treatment of the clock paradox in special relativity is unrealistic anyway, since the theory only concerns uniform motions and therefore cannot handle velocity changes, so one has to modify the theory. Einstein responded by letter in which he praised Petzoldt's philosophical take on relativity, yet he rejected Petzoldt's conclusions concerning the clock paradox by showing that any finite acceleration at turnaround during the round-trip can only influence the clock in a finite way and therefore can be neglected by minimizing the time of acceleration with respect to the time of uniform translation, so it &quot;must be concluded&quot; that the clock traveling on a polygonal path is retarded at reunion.

May 1914:&lt;ref name=rowe group=S /&gt; During a conversation with [[w:Ernst Gehrcke]] who claimed that the clock paradox contradicts the relativity principle, Einstein replied that clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are irrelevant for the amount of the time difference, their presence nevertheless cause B to fall behind (&quot;accelerated motions are absolute in the theory of relativity&quot;).

1916:&lt;ref name=einstein16 /&gt; In a lecture of which only an abstract was published, Einstein spoke about the &quot;clock paradox of special relativity from the standpoint of general relativity.&quot;

September 1918:&lt;ref name=einadl /&gt; In a letter to Einstein, [[w:Friedrich Adler (politician)|Friedrich Adler]] (while in prison for the [[w:Assassination of Karl von Stürgkh]]) claimed that the clock paradox which he described on a circular round-trip contradicts the special relativity principle, and also referred to the similar opinions of [[w:#a|Berg and Petzoldt]]. Einstein responded by letter and explained that there is no contradiction as one of them accelerates; he then showed that general relativity makes both inertial frame K and accelerated frame K' equally justified, explaining the time difference in K' by combining the influence of velocity and gravitational potential, concluding that &quot;Berg and Petzoldt were wrong&quot;.

November 1918:&lt;ref name=einstein18 /&gt; In a fictitious dialogue between a relativity critic and a relativity apologist written by Einstein, the &quot;critic&quot; said that special relativity must predict differential clock aging in round-trip experiments, which was confirmed by the &quot;relativist&quot; who regretfully noted that even some pro-relativity authors tried to &quot;circumvent this unavoidable result&quot;. Yet the critic claimed that this leads to a contradiction: From the viewpoint of K, clock U1 is at rest while the clock U2 was in motion and therefore returns being retarded with respect to U1, but from the viewpoint of K', clock U2 is at rest while clock U1 was in motion and therefore returns being retarded with respect to U2, which was rebutted by the relativist by pointing out the acceleration of U2. Then the critic claimed that this problem &quot;rises again from the dead&quot; in general relativity which allows to symmetrically treat both K and K', which was rebutted by the relativist using the equivalence principle: In K', the rate increase of U1 during turnaround period 3) is &quot;the double&quot; of its velocity time dilation in the inertial periods 2) and 4). 

December 1918:&lt;ref name=einstein18b /&gt; In a letter to Einstein, [[w:Max Jakob]] doubted the result from Einstein's dialogue, according to which the advance of U1 in period 3) is the double of its retardation during periods 2) and 4). Einstein responded by letter, in which he used the gravitational time dilation factor &lt;math&gt;1+\Phi/c^{2}&lt;/math&gt; in K' in order to show that U1 at distance &lt;math&gt;l&lt;/math&gt;  is advancing by &lt;math&gt;\Phi/c^{2}=2vl/c^{2}&lt;/math&gt; in period 3), which is indeed the double of approximated delay &lt;math&gt;vl/c^{2}&lt;/math&gt; caused by velocity time dilation during periods 2) and 4).

1920:&lt;ref name=einstein20 /&gt; In conversations with [[w:Alexander Moszkowski]] between 1919 and 1920, Einstein argued that differential aging of the twins is rather a paradox of feeling, not a paradox of thought, because the latter would only arise if there were no reason for the asymmetric aging. The reason in special relativity lies in the fact that one of them suffered accelerations, while a deeper understanding of that question is obtained by using general relativity. Einstein argued that our &quot;common sense&quot; is located in the realm of feeling and analogy drawn from our ordinary experience; since there is no analogy to the example of the twins in our experience, it might appear paradoxical to the common sense, while it appears logical and necessary in light of intensified abstraction of the trained scientific mind.

August 1920:&lt;ref name=rowe group=S /&gt; At an anti-relativity event organized by the right-wing agitator [[w:Paul Weyland]] during which [[w:antisemitic]] leaflets were distributed and [[w:swastika]]s offered at the entrance, a lecture was given by Gehrcke re-iterating his criticism of the twin paradox, claiming that the stationary first organism is old or even dead at reunion while the second organism was in motion and therefore stayed young, but from the standpoint of the second organism he himself is old or even dead while the first organism was in motion and stayed young, thus relativity is either contradictory or it leads to different realities and physical [[w:solipsism]]. Einstein who was present at that event, directly responded in a newspaper article;&lt;ref name=einst20 /&gt; after suspecting antisemitic motives of his critics, he specifically addressed Gehrcke's objections regarding the &quot;well known example of the clocks (or twins)&quot;, remarking that the charge of solipsism will be &quot;greeted by the experts as a joke&quot;, and characterized the claim that relativity requires mutual retardation of two co-located clocks as a &quot;deliberate attempt to misinform the lay public&quot;.

1922:&lt;ref name=morand /&gt;&lt;ref name=nord /&gt; [[w:Paul Painlevé]], Einstein and Langevin discussed the clock paradox at a meeting in Paris. Painlevé imagined a clock on a train that performs a round-trip with constant speed and returns being retarded with respect to the station clock, yet he claimed that the relativity principle also allows to say that the station clock performed the round-trip and returned being retarded with respect to the train clock, in contradiction to the previous result. Einstein replied that the relativity principle cannot be applied since the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two systems having velocities in opposite direction is not an inertial frame; there is no reciprocity between a frame that changes direction and one that doesn't. Langevin consequently gave a detailed analysis in terms of the Lorentz transformation.

1954:&lt;ref name=einstein54 /&gt; In a letter, Einstein explained the &quot;well known clock paradoxon&quot; using two clocks &lt;math&gt;B_{1}&lt;/math&gt; and &lt;math&gt;B_{2}&lt;/math&gt;; clock &lt;math&gt;B_{2}&lt;/math&gt; has to reverse its speed in order to come back, thus it was initially at rest in inertial frame &lt;math&gt;S_{2}&lt;/math&gt; and then at rest in &lt;math&gt;S_{2}^{+}&lt;/math&gt;, whereas &lt;math&gt;B_{1}&lt;/math&gt; constantly remains at rest in &lt;math&gt;S_{1}&lt;/math&gt;, which explains the asymmetry between them.

==Historical references==

&lt;references&gt;

&lt;ref name=einstein05&gt;See p. 904f in: {{Citation |author=Einstein, A. |date=1905 |title=Zur Elektrodynamik bewegter Körper|journal=Annalen der Physik |volume=322 |issue=10 |pages=891–921 |doi=10.1002/andp.19053221004|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 2, Document 23}}. See also: [https://www.fourmilab.ch/etexts/einstein/specrel/www/ English translation at fourmilab].&lt;/ref&gt;

&lt;ref name=einstein11a&gt;See p. 10. in: {{Citation |author=Einstein, A. |title=Die Relativitäts-Theorie|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |issue=1-2|pages=1–14 |date=27 November 1911|orig-date=Lecture 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n11/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 17}}.&lt;br /&gt; The publication date 27 November 1911 can be seen on the [https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n5/mode/2up Title page and TOC of issue 1-2].&lt;/ref&gt;

&lt;ref name=einstein3&gt;Discussion between Einstien, Müller, Lämmel and others after the Zürich lecture: {{Citation |author=Einstein, A.; Müller, F., Lämmel, R.|title=Diskussion zu &quot;Die Relativitäts-Theorie&quot;|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |pages=II-IX |date=January 1912|orig-date=Lecture on 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n587/mode/2up|quote=Reprinted in  ''The Collected Papers of Albert Einstein'', Vol. 3, Document 18, and in the corresponding English translation volume}}&lt;br /&gt; While the discussion already happened on January 1911, the publication followed one year later in January 1912 in the session proceedings (Sitzungsberichte) of the third issue, see [https://www.ngzh.ch/publikationen/vjs/56/3 Full issue Nr. 3] with [http://www.ngzh.ch/archiv/1911_56/56_1-2/56_3.pdf Title page and TOC] and the [http://www.ngzh.ch/archiv/1911_56/56_3/56_30.pdf Sitzungsberichte including Einstein's discussion on pp. II-IX]. &lt;/ref&gt;

&lt;ref name=einst12manu&gt;See p. 46 in: {{Citation |author=Einstein, A. |date=1912 |chapter=Document 1: Einstein's manuscript on the special theory of relativity|title=The collected papers of Albert Einstein|volume=4|pages=3-108|trans-chapter=See also the English translation in the corresponding translation volume}}&lt;/ref&gt;

&lt;ref name=einstlor&gt;{{Citation|author=Einstein, A.|date=1914|title=Review of &quot;Lorentz, H. A. – Das Relativitätsprinzip&quot; |journal=Die Naturwissenschaften|volume=2|pages=1018|url=https://archive.org/details/CAT31421305002/page/1018/mode/2up|quote=Reprinted in  ''The Collected Papers of Albert Einstein'', Vol. 6, Document 11}}&lt;/ref&gt;

&lt;ref name=einstpetz&gt;{{Citation |author=Einstein, A. |date=1914 |chapter=Document 5: Letter from Einstein to Petzoldt|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}&lt;/ref&gt;

&lt;ref name=einstein16&gt;See p. 423f in: {{Citation |author=Einstein, A. |date=1916 |title=Announcement of Einstein's lecture &quot;Über einige anschauliche Überlegungen aus dem Gebiete der Relativitätstheorie&quot;|journal=Berliner Sitzungsberichte|pages=423|volume=1916 (part 1)|url=https://archive.org/details/sitzungsberichte1916deutsch/page/423/mode/2up}}&lt;/ref&gt;

&lt;ref name=einadl&gt;Letter exchange between Einstein and Adler in which the critique on the clock paradox by Berg (1910) and Petzoldt (1914) was mentioned, together with the general relativity solution in terms of the gravitational potential, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Adler's letter in Document 620 and Einstein's reply in Document 628|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}&lt;/ref&gt;

&lt;ref name=einstein18&gt;Einstein discussed in terms of inertial frames (special relativity) on pp. 697f; accelerated frames (general relativity) on pp. 698f.; distant masses (Mach's principle) on pp. 700f. in: {{citation |author=Einstein, A.|title=Dialog über Einwände gegen die Relativitätstheorie|date=November 1918|volume=6|issue=48|journal=Die Naturwissenschaften|pages=697-702|url=https://archive.org/details/sim_naturwissenschaften_1918-11-29_6_48|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 7, Document 13}}; See also English translation [[:s:Translation:Dialog about Objections against the Theory of Relativity|Dialog about Objections against the Theory of Relativity]] on Wikisource.&lt;/ref&gt;

&lt;ref name=einstein18b&gt;Letter exchange between Max Jakob and Einstein from December 1918, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Jakob's letter in Document 661c and Einstein's reply in Document 663a|title=The collected papers of Albert Einstein|volume=10|pages=189-190}}&lt;/ref&gt;

&lt;ref name=einstein20&gt;Interview of Einstein by Moszkowski, see p. 204f. in: {{citation |author=Moszkowski, A.|title=Einstein. Einblicke in seine Gedankenwelt|orig-date=Copyright date 1920 |date=1921|place=Hamburg|url=https://www.archive.org/details/einsteineinblick00moszuoft}}; See also English translation by H. L. [[Henry Brose|Brose]] (1921): [https://archive.org/details/einsteinsearch00moszrich Einstein, the searcher], p. 206&lt;/ref&gt;

&lt;ref name=einst20&gt;{{Citation|author=Einstein, A.|date=27 August 1920|journal=Berliner Tageblatt|title=Meine Antwort - Ueber die anti-relativitätstheoretische G. m. b. H.|issue=402|pages=1-2|url=https://www.deutsche-digitale-bibliothek.de/newspaper/item/YH65KFT53MOG4SMXDXK4IVUPTR3QRY7Q?issuepage=1|quote=Reprinted in &quot;The Collected Papers of Albert Einstein&quot;, Vol. 7, Document 45}}&lt;/ref&gt;

&lt;ref name=einstein54&gt;Letter from Einstein to N. V. Pope from March 1954; [[w:Albert Einstein Archives]], Object number 27-88 ([https://ein-web.adlibhosting.com/aea/Details/archive/110021626 Online dataset]); Scanned version as [https://groups.google.com/group/npachat/attach/d3121322d37764f2/Einstein%20letter.doc?part=0.1 Word document on Usenet] published by [https://groups.google.com/g/npachat/c/Haoib97d6OA/m/8mR30yITEtMJ Pope himself])&lt;/ref&gt;

&lt;ref name=morand&gt;Discussion between Painlevé, Einstein, and Langevin on p. 316ff in: {{citation |author=Morand, M.|title=Einstein au collège de france|date=April 1922|journal=La Nature|volume=50|issue=2511|pages=315-320|url=http://cnum.cnam.fr/CGI/fpage.cgi?4KY28.102/319/100/620/5/613}}&lt;/ref&gt;

&lt;ref name=lammel&gt;{{Citation|author=Lämmel, R.|date=28 April 1911|title=Die Relativitäts-Lehre|journal=Neue Zürcher Zeitung|volume=117|pages=1|url=https://www.e-newspaperarchives.ch/?a=d&amp;d=NZZ19110428-01.2.4.1}}; English translation of the part concering the twin pardox at [[:v:History of Topics in Special Relativity/Twin paradox#Lämmel 1911-Hum|Wikiversity:Early history of the twin paradox - Lämmel]]&lt;/ref&gt;

&lt;ref name=lammel2&gt;See p. 84ff in: {{Citation|author=Lämmel, R.|date=1921|orig-date=Preface December 1920|title=Die Grundlagen der Relativitätstheorie|place=Berlin|publisher=Springer|url=https://archive.org/details/diegrundlagende00lmgoog}}&lt;/ref&gt;

&lt;ref name=langevin1&gt;He derived differential aging from the proper time integral; pointed out that this demonstrates the &quot;absolute nature of acceleration&quot; with respect to an aether, see: {{citation |author=Langevin, P.|title=[[:s:fr:L’Évolution de l’espace et du temps|L’Évolution de l’espace et du temps]]|journal=Scientia |volume=X |pages=31–54 |date=July 1911|orig-date=Lecture 10 April 1911}}; English translation [[:s:en:Translation:The Evolution of Space and Time|The Evolution of Space and Time]] on Wikisource&lt;/ref&gt;

&lt;ref name=langevin2&gt;See p. 329 in: {{citation |author=Langevin, P. |title=Le temps, l'espace et la causalité dans la physique moderne |journal=Bulletin de la Société française de philosophie |volume=12 |orig-date=Lecture October 1911|date=1912|pages=1-28|url=http://ahp.li/1f7fc22d283fdf0deeca.pdf}}&lt;/ref&gt;

&lt;ref name=wiechert11&gt;See p. 745f. general description and proper time; 757f. space travel; in: {{Citation |author=Wiechert, E. |date=September 1911|orig-date=Lectures March-May 1911, submitted 26 July|title=[[:s:de:Relativitätsprinzip und Äther|Relativitätsprinzip und Äther]]|journal=Physikalische Zeitschrift |volume=12 |issue=17-18 |pages=[https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/741 689-707] published September 1; [https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/789 737–758] published September 15}}&lt;/ref&gt;

&lt;ref name=wiechert15&gt;See p. 46 (Einstein, Langevin, Wiechert) and pp. 51f (Laue versus Wiechert) in: {{citation |author=Wiechert, E.|contribution=Die Mechanik im Rahmen der allgemeinen Physik| title=Die Kultur der Gegenwart: Physik|volume=3.3.1|date=1915 |orig-date=Submitted July 1914|pages=1–78|contribution-url=https://www.archive.org/details/physikunterredak00warbuoft}}&lt;/ref&gt;

&lt;ref name=wiechert20&gt;See p. 46f in: {{citation |author=Wiechert, E.|title=Der Äther im Weltbild der Physik|orig-date=Presented December 1920|date=1921|journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse|pages=29-70|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00250586X}}&lt;/ref&gt;

&lt;ref name=wiechert21&gt;See p. 25ff in: {{citation |author=Wiechert, E.|title=[[:s:de:Prinzipielles über Äther und Relativität|Prinzipielles über Äther und Relativität]]|date=1922|orig-date=Lecture September 1921|journal=Physikalische Zeitschrift|volume=23|pages=25-28}}&lt;/ref&gt;

&lt;ref name=muller&gt;See p. 9 in: {{Citation|author=Müller, F.|date=October 1911|journal=Berliner Tageblatt|title=[[:s:de:Das Zeitproblem (1911)|Das Zeitproblem]]|pages=[https://www.deutsche-digitale-bibliothek.de/newspaper/item/2QKOIOLGNVQILTCEZQOGQPLTRVLPM5PZ?query=zeit&amp;issuepage=9 Part 1 published 16 October 1911] and [https://www.deutsche-digitale-bibliothek.de/newspaper/item/IO44I6QBC4SVV5YUKUDSGXYIPQUXXBN5?query=zeit&amp;issuepage=11 Part 2 published 23 October 1911]}}&lt;/ref&gt;

&lt;ref name=gruner&gt;See p. 253f in: {{Citation |author=Gruner, P. |title=[[:s:fr:Rapport sur la dernière discussion concernant le principe de la relativité et l’éther|Rapport sur la dernière discussion concernant le principe de la relativité et l’éther]] |journal=Archives des sciences physiques et naturelles |volume=33|issue=4 |pages=252-254 |date=March 1912}}&lt;/ref&gt;

&lt;ref name=laue3&gt;See p. 113f in: {{citation |author=Laue, M. v. |title=Das Relativitätsprinzip |journal=Jahrbücher der Philosophie |volume=1 |date=1913 |pages=99–128}}; {{icon|wikisource}} See also English translation of [[:s:Translation:The Principle of Relativity (Laue, Philosophy)|The Principle of Relativity]] on Wikisource&lt;/ref&gt;

&lt;ref name=weyl&gt;See p. 147f. in: {{Citation |author=Weyl, H. |date=March 1918|title=Raum-Zeit-Materie (first edition)|publisher=Berlin: Springer|url=https://archive.org/details/RaumZeitMaterieVolIMeinerFrauGewidmet}}; English translation of the 4th edition by H. [[Henry Brose|Brose]] (1921): [https://www.gutenberg.org/ebooks/43006 Space—Time—Matter], pp. 278f.&lt;/ref&gt;

&lt;ref name=gbaum&gt;See footnote on p. 507 in: {{Citation|author=Grünbaum, F. |title=Über einige ideelle Versuche zum Relativitätsprinzip|journal=Physikalische Zeitschrift|volume=12|pages=500–509|date=1911|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/540}}&lt;/ref&gt;

&lt;ref name=laue1&gt;Laue introduces the word &quot;paradox&quot;, alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource&lt;/ref&gt;

&lt;ref name=laue2&gt;See p. 42f. for general description; p. 58f. in terms of proper time; in: {{Citation |author=Laue, M. v. |orig-date=Preface December 1912|date=1913 |title=Das Relativitätsprinzip (Second Edition) |publisher=Vieweg |place=Braunschweig|url=https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE4597082}}; See also English translation [[:s:Translation:The Principle of Relativity (Laue 1913)|The Principle of Relativity, Second edition, Part III]] on Wikisource&lt;/ref&gt;

&lt;ref name=laue3&gt;See p. 113f in: {{citation |author=Laue, M. v. |title=Das Relativitätsprinzip |journal=Jahrbücher der Philosophie |volume=1 |date=1913 |pages=99–128}}; {{icon|wikisource}} See also English translation of [[:s:Translation:The Principle of Relativity (Laue, Philosophy)|The Principle of Relativity]] on Wikisource&lt;/ref&gt;

&lt;ref name=berg&gt;See p. 369f in: {{Citation |author=Berg, O. |date=1910 |title=Das Relativitätsprinzip der Elektrodynamik |journal=Abhandlungen der Fries'schen Schule |volume=3 |issue=2|pages=333-382 |url=http://hdl.handle.net/2027/hvd.hnuynk?urlappend=%3Bseq=351}}&lt;/ref&gt;

&lt;ref name=camp&gt;See p. 123f in: {{Citation |author=Campbell, N. |title=Relativitätsprinzip und Äther: Eine Entgegnung an Herrn Wiechert |journal=Physikalische Zeitschrift |volume=13 |pages=120-128 |issue=3|orig-date=Submitted December 1911|date=February 1912|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/150}}. The is based on an English manuscript translated by Max Iklé, and Campbell's first name was Germanised as &quot;Normann&quot;.&lt;/ref&gt;

&lt;ref name=seel&gt;{{Citation|author=Seeliger, R.|title=Review of &quot;P. Gruner – Rapport sur la dernière discussion concernant le principe de la relativité et l'éther&quot;|journal=Die Fortschritte der Physik|volume=68|issue=2|pages=336|date=1913|url=https://books.google.com/books?id=fSJGAQAAMAAJ&amp;pg=PA336}}&lt;/ref&gt;

&lt;ref name=study&gt;See footnote on p. 111 in: {{citation |author=Study, E. |title=Vorlesungen über ausgewählte Gegenstände der Geometrie |date=June 1911|url=https://archive.org/details/vorlesungenber00studuoft|publisher=B.G. Teubner|place=Leipzig}} &lt;/ref&gt;

&lt;ref name=robb1&gt;See pp. 356ff. in: {{Citation|author=Robb, A.|date=1914|title=A theory of time and space|place=Cambridge|publisher=University Press|url=https://archive.org/details/theoryoftimespac00robbrich}} &lt;/ref&gt;

&lt;ref name=robb2&gt;See §12 in: {{citation |author=Robb, A. A.|title=The Straight Path|date=1920 |journal=Nature|pages=599|volume=104|issue=2623|url=https://archive.org/details/sim_nature-uk_1920-02-05_104_2623/page/598/mode/2up}}&lt;/ref&gt;

&lt;ref name=edding2&gt;See p. 22 in: {{Citation |author=Eddington, A. S. |date=1922 |title=The theory of relativity, and its influence on scientific thought |publisher=Oxford Clarendon Press |url=https://archive.org/details/cu31924005748573}}&lt;/ref&gt;

&lt;ref name=rogers&gt;{{citation |author=Rogers, R. A. P.|title=The Time-Triangle and Time-Triad in Special Relativity|date=November 1922|journal=Nature|volume=110|issue=2769|pages=698–699|url=https://archive.org/details/sim_nature-uk_1922-11-25_110_2769/page/698/mode/2up}}&lt;/ref&gt;

&lt;ref name=lorentz1&gt;See pp. 37f, 55ff in: {{citation |author=Lorentz, H. A.|date=1913|title=Het relativiteitsbeginsel : drie voordrachten gehouden in Teyler's stichting|publisher=De Erven Loosjes |place=Haarlem|url=https://resolver.kb.nl/resolve?urn=MMKB24:063387000:00005}}; German translation on pp. 31f, 47f in: {{citation |author=Lorentz, H. A.|date=1914| title=Das Relativitätsprinzip. Drei Vorlesungen gehalten in Teylers Stiftung zu Haarlem|publisher=B.G. Teubner |place=Leipzig and Berlin|url=https://archive.org/details/bub_gb_89PPAAAAMAAJ}}; See also the transcription [[:s:de:Das Relativitätsprinzip (Lorentz)|Das Relativitätsprinzip]] on German Wikisource and the English translation [[:s:Translation:The Principle of Relativity (Lorentz)|The Principle of Relativity]] on English Wikisource&lt;/ref&gt;

&lt;ref name=lorentz3&gt;See §12 in: {{citation |author=Lorentz, H. A.|title=Considérations élémentaires sur le principe de relativité|date=1914 |journal=Revue générale des sciences pures et appliquées|pages=179-186|url=https://archive.org/details/revuegnraled25pari/page/178/mode/2up}}&lt;/ref&gt;

&lt;ref name=bloch&gt;See pp. 67 ff. in: {{Citation | author=Bloch, W.| date=September 1918|title=Einführung in die Relativitätstheorie| publisher=B. G. Teubner |url=https://hdl.handle.net/2027/njp.32101040276907}}&lt;/ref&gt;

&lt;ref name=bloch2&gt;See pp. 69ff. (special relativity) and 102ff. (general relativity) in: {{Citation | author=Bloch, W.| date=1920 |title=Einführung in die Relativitätstheorie (second edition)| publisher=B. G. Teubner |url=https://www.archive.org/details/einfhrungindier00blocgoog}}&lt;/ref&gt;

&lt;ref name=bollert1&gt;See p. 6 (special relativity), pp. 24-26 (EP) in: {{citation |author=Bollert, K.|title=Einstein’s Relativitätstheorie und ihre Stellung im System der Gesamterfahrung |date=April 1921|publisher=Steinkopff|url=https://archive.org/details/dbc.wroc.pl.001504}}&lt;/ref&gt;

&lt;ref name=born&gt;See pp. 190f. (special relativity), 250f (EP) in: {{Citation | author=Born, M.| date=1921 |title=Die Relativitätstheorie Einsteins und ihre physikalischen Grundlagen (Second edition)| publisher=Springer | place=Berlin|url=https://hdl.handle.net/2027/mdp.39015017387310}}; The [https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE5426498 first edition (1920)] of Born's book didn't include the twin paradox. English translation of the third edition by H. Brose (1924): [https://archive.org/details/einsteinstheoryo00born Einstein's theory of relativity]&lt;/ref&gt;

&lt;ref name=pauli&gt;See p. 558f (general description); p. 624f (proper time); p. 713f (accelerated frames); in: {{Citation |author=Pauli, W. |date=1921 |journal=Encyclopädie der Mathematischen Wissenschaften|title=Die Relativitätstheorie|pages=539–776|volume=5|issue=2 |url=http://resolver.sub.uni-goettingen.de/purl?PPN360709672}}; English translation by G. Field (1958): [https://books.google.com/books?id=rc3DAgAAQBAJ Theory of Relativity]&lt;/ref&gt;

&lt;ref name=thirring&gt;See p. 209ff in: {{citation |author=Thirring, H.|title=Über das Uhrenparadoxon in der Relativitätstheorie|date=April 1921|journal=Naturwissenschaften|volume=9|issue=18|pages=209-212|url=https://archive.org/details/sim_naturwissenschaften_1921-04-01_9_13/mode/2up}}&lt;/ref&gt;

&lt;ref name=sommerfeld&gt;See p. 71 in: {{citation |author=Sommerfeld, A. |date=May 1913|chapter=Remarks on Minkowski's &quot;Space and Time&quot;|title=Das Relativitätsprinzip|editor=Otto Blumenthal|pages=69-73|url=https://www.archive.org/details/dasrelativittsp00minkgoog}}&lt;/ref&gt;

&lt;ref name=kopff&gt;See pp. 45ff (special relativity and proper time); pp. 117ff (EP); pp. 189ff (Mach's principle), in: {{citation |author=Kopff, A.|title=Grundzüge der Einsteinschen Relativitätstheorie |date=February 1921|publisher=S. Hirzel|place=Leipzig|url=https://www.archive.org/details/grundzgedereins00kopfgoog}}; English translation by H. Levy (1923): [https://hdl.handle.net/2027/mdp.39015017188817 The mathematical theory of relativity].&lt;/ref&gt;

&lt;ref name=becqu1&gt;See p. 48ff (proper time), p. 240f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Le Principe de relativité et la théorie de la gravitation|Le Principe de relativité et la théorie de la gravitation]] |date=1922 |publisher=Gauthier-Villars|place=Paris}}; See also p. 57ff (proper time), p. 177f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Exposé élémentaire de la théorie d’Einstein et de sa généralisation|Exposé élémentaire de la théorie d’Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}&lt;/ref&gt;

&lt;ref name=nord&gt;Discussion between Painlevé, Einstein, and Langevin on pp. 146ff in: {{citation |author=Nordmann, C.|title=[[s:fr:Einstein expose et discute sa théorie|Einstein expose et discute sa théorie]]|date=May 1922|journal=Revue des deux mondes|volume=IX|pages=129-166}}&lt;/ref&gt;

&lt;/references&gt;

==Secondary sources==

&lt;references group=S&gt;

&lt;ref name=miller&gt;{{Citation |author=Miller, A. I. |date=1981 |title=Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911) |place=Reading |publisher=Addison–Wesley |isbn=978-0-201-04679-3}}; See section 7.4.13 (Langevin, Wiechert, Laue, Einstein), footnotes 29-34 of chapter 7 (Petzoldt, Sommerfeld, Bergson, Einstein)&lt;/ref&gt;

&lt;ref name=lange&gt;{{Citation|author=Lange, L.|date=1927|title=The clock paradox of the theory of relativity|journal=The American Mathematical Monthly|volume=34|issue=1|pages=22-30|jstor=2299914}}&lt;/ref&gt;

&lt;ref name=pes&gt;{{Citation |author=Pesic, P. |date=2003 |title=Einstein and the twin paradox |journal=European Journal of Physics |volume=24 |issue=6 |pages=585–590 |doi=10.1088/0143-0807/24/6/004}}&lt;/ref&gt;

&lt;ref name=during&gt;{{Citation |author=During, É. |date=2014 |title=Langevin ou le paradoxe introuvable |journal=Revue de métaphysique et de morale |volume=84 |pages=513-527 |doi=10.3917/rmm.144.0513|doi-access=free}}; See pp. 515f (Langevin), 520f. (Einstein, Laue, Weyl, Painlevé).&lt;/ref&gt;

&lt;ref name=debs&gt;{{Citation |author=Debs, T. A., &amp; Redhead, M. L. |title=The twin paradox and the conventionality of simultaneity |date=1996 |journal=American Journal of Physics |volume=64|issue=1| pages=384-392 |doi=10.1119/1.18252}}&lt;/ref&gt;

&lt;ref name=alizzi&gt;{{Citation |author=Alizzi, A., Sen, A., &amp; Silagadze, Z. K.|title=Do moving clocks slow down? |year=2022 |journal=European Journal of Physics |volume=43|issue=6|pages=065601 |doi=10.1088/1361-6404/ac93ca|arxiv=2209.12654}}; Appendix B with reference to Lange and Halsbury&lt;/ref&gt;

&lt;ref name=beng&gt;{{Citation |author=Benguigui, L. G. |date=2020 |title=A Tale Of Two Twins: The Langevin Experiment Of A Traveler To A Star |publisher=World Scientific|isbn=9789811219115}}; See early solutions (Einstein, Langevin, Lorentz, Born/Kopff) and the Bergson controversy. A shorter version appeared in {{arxiv|1212.4414}}.&lt;/ref&gt;

&lt;ref name=rowe&gt;{{Citation|author=Rowe, D. E.|date=2006|title=Einstein's allies and enemies: Debating relativity in Germany 1916–1920|journal=Interactions: Mathematics, Physics and Philosophy|pages=231-280|publisher=Springer|doi=10.1007/978-1-4020-5195-1_8}}; Covering the criticism of Gehrcke starting with 1912; discussion between Einstein and Gehrcke in 1914; Einstein's dialogue (1918) as response to antirelativists; the Weyland event in 1920 and Einstein's response.&lt;/ref&gt;

&lt;ref name=weiss&gt;Weiss, W. (Physics FAQ): [https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html The Twin Paradox: The Equivalence Principle Analysis]&lt;/ref&gt;

&lt;ref name=cuvaj&gt;{{Citation |author=Cuvaj, C. |date=1971 |title=Paul Langevin and the theory of relativity|journal=Japanese studies in the history of science|volume=10| pages=113-142|url=http://www.isc.meiji.ac.jp/~sano/hssj/pdf/Cuvaj_C-1972-Langevin_Relativity-JSHS-No_10-pp113-142.pdf}}&lt;/ref&gt;

&lt;ref name=koks&gt;Koks, D. (2018): [https://math.ucr.edu/home/baez/physics/Relativity/SR/sr-gr.html Physics FAQ: Where is the Boundary between Special and General Relativity?]&lt;/ref&gt;

&lt;/references&gt;

[[Category:History of special relativity]]

[[Category:Paradoxes]]</text>
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==Early history of the twin paradox==

{{Lorentzbox|Text={{center|Date of article creation: 9 November 2023; Last major revision: 2 March 2026}}}}

a) When was the [[:w:twin paradox]] applied to life forms and human beings?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref group=S name=pes /&gt;&lt;ref group=S name=during /&gt; report that {{slink||Einstein 1911-HU}} discussed the aging of living organisms, and that {{slink||Langevin 1911-HU}} and {{slink||Wiechert 1911-HU}} explicitly discussed the aging of human beings. 

:*More details in sections {{slink||Human beings in 1911|Twins from 1911 to 1920}}, including newspaper articles from 1911 written by {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}} that clearly show that Einstein was the first to explicitly discuss the aging of human beings as well.

b) Who was the first to formulate the principle of maximal proper time along straight worldlines, upon which differential aging in the standard twin paradox is based?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref group=S name=during /&gt; mention Langevin (1911), Laue (1911). 

:*More details in section {{slink||Maximal proper time}} with the contributions of Langevin (1911), Wiechert (1911), Study (1911), Laue (1911-13).

c) Who was the first to formulate [[w:Triangle inequality#Reversal in Minkowski space|inverse triangle inequality]] in Minkowski space, which represents the simplest version of the twin paradox?

:*See details in section {{slink||Triangle inequality}} with the contributions of Robb (1914-20), Eddington (1922), Rogers (1922).

d) Who was the first to show that any influence of proper acceleration on clocks can be neglected in the computation of the twin paradox from the viewpoint of the stay-at-home twin?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref group=S name=pes /&gt; mention Einstein (1911), Laue (1913). 

:*More details in section {{slink||Negligibility of proper acceleration}} with the contributions of Einstein (1911), Wiechert (1911), Laue (1913), Lorentz (1913).

e) Who was the first to introduce the three clock/brother example that completely removes acceleration from the clock/twin paradox?

:*Historical accounts&lt;ref group=S name=debs /&gt;&lt;ref group=S name=alizzi /&gt; date it back to Lange (1927) and Lord Halsbury (1957).

:*More details in section {{slink||Relay (three brothers) experiment}} with the contributions of Grünbaum (1911) and Wiechert (1920-22).

f) Who was the first to use acceleration as an asymmetry indicator?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref name=cuvaj group=S /&gt;&lt;ref group=S name=pes /&gt; mention Langevin (1911), Einstein (1918). 

:*More details in section {{slink||Acceleration as asymmetry indicator}} with the contributions of Langevin (1911), Sommerfeld (1913), Lorentz (1913), Einstein (1914-20).

g) Who was the first to use different frame distribution as asymmetry indicator as an asymmetry indicator?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref group=S name=pes /&gt; mention Laue (1911-13). 

:*More details in section {{slink||Frame distribution as asymmetry indicator}} with the contributions of Laue (1911-13), Bloch (1918).

h) Who was the first to describe the perspective of the traveler?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref group=S name=beng /&gt; mention Langevin (1911), Lorentz (1914), Einstein (1918). 

:*More details in section {{slink||Perspective of the traveler}} with the contributions of Langevin (1911), Lorentz (1913-14), Einstein (1918), Thirring (1921).

i) Who was the first to describe a round-trip experiment in curved spacetime?

:*See section {{slink||Curved spacetime}} with the contribution of Becquerel (1922).

j) Who was the first to denote the round-trip experiment as paradoxical?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref group=S name=during /&gt; point to Laue (1911).

:*See section {{slink||Paradoxical?}} for details.

k) Who was the first to misunderstand the twin paradox?

:*See section {{slink||Misunderstandings}} with the contributions of Berg (1910), Wiechert (1911), Campbell (1911/12), Gruner (1912).

l) What were Einstein's contributions?

:*See section {{slink||Einstein's contributions}}.

==Human beings in 1911==

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

![[w:Albert Einstein|Einstein]] 

|-

|{{anchor|Einstein 1905}}In 1905&lt;ref name=einstein05 /&gt; he showed that a clock moving on a round-trip away from A and back along a polygonal or curved path, is retarded with respect to a clock stationary at A by approximately &lt;math&gt;\tfrac{1}{2}t(v/V)^{2}&lt;/math&gt; at reunion. For example, a clock on the equator is retarded with respect to a clock on the pole. He described this consequence as being &quot;peculiar&quot; (German: eigentümlich).

{{anchor|Einstein 1911-HU}}In a lecture given on January 1911&lt;ref name=einstein11a /&gt; (published in November), he extended this &quot;funny&quot; (German: drollig) experiment to living organisms:

{| 

! width=55% | Einstein wrote

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Wenn wir z. B. einen lebenden Organismus in eine Schachtel hineinbrächten und ihn dieselbe Hin- und Herbewegung ausführen lassen wie vorher die Uhr, so könnte man es erreichen, dass dieser Organismus nach einem beliebig langen Fluge beliebig wenig geändert wieder an seinen ursprünglichen Ort zurückkehrt, während ganz entsprechend beschaffene Organismen, welche an den ursprünglichen Orten ruhend geblieben sind, bereits längst neuen Generationen Platz gemacht haben. Für den bewegten Organismus war die lange Zeit der Reise nur ein Augenblick, falls die Bewegung annähernd mit Lichtgeschwindigkeit erfolgte!

| style=&quot;padding: 0px 20px 0px 20px;&quot; |For example, if we put a living organism in a box and make it undergo the same back and forth movement as the clock before, we could achieve that this organism returns to its original location with arbitrary little change after a flight of arbitrary length, whereas completely identical organisms that remained at rest in the original location have long since made room for new generations. To the moving organism, the long journey was only a moment if the movement happened close to the speed of light!

|}

{{Lorentzbox|Text=Two participants of that lecture, {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}}, report that Einstein also talked about the aging of ''human beings''.}}

|-

!{{anchor|Lämmel 1911-HU}}[[w:Rudolf Lämmel|Lämmel]]

|-

|He attended Einstein's 1911 lecture and gave a popular report about it in the Swiss newspaper &quot;[[w:Neue Zürcher Zeitung|Neue Zürcher Zeitung]]&quot; published on 28 April 1911,&lt;ref name=lammel /&gt; including additional details. Regarding the round-trip clock experiment he wrote:

{| 

! width=50% | Lämmel wrote

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Bewegt sich eine Uhr mit Lichtgeschwindigkeit längs einer Geraden, auf der gerichtete Uhren stehen, so scheint die bewegte Uhr, beurteilt vom Standpunkt der ruhenden aus, im oben stizzierten Sinn, stillzustehen. Kehrt die Uhr, nach einem Ruck, mit Lichtgeschwindigkeit wieder zurück zur Zentral-Uhr, so ist, nach Einstein, für den Beobachter bei der Zentral-Uhr die Sache so, als ob ein mit der bewegten Uhr mitgeführter Beobachter (samt dessen Uhr) nicht gealtert hätte. Hinge also des letzteren Alter von den Angaben des ruhenden Beobachters ab, so könnte der von einer großen Reise ins Weltall zurückkehrende Beobachter bei der Zentral-Uhr spätere Generationen antreffen – er selber hätte nicht gealtert. Welche Bedeutung diese ''ad absurdum'' geführte Gedankenspielerei etwa hat, läßt sich heute nicht absehen – vielleicht, ja wahrscheinlich ist sie ohne jeden Einfluß auf die tatsächlichen Verhältnisse. Aber man sieht dabei immerhin, daß die Physik imstande ist, die kühnsten Träume der Phantasie noch – zu überbieten.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Let a clock be moving at speed of light along a line on which regulated clocks are standing, then the moving clock's hand appears to be standing still (in the sense described above) as judged from the standpoint of the resting one. If the clock, after one jolt, comes back with light speed to the central clock, then according to Einstein the matter presents itself to the observer at the central clock, as if the observer comoving with the clock (together with his clock itself) hasn't been grown older. Thus if the age of the latter would depend on the indications of the resting observer, the observer returning from a great journey into space could meet later generations at the central-clock – he himself hasn't been grown older. The importance of this play of thought led ''ad absurdum'' cannot be seen today – maybe, or even probably, it is without any influence on the actual situations. Though at least one can see that physics is able to – surpass – even the boldest dreams and fantasies.

|}

Lämmel in December 1920 (published 1921)&lt;ref name=lammel2 /&gt; again alluded to Einstein's lectures in Zürich (possibly the one from 1911, and maybe also later ones), describing a discussion between himself and Einstein. After Einstein concluded that the travelers who came back after their journey will probably meet their former contemporaries as old men while they themselves could have been away for only a few years, Lämmel objected that this conclusion is only drawn with respect to rods and clocks, but not with respect to living beings. Einstein responded though, that all processes in the blood, in the nerves etc. are eventually periodical oscillations, i.e. motions. Yet to any such motion the relativity principle applies, thus the conclusion regarding the unevenly rapid aging it permissive.

{{Lorentzbox|Text=While the official publication of Einstein's January lecture ({{slink||Einstein 1911-HU}}) mentions the aging of organisms, Lämmel recalls the reference to the aging of a human space traveler (&quot;observer returning from a great journey into space&quot;). This means that Einstein was the first to use human beings in the clock/twin paradox on January 16 which was first published by Lämmel on April 28, 1911. In comparison, {{slink||Langevin 1911-HU}} used space travelers in a lecture on April 10 with publication in July, and {{slink||Wiechert 1911-HU}} used space travelers in lectures held between March 25 and May 23 with publication in July/September. It seems very unlikely that before April 28, Lämmel became somehow aware of the content of Langevin's or Wiechert's lectures held a few weeks earlier, in order to use them in his description of Einstein's lecture.}}

|-

!{{anchor|Langevin 1911-HU}}[[w:Paul Langevin|Langevin]]

|-

|On 10 April 1911, published July 1911,&lt;ref name=langevin1 /&gt; he held a now famous lecture popularizing the clock/twin paradox which he derived from the proper time integral as described in {{slink||Langevin 1911-PT}}. He demonstrated that a moving radioactive sample of radium is less evolved and less aged and therefore more active at return then the ones that remained in the laboratory. He also used light signals and the Doppler effect to visualize the effect. The most famous part concerned his description of the aging of human space travelers:

{| 

! width=50% | Langevin wrote

! [[:s:Translation:The Evolution of Space and Time|English Wikisource translation]]

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Cette remarque fournit le moyen, à celui d’entre nous qui voudrait y consacrer deux années de sa vie, de savoir ce que sera la Terre dans deux cents ans, d’explorer l’avenir de la Terre en faisant dans la vie de celle-ci un saut en avant qui pour elle durera deux siècles et pour lui durera deux ans, mais ceci sans espoir de retour, sans possibilité de venir nous informer du résultat de son voyage puisque toute tentative du même genre ne pourrait que le transporter de plus en plus avant.

Il suffirait pour cela que notre voyageur consente à s’enfermer dans un projectile que la Terre lancerait avec une vitesse suffisamment voisine de celle de la lumière, quoique inférieure, ce qui est physiquement possible, en s’arrangeant pour qu’une rencontre, avec une étoile par exemple, se produise au bout d’une année de la vie du voyageur et le renvoie vers la Terre avec la même vitesse. Revenu à la Terre ayant vieilli de deux ans, il sortira de son arche et trouvera notre globe vieilli de deux cents ans si sa vitesse est restée dans l’intervalle inférieure d’un vingt-millième seulement à la vitesse de la lumière. Les faits expérimentaux les plus sûrement établis de la physique nous permettent d’affirmer qu’il en serait bien ainsi.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |This remark provides the means for any among us who wants to devote two years of his life, to find out what the Earth will be in two hundred years, and to explore the future of the Earth, by making in his life a jump ahead that will last two centuries for Earth and for him it will last two years, but without hope of return, without possibility of coming to inform us of the result of his voyage, since any attempt of the same kind could only transport him increasingly further.

For this it is sufficient that our traveler consents to be locked in a projectile that would be launched from Earth with a velocity sufficiently close to that of light but lower, which is physically possible, while arranging an encounter with, for example, a star that happens after one year of the traveler's life, and which sends him back to Earth with the same velocity. Returned to Earth he has aged two years, then he leaves his ark and finds our world two hundred years older, if his velocity remained in the range of only one twenty-thousandth less than the velocity of light. The most established experimental facts of physics allow us to assert that this would actually be so.

|}

{{Lorentzbox|Text=Reading his lecture in full, one finds the word &quot;paradoxical&quot; only in relation to the constancy of light speed, not on relation to the round-trip clock experiment.}}

|-

!{{anchor|Wiechert 1911-HU}}[[w:Emil Wiechert|Wiechert]]

|-

|In lectures on 25 March and 23 May 1911, submitted July and published September 1911,&lt;ref name=wiechert11 /&gt; he described the round-trip clock experiment with two equal clocks regulated to the same rate and brought to the same pointer position, or by introducing the same chemical process two times, or by introducing ''two life forms that began their life at the same time''. At the end of his paper he applied this to human travelers:

{|

! width=50% | Wiechert wrote

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Nehmen wir aber wieder eine Relativgeschwindigkeit an, die bis auf 3 Proz. der Lichtgeschwindigkeit nahekommt, dann wird das Verhältnis der empfundenen Zeitlängen wie 4:1. Das Bild mag etwas weiter noch ausgemalt werden. Denken wir uns, daß ein Beobachter durch den Raum unseres Sternhimmels mit dieser Geschwindigkeit in einer Kreisbahn mit einem Radius von 16 Lichtjahren fährt, dann wird er nach unserer Zeitrechnung nach je 100 Jahren wieder an unserem Sonnensystem vorüberkommen. In seinem Gefährt wird dabei die Zentrifugalkraft so auf ihn einwirken, daß sie gemäß den Relativitätsgesetzen der Einwirkung der Schwerkraft auf uns Erdenbewohner gleich erscheint. Es sind also die wirkenden Kräfte nur so groß, daß der Phantasie die Möglichkeit geboten wird, den Reisenden als menschliches Wesen zu denken. Da hier dauernd &lt;math&gt;\sqrt{1-v^{2}/c^{2}}&lt;/math&gt; ist, fließt die Eigenzeit für den Reisenden viermal langsamer dahin, als für die Bewohner der Gestirne. Wenn er also nach 100 unserer Jahre wieder zu unserem Sonnensystem zurückkehrt, wird er sich selbst nur um 25 Jahre gealtert fühlen. Erreicht er nach der Entwicklung seines Körpers und nach seiner Zeitempfindung ein Alter von 75 Jahren, so entspricht dies doch einer dreimaligen Wiederkehr zu unserem Sonnensystem, also 300 unserer Erdenjahre.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Yet if we again assume a relative velocity approximating the speed of light by 3 percent, then the ratio of the experienced duration of time becomes 4:1. This image can be further extended. Let's imagine that an observer travels with that velocity on a circular path at a radius of 16 light years through the space of our galaxy, then according to our time calculation he passes by our solar system every 100 years. In his vehicle the centrifugal force will act on him in such a way, that in accordance with the relativity laws it will appear to be equal to the force of gravity acting upon the inhabitants of Earth. Thus the acting forces are only thus big, in order to give our fantasy the possibility to imagine the traveler as a human being. Since we have &lt;math&gt;\sqrt{1-v^{2}/c^{2}}&lt;/math&gt; throughout, proper time flows four times slower for the traveler than for the inhabitants of the stars. Thus when he comes back to our solar system after 100 of our years, he will feel to have aged only by about 25 years. If he reaches an age of 75 years according to the development of his body and his own time experience, then this corresponds to a threefold return to our solar system, i.e. 300 of our Earth years.

|}

{{Lorentzbox|Text=a) Wiechert (1915)&lt;ref name=wiechert15 /&gt; later provided a short historical survey of the clock/twin paradox. He referred to the fact that already {{slink||Einstein 1905}} considered the case of two clocks (&quot;Einstein's clock experiment&quot;), and even though [[w:Hermann Minkowski|Minkowski]] himself didn't consider the case, his proper time formula provides the result in a straight forward manner. The latter was done by himself in lectures on 25 March and 23 May 1911, as well as by Langevin published in July 1911. Wiechert pointed out that he himself and Langevin used &quot;humorist&quot; examples in order to clarify the situation: While Wiechert argued that one has to make a journey in order to stay young, Langevin argued that one has to romp about in a laboratory in order to stay young. Both of them used human beings, arguing that their physical and mental life should have been influenced in the same way as any other process in nature.

b) The dates given by Wiechert (1915) are not complete. The correct ones are:

*Langevin's lecture on 10 April 1911, published in July.

*Wiechert's lectures on 25 March and 23 May 1911, submitted on July 26, published in September.

*He was still unaware of Einstein's lecture from January 1911, published in November 1911.}}

|-

!{{anchor|Müller 1911-HU}}[[w:Fritz Müller-Partenkirchen|Müller]]

|-

|The freelance writer and law student Fritz Müller (who was later known as [[w:Fritz Müller-Partenkirchen|Müller-Partenkirchen]]) attended Einstein's lecture and wrote a popular report about it in the German newspaper &quot;[[w:Berliner Tageblatt|Berliner Tageblatt]]&quot; on 16th and 23rd October 1911,&lt;ref name=muller /&gt; in which he gave further details (compare with {{slink||Lämmel 1911-HU}}). Regarding the clock/twin paradox he wrote:

{| 

! width=50% | Müller wrote

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Zwei gleichgehende Uhren sollen je einen Beobachter haben und nebeneinander ruhen. Nun soll die eine mit ihrem Beobachter plötzlich mit Lichtgeschwindigkeit in den Weltenraum hinausreisen. Vorher haben die beiden vereinbart, sich alle Sekunden mit einem Lichtsignal die Zeit zu telegraphieren. [...] In unserem Grenzfall, wo die Reise mit Lichtgeschwindigkeit vor sich geht, müßte der ruhende Beobachter erklären, jene andere Uhr käme in der Zeit überhaupt nicht voran. Die Zeit stünde dort still. Tatsächlich kommen die Einsteinschen Gleichungen zu diesem Resultat. Für den mit der Uhr reisenden Beobachter, sagt Einstein, gelte dasselbe. Das heißt, im Urteil des Zurückbleibenden würde jener niemals alt. „Und wenn er auf einer gebrochenen Reiselinie wieder an seinen Ausgangspunkt zurückkehrte?&quot; fragt man den Vortragenden in der Diskussion. – „So bliebe er in unserem Urteil so jung wie bei der Ausreise,&quot; erwidert Einstein mit vollem Ernst, „selbst wenn wir Zurückgebliebenen inzwischen Männer mit weißen Bärten geworden sind – die Gleichungen liefern für jede Richtung der Bewegung, auch für eine gebrochene Bewegung, unerschütterlich die selben Resultate.&quot; – Wir sehen einander an. Das klingt märchenhaft. Märchenhaft? Gewiß, die alten Märchen vom Mönch von Heisterbach, vom Rip van Winkle, von Urashima Taro steigen auf. Merkwürdig, wie die Volksphantasie bei den Deutschen, bei den Amerikanern, bei den Japanern in der gleichen Richtung gearbeitet hat – alle drei Märchen erzählen ja von Leuten, deren Leben still steht, viele hundert Jahre lang, während die andern altern. So fanden sie bei ihrer Rückkehr ein anderes Land und eine andere Generation.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Two synchronous clocks at rest next to each other, shall each be accompanied by an observer. Now one of them, together with its observer, suddenly travels into space at the speed of light. Previously, both have arranged that every second they telegraph their time to each other using light signals. [...] In our limiting case where the journey happens at light speed, the resting observer would have to declare that the other clock would not proceed in time at all. Time would stand still at this place. Einstein's equations indeed produce this result. As to the observer traveling with the clock, says Einstein, the same is true. That means in the judgment of the remaining one, the other one would never become old. Then the lecturer [i.e. Einstein] was asked in the discussion: &quot;And if he comes back to his starting point on a curved travel path?&quot;, to which Einstein replied in full earnest: &quot;Then in our judgment he would remain as young as he was at departure, even if we remaining ones became men with white beards in the meantime, the equations unshakably give the same result in every direction of motion, also for curved motion&quot;. We look at each other. That sounds fabulous. Fabulous? Of course, the old fairy tales of [[w:Heisterbach Abbey|w:The monk of Heisterbach]] or [[w:Rip Van Winkle]] or [[w:Urashima Tarō]] come forward. Strange, how the folk fantasy of the Germans, the Americans, the Japanese worked in the same direction, all three fairy tales indeed tell about people whose life stands still, many hundred years long, while the other ones grow old. Thus they found another country and another generation when they returned.

|}

{{Lorentzbox|Text=Müller's account confirms {{slink||Lämmel 1911-HU}} that Einstein indeed mentioned human beings, but his description also suggests that Einstein was the first to use mutually sent light signals. However, as this was published in October, it cannot be excluded that Müller's description of light signals was influenced by {{slink||Langevin 1911-HU}}, published in July, in which light signals were used as well.}}

|}

==Twins from 1911 to 1920==

We now provide a list of authors who employed ''twins'', i.e. ''two'' life forms or humans that initially were of ''same age'' when the round-trip began:

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Date !! Description

|-

|[[w:Emil Wiechert|Wiechert]]&lt;ref name=wiechert11 /&gt;

|1911

|Two life forms that begin their life at the ''same time'' (German: &quot;Zwei Lebewesen [..] die ihr Leben gleichzeitig beginnen&quot;), of which the moving one returns retarded in its progression with respect to the stationary one.

|-

|[[w:Paul Gruner|Gruner]]&lt;ref name=gruner /&gt;

|1912

|Two persons of ''same age'' (French: &quot;deux personnes du même âge&quot;), of which the moving one returns less developed than stationary one.

|-

|[[w:Max von Laue|Laue]]&lt;ref name=laue3 /&gt;

|1913

|The moving life form returns younger than its ''former agemates'' (German: &quot;ehemaligen Altersgenossen&quot;).

|-

|[[w:Hermann Weyl|Weyl]]&lt;ref name=weyl /&gt;

|Easter 1918

|

{| 

! width=50% | German original

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Von zwei Zwillingsbrüdern, die sich in einem Weltpunkt A trennen, bleibe der eine in der Heimat (d. h. ruhe dauernd in einem tauglichen Bezugsraum), der andere aber unternehme Reisen, bei denen er Geschwindigkeiten (relativ zur »Heimat«) entwickelt, die der Lichtgeschwindigkeit nahekommen; dann wird sich der Reisende, wenn er dereinst in die Heimat zurückkehrt, als merklich jünger herausstellen denn der Seßhafte.

|Suppose we have two twin-brothers who take leave from one another at a world-point A, and suppose one remains at home (that is, permanently at rest in an allowable reference-space), whilst the other sets out on voyages, during which he moves with velocities (relative to “home”) that approximate to that of light. When the wanderer returns home in later years he will appear appreciably younger than the one who stayed at home. 

|}

{{Lorentzbox|Text=Weyl was the first to ''explicitly use twins'' in relation to the round-trip experiment. The fourth edition (1920) of that book was translated from German into English and French in 1922.}}

|-

|[[w:Albert Einstein|Einstein]]&lt;ref name=einstein20 /&gt;

|1920/21

|{{Anchor|Einstein 1921-TW}}

{| 

! width=50% | German original

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; | Trifft A wieder bei B ein, so kann es sich ereignen, daß der beharrende Zwilling inzwischen 60 Erdjahre alt geworden ist, während der zurückkehrende nur 15 Jahre zählt, oder sich gar noch im Säuglingsstadium befindet. [..] Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls -Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. 

|If A then returns to B, it may happen that the twin who stayed at home is now sixty years old, whereas the wanderer is only fifteen years of age, or is perhaps only an infant still. [..] In the case of these two twins, Einstein declared, we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures. 

|}

{{Lorentzbox|Text=This was based on an interview of Einstein by Moszkowski. While the expression &quot;clock paradox&quot; was used since 1911/12 (see section {{slink||Paradoxical?}}), this seems to be the first time that it was rebranded as &quot;twin paradox&quot;. The copyright mark indicates 1920, while the title page indicates 1921. The translation from German into English also appeared in 1921.}}

|}

==Maximal proper time==

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Early examples

|-

|[[w:Paul Langevin|Langevin]]

1911

|{{anchor|Langevin 1911-PT}}In April 1911 (published July),&lt;ref name=langevin1 /&gt; he described the round-trip experiment without formulas using two portions of matter present at two events happening at the same place. The ''integration of proper time'' along the entire wordlines shows that the portion of matter that starts a closed cycle by receding and finally coming back, will have a ''smaller proper time'' than the one that stayed behind.

In October 1911 (published 1912),&lt;ref name=langevin2 /&gt; Langevin again showed that the portion of matter that described a closed cycle will have a ''smaller proper time'' &lt;math&gt;R&lt;/math&gt; than the one that stayed in an inertial frame, which is defined by the equation:

:&lt;math&gt;\begin{matrix}V^{2}\left(t-t_{0}\right)^{2}=d^{2}-R\\
\left[d^{2}=\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right]
\end{matrix}&lt;/math&gt; 

|-

|[[w:Emil Wiechert|Wiechert]]&lt;ref name=wiechert11 /&gt;

Lectures March-May 1911

submitted July

published September

|{{anchor|Wiechert 1911-PT}}Let two equal processes be observed in two equal material systems colocated in two moments (1) and (2), and let there velocities have been changed in arbitrarily different ways in the meantime. It follows that the ratio of advancement of those processes is given by the two intervals &lt;math&gt;\Delta\tau &lt;/math&gt; of their respective ''proper times''. He concluded that any round-trip clock experiment can be easily comprehended from that theorem by computation. The corresponding integral is:

:&lt;math&gt;\Delta\tau=\int_{1}^{2}d\tau=\int_{1}^{2}dt\sqrt{1-\frac{\mathfrak{v}^{2}}{c^{2}}}&lt;/math&gt; 

|-

|[[w:Eduard Study|Study]]&lt;ref name=study /&gt;

June 1911

|Minkowski's concept of worldlines implies that the straight path between two points of the same worldline is the ''longest'' among all paths between those points, if the path length on a worldline is defined by the related proper time.

{{Lorentzbox|Text=Study's book was purely mathematical without mentioning clocks or the round-trip experiment, alluding to his result only in a footnote.}}

|-

|[[w:Max von Laue|Laue]]

1911-13

|{{anchor|Laue 1911/12-PT}}In December 1911 (published 1912),&lt;ref name=laue1 /&gt; Laue showed without formulas that the round-trip experiment is represented by a curved worldline, which at worldpoint A decomposes into a row of curves, after which all of them will be re-united at worldpoint B to a single line. Of all curves connecting the points A and B having time-like direction throughout, the straight connection has the ''longest proper time.''

{{anchor|Laue 1912/13-PT}}In December 1912 (published 1913) in the second edition of this relativity book,&lt;ref name=laue1 /&gt; Laue described the proper time integral between events 1 and 2 of a slowly accelerated clock covering a broken line and a stationary clock covering a straight worldline. Of all worldlines covering 1 and 2, the straight line has the ''longest proper time''. Therefore the traveling clock in the round-trip experiment is retarded at reunion, because its curved worldline corresponds to a shorter proper time. This result he presented in terms of the following inequality, of which the right-hand side refers to the straight curve of the stationary clock, while all others possible curves are represented on left-hand side:

:&lt;math&gt;\tfrac{1}{c}\int_{1}^{2}\sqrt{du^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)}&lt;\tfrac{1}{c}\int_{1}^{2}du&lt;/math&gt; 

{{Lorentzbox|Text={{anchor|Sommerfeld 1913-PT}}Similar treatments can be found in the textbooks of [[w:Arnold Sommerfeld|Sommerfeld]] (1913),&lt;ref name=sommerfeld /&gt; [[w:Hermann Weyl|Weyl]] (1918),&lt;ref name=weyl /&gt; [[w:Wolfgang Pauli|Pauli]] (1921),&lt;ref name=pauli /&gt; [[w:August Kopff|Kopff]] (1921),&lt;ref name=kopff /&gt; [[w:Jean Becquerel|Becquerel]] (1922).&lt;ref name=becqu1 /&gt;}}

|}

==Triangle inequality==

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

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! Author !! Early examples

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|valign=top|[[w:Alfred Robb|Robb]]

1914-1920

|{{anchor|Robb 1914-TR}}In 1914&lt;ref name=robb1 /&gt; he showed that there are three types of triangles formed by intervals in Minkowski space, depending on whether one deals with &quot;separation lines&quot; (spacelike intervals), &quot;optical lines&quot; (lightlike intervals), or &quot;inertia lines&quot; (timelike intervals representing the path of nonaccelerated particles defined by &lt;math&gt;{\scriptstyle \left(x_{1}-x_{0}\right)^{2}+\left(y_{1}-y_{0}\right)^{2}+\left(z_{1}-z_{0}\right)^{2}-c^{2}\left(t_{1}-t_{0}\right)^{2}&lt;0}&lt;/math&gt;). As to a triangle formed by inertia lines, he showed that the sum of a certain two sides is ''less'' than that of the third one. 

{{Lorentzbox|Text=So the triangle inequality derived from time-like intervals in Minkowski space is ''[[w:Triangle inequality#Reversal in Minkowski space|inverse]]'' to the inequality in Euclidean space. This inverse inequality directly represents the most simple variant of the twin paradox: the traveler follows two sides of the time-triangle, while the stay-at-home observer follows the third side indicating maximal proper time.}}

[[File:RobbTriangle.svg|right|150px]]
In 1920&lt;ref name=robb2 /&gt; Robb gave a numerical example of the triangle ABC with time-like intervals (&quot;inertia lines&quot;) defined by coordinates

:&lt;math&gt;\begin{matrix} &amp; x &amp; y &amp; z &amp; t\\
A\  &amp; 0 &amp; 0 &amp; 0 &amp; 0\\
B\  &amp; 0 &amp; 0 &amp; 0 &amp; 10\\
C\  &amp; 4 &amp; 0 &amp; 0 &amp; 5
\end{matrix}&lt;/math&gt; 

which he plugged into 

:&lt;math&gt;\bar{s}^{2}=\left(t_{1}-t_{0}\right)^{2}-\left(x_{1}-x_{0}\right)^{2}-\left(y_{1}-y_{0}\right)^{2}-\left(z_{1}-z_{0}\right)^{2}&lt;/math&gt; 

from which he obtained the sides AB=10, AC=3, CB=3 and the inequality &lt;math&gt;AC+CB&lt;AB&lt;/math&gt;.

|-

|[[w:Arthur Eddington|Eddington]]&lt;ref name=edding2 /&gt;

1922

|He distinguished between the &quot;space-triangle&quot; for spacelike intervals, and the &quot;time-triangle&quot; for time-like intervals. The latter is measured with a clock from A to B and from B to C, with the sum of those readings ''is always less'' than the reading of a clock measuring directly from A to C. In the ordinary space-triangle any two sides are together greater than the third side; in the time-triangle two sides are together ''less'' than the third side. 

|-

|Rogers&lt;ref name=rogers /&gt;

1922

|He showed that the &quot;pure time-triangle&quot; C, A, B (in their proper time order) satisfies the relation &lt;math&gt;\cosh C=\tfrac{\alpha^{2}+\beta^{2}-\gamma^{2}}{2\alpha\beta}&lt;/math&gt;, where &lt;math&gt;\cosh C&lt;/math&gt; denotes the unit-scalar product of the vectors CA, CB, and &lt;math&gt;\alpha,\beta,\gamma &lt;/math&gt; the real and positive intervals BC, CA, AB. Since &lt;math&gt;\alpha&gt;\beta &lt;/math&gt; and &lt;math&gt;\cosh C&gt;1&lt;/math&gt;, it follows that &lt;math&gt;\alpha&gt;\beta+\gamma &lt;/math&gt;. That is, &quot;the greatest side of pure time-triangle is greater than the sum of the other two sides&quot;. It follows at once that the stationary value of the proper time integral is an &quot;absolute maximum&quot;.

|}

==Negligibility of proper acceleration==

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

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! Author !! Early examples

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|valign=top |[[w:Albert Einstein|Einstein]]

1905-1918


|In 1905,&lt;ref name=einstein05 /&gt; Einstein used velocity time dilation &lt;math&gt;\tau=t\sqrt{1-\left(\frac{v}{V}\right)^{2}}&lt;/math&gt; to derive the retardation of a clock performing a round-trip with constant speed &lt;math&gt;v&lt;/math&gt; along a polygonal path or a continuously curved line, without mentioning any influence of acceleration at turnaround.

{{anchor|Einstein 1911-VA}} In 1911 (published 1912),&lt;ref name=einstein3 /&gt; Einstein said that special relativity doesn't say anything about what happened to the clock's pointer position during the acceleration that changes the clock's direction along the round-trip, yet the influence of this change must be getting smaller the longer the clock ''is moving uniformly'', i.e. the longer one chooses the dimensions of the path.

{{anchor|Einstein 1912-VA}}In an unpublished manuscript on special relativity from 1912,&lt;ref name=einst12manu /&gt; he pointed out that any influence of acceleration during the round-trip experiment, can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygonal path. 

{{anchor|Einstein 1914a-VA}}In a letter from April 1914,&lt;ref name=einstpetz /&gt; Einstein showed that any ''finite'' acceleration at turnaround during the round-trip experiment can only influence the clock in a ''finite'' way, thus it can be neglected by minimizing the time of acceleration with respect to the time of uniform translation. So it ''must be concluded'' that the clock is retarded at reunion after traveling on a polygonal path.

{{anchor|Einstein 1914b-VA}}During a conversation in May 1914,&lt;ref name=rowe group=S /&gt; Einstein is reported to have replied that the accelerations during the round-trip are &quot;irrelevant for the amount of the time difference&quot;. (Compare with {{slink||Einstein 1914b-AC}}) 

{{anchor|Einstein 1918-VA}}In his famous &quot;Dialog about Objections against the Theory of Relativity&quot; from 1918,&lt;ref name=einstein18 /&gt; Einstein pointed out that any effect of velocity changes at turnaround must be limited, thus the traveling clock must be retarded at reunion due to time dilation if one makes the path AB and back along the round-trip long enough. (Compare with {{slink||Einstein 1918-AC}})

|-

|[[w:Emil Wiechert|Wiechert]]&lt;ref name=wiechert11 /&gt;

1911

|{{Anchor|Wiechert 1911-VA}}[[File:WiechertTwin.svg|110px|right]] He demonstrated that differential aging along the round-trip cannot be caused during the passage from one velocity to another (i.e. acceleration) at turnaround, because the same result also follows when ''both'' A and B experience the ''same velocity changes'' with respect to another frame, only with the difference that B has relative velocities &lt;math&gt;+u&lt;/math&gt;  and &lt;math&gt;-u&lt;/math&gt; for a long time, while A is brought after a short time from relative velocity &lt;math&gt;+u&lt;/math&gt; to relative rest at which it remains a long time, and then it is brought to relative velocity &lt;math&gt;-u&lt;/math&gt;  for a short time.

{{Lorentzbox|Text=He was probably the first to use an example in which both accelerate with same magnitude.}}

|-

|[[w:Max von Laue|Laue]]&lt;ref name=laue3 /&gt;

1913

|{{anchor|Laue 1913-VA}}He showed that the problem of the influence of acceleration at turnaround in the round-trip experiment, can be eliminated by ''arbitrarily'' enlarging the time in inertial motion.

{{Lorentzbox|Text=This is the same argument as given in {{slink||Einstein 1911-VA}}. The Einstein-Laue argument was also used by others such as [[w:Hans Thirring|Thirring]] (1921)&lt;ref name=thirring /&gt; or [[w:Max Born|Born]] (1921).&lt;ref name=born /&gt;}}

|-

|[[w:Hendrik Lorentz|Lorentz]]&lt;ref name=lorentz1 /&gt;

1913

|He pointed out that any effect of acceleration on the traveling clock at turnaround, can be separated from the time dilation effect since only the latter depends on the distance traversed along the round-trip.

{{Lorentzbox|Text=Similarly, [[w:Wolfgang Pauli|Pauli]] (1921) stated that the arising infinitesimal accelerations at turnaround are certainly independent of the total travel time and ''therefore easy to eliminate''.&lt;ref name=pauli /&gt;}}
|}

==Relay (three brothers) experiment==

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

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! Author !! Early examples

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|[[w:de:Fritz Grünbaum (Physiker)|Grünbaum]]&lt;ref name=gbaum /&gt;

1911

|He discussed a one-way time dilation experiment in which the first clock is set into motion from the origin and then moving to the second clock. He argued that one can avoid the problem of acceleration experienced by the first clock when set into motion, by replacing it with a ''third'' clock that is already in motion with constant velocity and is synchronized at the origin with the first clock.

{{Lorentzbox|Text=While Grünbaum didn't discuss round-trip experiments, his introduction of a third clock in order to avoid acceleration is the basis of the three-brother experiment.}}

|-

|valign=top|[[w:Emil Wiechert|Wiechert]]

1920-1922

|In 1920 (published 1921),&lt;ref name=wiechert20 /&gt; Wiechert explained how to completely remove acceleration from the round-trip experiment: Bodies A, B, C move undisturbed and non-accelerated in different directions. A and B pass each other at time (1), B and C pass each other at a later time (2), and C and A finally pass each other at an even later time (3). So in this setup, the condition of C is the continuation of the condition of B. On any of the three bodies one can count the oscillations of light of a certain spectral-line, in which case relativity predicts that the ''combined sum of all oscillations'' on B+C is smaller than the number of oscillations on A alone. Wiechert also held that one can replace the light oscillations by the life functions of human-like beings which live on A, B and C. For instance, while the inhabitants of B+C only had time for one meal, there were arbitrarily many generations on A who follow after each other by death and birth.

[[File:Wiechert1922a.png|180px|right]]
In 1921 (published 1922),&lt;ref name=wiechert21 /&gt; Wiechert extended his previous acceleration-free round-trip experiment to an arbitrary number of non-accelerated bodies &lt;math&gt;B_{1}&lt;/math&gt;, &lt;math&gt;B_{2}&lt;/math&gt;, ..., which constitutes a &quot;relay&quot; (German: Stafette) starting from body A and back again. The first B passes A and moves away, and after some time the last B comes back to A. Since any B body continues the fate of the previous one, all bodies &lt;math&gt;B_{1}&lt;/math&gt;, &lt;math&gt;B_{2}&lt;/math&gt;, ..., combined have emitted fewer oscillations than A alone during the relay race. Wiechert pointed out that instead of light oscillations one can also choose the aging of life forms.

{{Lorentzbox|Text=Such relay experiments were later independently rediscovered in English language papers&lt;ref name=debs group=S /&gt; such as by Lange (1927)&lt;ref group=S name=lange /&gt; in which the brothers synchronize their times when they pass each other (“three brother experiment”).}}
|}

==Acceleration as asymmetry indicator==

While it was known that any direct influence of [[w:proper acceleration]] on clocks can be neglected in the computation of the inertial frame of the stay-at-home twin (see previous section {{slink||Negligibility of proper acceleration}}), the very fact that only one of them is accelerating is still useful as an asymmetry argument in order to show that there is no contradiction to the relativity principle.

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

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! Author !! Early examples

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|[[w:Paul Langevin|Langevin]]&lt;ref name=langevin1 /&gt;

1911

|{{Anchor|Langevin 1911-AC}}He derived differential aging in the round-trip experiment using the proper time integral along worldlines (see {{slink||Langevin 1911-PT}}) and used acceleration as an asymmetry indicator: The result of the round-trip experiment is &quot;another example of the absolute character of acceleration&quot; in which the &quot;asymmetry occurred because only the traveler, in the middle of his journey, has undergone an acceleration that changes the direction of his velocity&quot;. 

|-

|[[w:Arnold Sommerfeld|Sommerfeld]]&lt;ref name=sommerfeld /&gt;

1913

|After he showed (see {{slink||Sommerfeld 1913-PT}}) that retardation of time in the round-trip experiment derived from the proper time integral rests on the assumption that the clock's rate ''only depends on its momentary velocity'' (now called &quot;clock hypothesis&quot;), he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle since one of the clocks has to be accelerated in order to come back, thus the retardation in the round-trip experiment does not demonstrate &quot;motion&quot;, but &quot;accelerated motion&quot;.

|-

|[[w:Hendrik Lorentz|Lorentz]]

1913&lt;ref name=lorentz1 /&gt;

|After he derived differential aging in the round-trip experiment from velocity time dilation and pointed out the negligibility of proper acceleration for the computation, he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle, since one of them changes velocity and accelerates; the relativity principle does not require symmetry between inertial and non-inertial observers.

|-

|valign=top|[[w:Albert Einstein|Einstein]]

1914-1920

|{{anchor|Einstein 1914b-AC}} During a conversation in 1914,&lt;ref name=rowe group=S /&gt; Einstein is reported to have said that moving clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are ''irrelevant'' for the ''amount'' of the time difference, their ''presence'' nevertheless cause B to fall behind (&quot;accelerated motions are absolute&quot;).

{{anchor|Einstein 1918-AC}}In his famous &quot;Dialog about Objections against the Theory of Relativity&quot; from 1918&lt;ref name=einstein18 /&gt;, Einstein pointed out the negligibility of velocity changes from the viewpoint of an inertial frame (see {{slink||Einstein 1918-VA}}). Then he used ''acceleration as an asymmetry indicator'' in order to show, that there is no contradiction to the relativity principle, because relativity only predicts the equivalence of non-accelerated inertial frames: &quot;only K is such a frame while K' is temporarily accelerated, thus the retardation of U2 with respect to U1 cannot be used to construe a contradiction against the theory.&quot; 

{{anchor|Einstein 1920-AC}}Einstein is reported to have said in an interview from 1920:&lt;ref name=einstein20 /&gt;
{| 

! width=50% | German original

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls-Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. Dieser Grund für das Jüngerbleiben des A ergibt sich vom Gesichtspunkt der speziellen Relativitätstheorie aus der Tatsache, daß das betreffende Geschöpf — und nur dieses — Beschleunigungen erlitten hat.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |In the case of these two twins,&quot; Einstein declared, &quot;we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures . This ground, which counts for the comparative youth of A, is given, from the point of view of the special theory of relativity, by the fact that the creature in question, and only this creature, has been subject to accelerations.&quot;
|}

In a discussion from 1922,&lt;ref name=morand /&gt; Einstein is reported to have said that there is no contradiction in the round-trip experiment (in terms of a train leaving the station and returning later): The relativity principle is not applicable to this case, because the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two frames having velocities in opposite direction is not an inertial frame. There is no reciprocity between a frame that changes direction and one that doesn't.

|}

==Frame distribution as asymmetry indicator==

Because any direct influence of proper acceleration on the traveling clock at turnaround can be neglected (see {{slink||Negligibility of proper acceleration}}), the importance of {{slink||Acceleration as asymmetry indicator}} is limited to the mere fact that it reveals that only the traveler was in a non-inertial frame as only he changed his inertial frames, thus instead of emphasizing the occurrence of proper acceleration at turnaround, it's possible to describe the asymmetry more geometrically by emphasizing the different distribution of inertial frames of the twins along their worldlines.

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

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! Author !! Early examples

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|valign=top|[[w:Max von Laue|Laue]]

1911-1913

|{{Anchor|Laue 1911/12-VA}} In 1911/12,&lt;ref name=laue1 /&gt; he pointed out that during the time of separation, that clock is most advanced which was at rest in an inertial frame all the time; namely there is ''always one, and only one inertial frame'', in which the locations of separation and re-encounter lie in the same geometric point. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1911/12-PT}}).

In 1912/13,&lt;ref name=laue2 /&gt; he argued that in the round-trip experiment, we indeed can decide, which one of the clocks was steadily at rest in one and the same reference system, and which one was in the meantime at rest in two or more such systems. Among them there is of course a real physical difference. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1912/13-PT}}).

In 1913&lt;ref name=laue3 /&gt; Laue pointed out:

{| 

! width=50% | German original

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; | Aber nach unseren Voraussetzungen ruht während der Zeit der Trennung die erste Uhr in ''einem'' berechtigten Bezugssystem, die zweite hingegen ruht zwar sowohl bei der Hin- wie bei der Rückbewegung in berechtigten Bezugssystemen, aber notwendig in ''zwei verschiedenen. Deshalb'' unterscheiden sich beider Schicksale physikalisch. Ließe man die zweite Uhr in der ihr anfangs erteilten Bewegung und schickte man ihr dafür die erste Uhr nach einiger Zeit mit größerer Geschwindigkeit nach, so würde beim Zusammentreffen die erste gegen die zweite zurückgeblieben sein; denn jetzt hat die erste während der Trennung in zwei verschiedenen Systemen geruht. (Footnote: Dem naheliegenden Einwand, daß wir über den Gang einer Uhr während eines Geschwindigkeits''wechsels'' nichts aussagen können, begegnet man am einfachsten mit dem Hinweis, daß man die Zeiten der gleichförmigen Bewegung ''beliebig'' groß gegen die der Beschleunigung machen kann.)

| style=&quot;padding: 0px 20px 0px 20px;&quot; | However, by our presuppositions, one clock is at rest in ''one'' valid reference system during the time of separation, while the second one is at rest in valid reference systems both during the forward- and the backward motion, but necessarily in ''two different ones. Therefore'' the two fates differ physically. If the second clock remains in the motion which was given to it at the start, and if after some time it is followed by the first clock with greater velocity, then the first one would be retarded with respect to the second one at the encounter; since now it was the first one that was at rest in two different systems during the separation. (Footnote: The objection which is near at hand, that we cannot say anything about the rate of a clock during a velocity ''change'', can be met most simply by the allusion, that we can render the times of uniform motion ''arbitrarily'' great with respect to acceleration..)
|}

|-

|[[w:Werner Bloch|Bloch]]&lt;ref name=bloch /&gt;

September 1918

|{{anchor|Bloch 1918-VA}} He represented the frames with three movable slots K, K' and K”, provided with hooks on which one can hang clocks at the origins of K and K'; while one clock always hangs on a hook of slot K, the other clock moved away with K' and after some time was transferred (neglecting any effect of acceleration) by a mechanical device to slot K” that moves in the other direction, by which it comes back; there is no contradiction to the relativity principle, as one clock rested in one inertial frame while the other one rested in two such frames.

|}

==Perspective of the traveler==

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

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! Author !! Early examples

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|[[w:Paul Langevin|Langevin]]&lt;ref name=langevin1 /&gt;

1911

|{{anchor|Langevin 1911-LI}}[[Image:rstd4.gif|170px|right]] After deriving differential aging from the proper time integral in {{slink||Langevin 1911-PT}} and using human beings in {{slink||Langevin 1911-HU}}, he described the perspectives of both observers using light signals and the Doppler effect. When they separate they see each other live 200 times slower, while at return they see each other live 200 times faster. So ''from the explorer's viewpoint'', in the first year he sees the Earth perform the actions of two days, while in the second year he sees the Earth perform the actions of two centuries. The asymmetry can be seen by noticing, that the observer on Earth in 200 years sees the explorer performs the actions of 1 year. Then the explorer turns around, after which the observer on Earth in 2 days sees the projectile perform the actions of another year.

{{Lorentzbox|Text=Langevin used &lt;math&gt;v=c\left(1-\tfrac{1}{20000}\right)&lt;/math&gt;, producing Lorentz factor &lt;math&gt;\gamma\approx100&lt;/math&gt; and Doppler factor &lt;math&gt;\sqrt{\tfrac{c+v}{c-v}}\approx200&lt;/math&gt;.}}

|-

|[[w:Hendrik Lorentz|Lorentz]]

Lectures published in 1913&lt;ref name=lorentz1 /&gt;

Similar treatment in 1914&lt;ref name=lorentz3 /&gt;

|{{anchor|Lorentz 1913/14-LI}}Described the round-trip experiment in terms of inertial observer A (equipped with clock K) and traveling observer B (equipped with clock K'). In the frame of A, clock K' is retarded with respect to K at reunion due to time dilation. He then described the perspective of the traveling observer B by using two-way propagation of light from K' to K and back to K', leading to three periods defined by the moment of B's turnaround: In the first period the light signals return to K' before turnaround; in the second period the signals are emitted before turnaround and return after turnaround; in the third period emission and return of the signals are both happening after turnaround. Lorentz showed that K is time dilated by a factor of &lt;math&gt;\sqrt{1-v^{2}/c^{2}}&lt;/math&gt; with respect to K' in the first and third period, but in the second period K is ticking ''faster'' than K' by a factor of &lt;math&gt;\sqrt{\tfrac{c+v}{c-v}}&lt;/math&gt; which overcompensates the dilation in the other periods and explains, even from the perspective of B, why K' is retarded with respect to K at reunion.

{{Lorentzbox|Text=In a review of the German translation of Lorentz's book, Einstein (1914) didn't directly mention Lorentz's treatment of the twin paradox, but he wrote that nobody who is seriously interested in relativity should neglect to read that book.&lt;ref name=einstlor /&gt; [[w:Wolfgang Pauli|Pauli]] (1921) refers to Lorentz's book as one of three papers that analyze the twin paradox more closely.&lt;ref name=pauli /&gt;}}

|-

|valign=top| [[w:Albert Einstein|Einstein]]

1916-1920

|{{anchor|Einstein 1916-EP}}In a lecture from 1916,&lt;ref name=einstein16 /&gt; of which only an abstract was published, Einstein spoke about the &quot;clock paradox of special relativity from the standpoint of [[w:general relativity]].&quot;

{{anchor|Einstein 1918a-EP}}In a letter from September 1918,&lt;ref name=einadl /&gt; Einstein showed that general relativity makes the inertial frame K and and the accelerated frame K' of the clocks in the round-trip experiment &quot;equally justified&quot;, explaining the time difference in K' by combining the influence of velocity and gravitational potential on clocks.

{{anchor|Einstein 1918-EP}}In his famous &quot;Dialog about Objections against the Theory of Relativity&quot; from November 1918,&lt;ref name=einstein18 /&gt; aimed at clarifying misconceptions of the clock paradox, he explained that there is no paradox in special relativity because there is no symmetry between clock U1 at rest in inertial frame K and clock U2 at rest in accelerated frame K' (see {{slink||Einstein 1918-AC}}). Yet [[w:general relativity]] and the [[w:equivalence principle]] allow the treatment of this problem also from the standpoint of frame K', where clock U2 remains at rest all of the time while U1 makes the following movements: (1) It is accelerated by a homogeneous gravitational field in the negative direction, (2) it moves with constant velocity &lt;math&gt;-v&lt;/math&gt;, (3) it is accelerated in the positive direction until it turns around and comes by with constant velocity &lt;math&gt;+v&lt;/math&gt;, (4) it moves with velocity &lt;math&gt;+v&lt;/math&gt;, (5) it is accelerated in the negative direction until it stops. Clock U1 is retarded with respect to U2 in periods 2) and 4) due to velocity time dilation, but this retardation is overcompensated by the faster rate of U1 during period 3), because U1 is at a higher gravitational potential. He argued that the computation (which he didn't provide) shows that the advance of U1 in period 3) is double its retardation during periods 2) and 4). Einstein concluded that by this consideration &quot;the paradox is completely resolved&quot;. Using [[w:Mach's principle]], he pointed out that the gravitational field in K' might be induced by the masses of the universe that are accelerated in this frame.

{{anchor|Einstein 1918b-EP}}In a letter to Einstein from December 1918, [[w:Max Jakob|Jakob]] doubted the result that the advance in period 3) is double the retardation during periods 2) and 4). Einstein responded by letter,&lt;ref name=einstein18b /&gt; in which he used the gravitational time dilation factor &lt;math&gt;1+\Phi/c^{2}&lt;/math&gt; in K' in order to show that U1 at distance &lt;math&gt;l&lt;/math&gt; is advancing by &lt;math&gt;\Phi/c^{2}=2vl/c^{2}&lt;/math&gt; in period 3), which is indeed the double of approximated delay &lt;math&gt;vl/c^{2}&lt;/math&gt; caused by velocity time dilation during periods 2) and 4).

{{anchor|Einstein 1921-EP}}Einstein is reported to have said in an interview from 1920,&lt;ref name=einstein20 /&gt; that while acceleration explains the age difference between the stationary twin B and the traveling twin A in terms of special relativity (see {{slink||Einstein 1920-AC}}), the &quot;proper&quot; description in terms of general relativity is as follows:

{| 

! width=50% | German original

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; | Eine tiefere Erfassung des Grundes ist indeß nur auf dem Boden der „Allgemeinen Relativitätstheorie&quot; zu erlangen, die uns erkennen läßt, daß von A aus beurteilt ein Zentrifugalfeld existiert, von B aus betrachtet aber nicht; und dieses Feld hat einen Einfluß auf den relativen Ablauf und die Raschheit der Lebensvorgänge.

| style=&quot;padding: 0px 20px 0px 20px;&quot; | A proper grasp of the reason is furnished only when we adopt the general theory of relativity, which tell us that, from the point of view of A, a centrifugal field exists, whereas it is absent from the point of view of B. This field exerts an influence on the relative rate of happening of the events of life.&quot;
|}

{{Lorentzbox|Text=a) Einstein's explanation was quickly adopted in the textbooks of [[w:Werner Bloch|Bloch]] (1920),&lt;ref name=bloch2 /&gt; [[w:Wolfgang Pauli|Pauli]] (1921),&lt;ref name=pauli /&gt; [[w:August Kopff|Kopff]] (1921),&lt;ref name=kopff /&gt; [[w:Karl Bollert|Bollert]] (1921),&lt;ref name=bollert1 /&gt; [[w:Max Born|Born]] (1921),&lt;ref name=born /&gt; expressing the view that general relativity is &quot;necessary&quot; to provide the &quot;complete&quot; solution of the twin paradox.
b) From a modern standpoint, however, Einstein's explanation has nothing to do with general relativity, but is rather an application of accelerated frames and &quot;pseudo&quot;-gravitational fields to flat Minkowski space of ''special'' relativity.&lt;ref name=weiss group=S /&gt;}}

|-

|[[w:Hans Thirring|Thirring]]&lt;ref name=thirring /&gt;

April 1921

|{{anchor|Thirring 1921-DS}}[[Image:Twin Paradox Minkowski Diagram.svg|right|200px]]
He described the round-trip experiment by using two platforms K (clock A) and K' (clock B) each equipped with rows of clocks. He first demonstrated the symmetry of time dilation and the mutual relativity of simultaneity on the platforms and its effect on clock synchronization. The K clocks that B passes are all advanced because of &lt;math&gt;t'=\gamma\left(t-vx/c^{2}\right)&lt;/math&gt;, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation &lt;math&gt;t'-t'_{0}=\gamma\left(t+vx/c^{2}\right)&lt;/math&gt; leading to the effect of clock desynchronization, where &lt;math&gt;t'_{0}&lt;/math&gt; is a constant depending on which clock one uses as standard for the new synchronization. He graphically showed using Minkowski diagrams, that this simultaneity jump due to desynchronization amounts to double the velocity time dilation during the inertial phases, explaining why A is more advanced than B at reunion. 

{{Lorentzbox|Text=Using clock B as synchronization standard, Thirring's constant is given by &lt;math&gt;t'_{0}=2l\gamma v/c^{2}=2t\gamma v^{2}/c^{2}&lt;/math&gt; with &lt;math&gt;l=vt&lt;/math&gt; as position of turnaround. A similar explanation was subsequently given by Langevin (1922).&lt;ref name=morand /&gt;}}

|}

==Curved spacetime==

While the previous examples are defined in flat Minkowski spacetime and therefore can be fully discussed in terms of special relativity, [[general relativity]] is required when [[:w:spacetime curvature]] in the presence of mass and energy cannot be neglected any more.&lt;ref name=koks group=S /&gt; 

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Early examples

|-

|[[w:Jean Becquerel|Becquerel]]&lt;ref name=becqu1 /&gt;

1922

|After defining gravitational time dilation &lt;math&gt;d\tau=\sqrt{1-\tfrac{2GM}{c^{2}r}}dt&lt;/math&gt; in terms of the [[w:Schwarzschild metric]] around a material center, he discussed the following round-trip experiment: There are two identical clocks A and B placed next to each other, at a point very far from the material center, initially marking the same time &lt;math&gt;t&lt;/math&gt;. Let us transport clock A to a point where the field is more intense, at a distance &lt;math&gt;r&lt;/math&gt; from the center; this clock will measure time &lt;math&gt;\int d\tau&lt;/math&gt; which is shorter than &lt;math&gt;\int dt&lt;/math&gt;, thus it will run more slowly. If we bring clock A back to clock B, we will have to note that it is retarded with respect to B.

|}

==Paradoxical?==

{| class=wikitable style=&quot;background-color:white;&quot;

! width=50% | German original of [[w:Max von Laue|Laue]] (1911/12):&lt;ref name=laue1&gt;Laue introduces the word &quot;paradox&quot;, alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource&lt;/ref&gt;

! English translation

|-

|Unter all den paradox erscheinenden Folgerungen aus der Zeittransformation der Relativitätstheorie gibt es wohl keine, gegen welche sich der natürliche Menschenverstand bei jedem, der der Sache noch ungewohnt ist, so sehr sträubt, wie gegen die, daß die Zeitangabe einer Uhr von ihrem Bewegungszustand abhängen soll. Schon in seiner grundlegenden Arbeit hat Einstein diese Paradoxie auf die Spitze getrieben in einem Gedankenexperiment, welches neuerdings von Langevin in einem auch sonst sehr lesenswerten Vortrage besonders hübsch erläutert worden ist.

|Of all apparently paradox consequences that stem from the time-transformation of the theory of relativity, there is probably none against which the common sense of anyone who is still unfamiliar with the matter is more reluctant, than the one according to which the time indication of a clock shall be dependent on its state of motion. Already in his fundamental paper, Einstein has driven this paradox to the extreme by a thought experiment, recently explained in a very nice way by Langevin in a lecture that is also very readable in other respects.

|-

|colspan=2|{{Lorentzbox|Text=Laue was probably the first to denote the round-trip experiment as paradoxical (even though he pointed out that there are no real contradictions). Subsequently, [[:w:Paul Gruner|Gruner]] (1912)&lt;ref name=gruner /&gt; and others including Einstein (1918)&lt;ref name=einstein18 /&gt; explicitly used the expression &quot;clock paradox&quot; (French: Paradoxe des horloges, German: Uhrenparadoxon), whereas [[w:Rudolf Seeliger|Seeliger]] (1913)&lt;ref name=seel /&gt; spoke of the &quot;familiar Einstein-Langevinian paradox&quot; (German: &quot;bekannte Einstein-Langevinsche Paradoxon&quot;).}}

|}

==Misunderstandings==

{| class=wikitable style=&quot;background-color:white;&quot; 

! width=50% padding=10 | German original by [[w:Otto Berg (scientist)|Berg]] (1910):&lt;ref name=berg /&gt;

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Im Punkte &lt;math&gt;x = 0&lt;/math&gt; des Systems S befinde sich eine Uhr, eine andere im Punkte &lt;math&gt;x'=0&lt;/math&gt; von S'. Diese zweite bewege sich mit S' bis zum Punkte &lt;math&gt;x = a&lt;/math&gt;, kehre dort um und bewege sich nun mit der Geschwindigkeit &lt;math&gt;v&lt;/math&gt; zurück bis zum Punkte &lt;math&gt;x= 0&lt;/math&gt;. Welche Zeit müssen beide Uhren in dem Moment angeben, wo sie sich wieder treffen? Wir beantworten diese Frage zunächst vom Standpunkt des Beobachters in S. Die Uhr in &lt;math&gt;x' = 0&lt;/math&gt; hat sich mit der Geschwindigkeit &lt;math&gt;v&lt;/math&gt; bis zum Punkte &lt;math&gt;x = a&lt;/math&gt; bewegt; dazu brauchte sie die Zeit &lt;math&gt;\tau=\tfrac{a}{v}&lt;/math&gt;. Zum Rückweg ist dieselbe Zeit nötig. Nach der Zeit &lt;math&gt;2\tau=2\tfrac{a}{v}&lt;/math&gt; ist die Uhr also wieder im Punkte &lt;math&gt;x = 0&lt;/math&gt; angelangt. Wir stellen uns nun auf den Standpunkt des Beobachters in S'. Für diesen führt nach dem Relativitätsprinzip das System S genau dieselben Bewegungen aus wie das System S' für den Beobachter in S, nur in entgegengesetzter Richtung. Die Zeit bis zum Zusammentreffen beider Uhren ist also im System S' ebenfalls gegeben durch &lt;math&gt;2\tau=2\tfrac{a}{v}&lt;/math&gt;. Betrachtungen, die auf anschauliche Vorstellungen, wie Nachgehen von Uhren, gestützt sind, führen hier leicht zu Irrtümern, von denen auch die Fachlitteratur nicht frei ist.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |There is a clock at point &lt;math&gt;x=0&lt;/math&gt; of system S, and another one at point &lt;math&gt;x'=0&lt;/math&gt; of S'. The second one moves together with S' until point &lt;math&gt;x=a&lt;/math&gt;, turns around and now moves back with speed &lt;math&gt;v&lt;/math&gt; to point &lt;math&gt;x=0&lt;/math&gt;. Which time must both clocks indicate at the moment at which they encounter again? We answer this question at first from the standpoint of the observer in S. The clock at &lt;math&gt;x=0&lt;/math&gt; has been moving with speed &lt;math&gt;v&lt;/math&gt; until point &lt;math&gt;x=a&lt;/math&gt;, for which it required time &lt;math&gt;\tau=\tfrac{a}{v}&lt;/math&gt;. The same time is required for the way back. After time &lt;math&gt;2\tau=2\tfrac{a}{v}&lt;/math&gt; the clock has thus arrived again at point &lt;math&gt;x=0&lt;/math&gt;. Let's now take the standpoint of the observer in S'. In his view in accordance with the relativity principle, system S is conducting exactly the same motions as those of system S' with respect to the observer in S, only in opposite direction. Thus the time until the meeting of both clocks is given by &lt;math&gt;2\tau=2\tfrac{a}{v}&lt;/math&gt; in system S' as well. Considerations based on illustrative notions, such as the retardation of clocks, easily lead to mistakes at this place, of which also the professional literature isn't free.

|-

|colspan=2|{{Lorentzbox|Text=Berg was probably the first to turn the relativity principle against asymmetric aging in the round-trip experiment, claiming that both clocks must indicate the same time at reunion. See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem.}}

|-

! width=50% | German original by [[w:Emil Wiechert|Wiechert]] (1911)&lt;ref name=wiechert11 /&gt;

! English translation

|-

|colspan=2| Even though he correctly derived differential clock aging in the round-trip experiment, he claimed that effects like time dilation are &quot;apparent&quot; if one admits Einstein's &quot;unconditional&quot; relativity principle in which there is no aether and all &quot;strides&quot; (i.e. non-accelerated motions) are physically equivalent, but they are &quot;real&quot; if one admits the existence of an aether in the framework of a &quot;conditional&quot; relativity principle in which all strides are physically non-equivalent or anisotropic. This led him to the following interpretation of the clock paradox:

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |[...] so muß am Schluß des Versuches B in seinem Fortschritt gegenüber A im Verhältnis &lt;math&gt;1:\sqrt{1-u^{2}/c^{2}}&lt;/math&gt; zurückgeblieben sein. Und dieses Zurückbleiben ist unbedingt reell, denn die beiden Gebilde A und B können ja unter gleichen Umständen unmittelbar beieinander verglichen werden. Hier ist es ganz sicher ausgeschlossen, an einen Schein zu glauben, der durch unsere Auffassung der Zeit bewirkt wird. So ist denn also auch die Folgerung unabwendbar, daß für den Verlauf der Weltvorgänge die Schreitungen nicht gleichwertig sind, ''und damit sind wir von neuem zu einem Schluß gekommen, welcher der Unbedingtheit des Relativitätsprinzipes durchaus widerspricht.'' [...] Man kann den Versuch noch mannigfach variieren, z. B. so, daß A ebenso wie B zwei verschiedene Schreitungen, &lt;math&gt;+u&lt;/math&gt; und &lt;math&gt;-u&lt;/math&gt;, nacheinander inne hat. Wird dann zu A der Wert &lt;math&gt;u_{1}&lt;/math&gt;, zu B  der Wert &lt;math&gt;u_{2}&lt;/math&gt;, zugeordnet, so muß der Vergleich von A  und B am Schluß des Versuches ergeben, daß B  oder A in seinem Fortschritt zurückgeblieben erscheint, je nachdem die Schreitungen &lt;math&gt;+u_{1}&lt;/math&gt;, &lt;math&gt;-u_{1}&lt;/math&gt;, oder &lt;math&gt;+u_{2}&lt;/math&gt;, &lt;math&gt;-u_{2}&lt;/math&gt; weiter auseinanderliegen. ''Vielleicht ist gerade diese Formulierung des Satzes besonders geeignet, um die Ungleichwertigkeit der verschiedenen Schreitungen klar und deutlich zu zeigen.''

| style=&quot;padding: 0px 20px 0px 20px;&quot; | [...] thus B's progress must be retarded with respect to A's in the ratio &lt;math&gt;1:\sqrt{1-u^{2}/c^{2}}&lt;/math&gt; at the end of the experiment. And this retardation is definitely real, since both bodies A and B indeed can be immediately compared side by side under the same conditions. Here it is certainly excluded to believe that this is an appearance due to our conception of time. Thus the consequence is unavoidable too, that the strides are not equivalent in the course of the world processes, ''and therefore we again came to a conclusion that completely contradicts the unconditionality of the relativity principle.'' [...] One can vary this experiment in many ways, for instance, so that A in the same way as B successively undergoes two different strides &lt;math&gt;+u&lt;/math&gt; and &lt;math&gt;-u&lt;/math&gt;. If we apply the value &lt;math&gt;u_{1}&lt;/math&gt; to A and &lt;math&gt;u_{2}&lt;/math&gt; to B, then the comparison of A and B at the end of the experiment must give the result, that B or A is retarded in its progress depending on whether the strides &lt;math&gt;+u_{2},-u_{2}&lt;/math&gt; or &lt;math&gt;+u_{1},-u_{1}&lt;/math&gt; are further apart. ''Probably it is precisely this formulation of the theorem that is particularly suitable to demonstrate the non-equivalence of the different strides clearly and explicitly.''

|-

|colspan=2|{{Lorentzbox|Text=This interpretation was directly rebutted by Laue (1911/12) who demonstrated the geometrical meaning of differential aging in Minkowski space, see sections {{slink||Laue 1911/12-PT|Laue 1911/12-VA}}, showing that there is no need to assume non-equivalance or anisotropy of motions. Laue added, that as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.&lt;ref name=laue1 /&gt;}}

|-

! width=50% | German original by [[w:Norman Robert Campbell|Campbell]] (November 1911, published 1912)&lt;ref name=camp /&gt;

! English translation

|-

|colspan=2|After describing the round-trip experiment (as given by Wiechert) according to which the traveling clock B is retarded when it returns with respect to stationary clock A, he abandoned differential clock aging as follows: 

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Dieser Schluß ist nicht richtig. Die Beziehung zwischen &lt;math&gt;t&lt;/math&gt;, der Ablesung an der Uhr auf A seitens des Beobachters auf A und &lt;math&gt;t'&lt;/math&gt;, der Ablesung an der Uhr auf B seitens des Beobachters auf A, ist (unter der Annahme, daß zu Beginn des Versuchs &lt;math&gt;t=t'&lt;/math&gt; ist)
:&lt;math&gt;t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)&lt;/math&gt;. 
Der Unterschied zwischen &lt;math&gt;t'&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt; ist eine Funktion von &lt;math&gt;z&lt;/math&gt; und &lt;math&gt;v&lt;/math&gt; allein. Wenn man diesen Größen ihre früheren Werte wiedergibt, indem man die beiden Uhren wieder zur Koinzidenz bringt, während sie relativ zueinander ruhen, so geht der Unterschied zwischen &lt;math&gt;t'&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt; wieder auf null zurück, gleichviel, welche Werte &lt;math&gt;z&lt;/math&gt; und &lt;math&gt;v&lt;/math&gt; während der Zwischenzeit gehabt haben mögen. Wenn an irgendeinem Punkte der Bahn die Geschwindigkeit von B relativ zu A eine endliche plötzliche Änderung erfährt, so erfährt auch der Wert von &lt;math&gt;v&lt;/math&gt; eine endliche plötzliche Änderung.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |This conclusion is not correct. The relationship between &lt;math&gt;t&lt;/math&gt; as the reading on the clock on A by the observer on A, and &lt;math&gt;t'&lt;/math&gt; as the reading on the clock on B by the observer on A, is given by (assuming that &lt;math&gt;t=t'&lt;/math&gt; at the beginning of the experiment)
:&lt;math&gt;t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)&lt;/math&gt;. 
The difference between &lt;math&gt;t'&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt; is a function of &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;v&lt;/math&gt; alone. If these quantities are given their previous values by bringing the two clocks back to coincidence during which they are at rest relative to one another, the difference between &lt;math&gt;t'&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt; goes back to zero, no matter what values &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;v&lt;/math&gt; may have had in the meantime. If at any point on the path the speed of B experiences a finite sudden change relative to A, then the value of &lt;math&gt;t'&lt;/math&gt; also undergoes a finite sudden change.

|-

|colspan=2|{{Lorentzbox|Text=So Campbell claims that any time difference during the outbound path is wiped out during the inbound path. His mistake is obvious: Campbell is confusing coordinate differences stemming from the Lorentz transformation of ''events'' (which indeed depend on position and direction) with differences in ''clock aging'' derived from the proper time integral (which is ''accumulative'' and independent of position and direction.)}}

|-

! width=50% | French original by [[w:Paul Gruner|Gruner]] (March 1912):&lt;ref name=gruner /&gt;

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |[...] deux personnes du même âge, se séparant dans des systèmes de « marche » très différents et retournant après un laps de temps assez long, constateront une différence d'âge très sensible. [...] le principe de relativité exige toujours la ''réciprocité parfaite'' des phénomènes entre deux systèmes qui possèdent un mouvement relatif. Si, dans l'exemple cité, les deux personnes du même âge se séparent avec une vitesse relative pour se retrouver plus tard, la constatation d'une différence d'âge sera parfaitement mutuelle : A dira positivement que B est resté en arrière dans son développement, et B affirmera avec le même droit que c'est A qui ne s'est pas développé assez vite. Ainsi le principe absolu de la relativité montre ses conséquences les plus extrèmes et il est clair que l'introduction de l’éther n'est plus en état de résoudre cette contradiction irréductible et inconcevable.

| style=&quot;padding: 0px 20px 0px 20px;&quot; | [...] two people of same age, separating into very different systems of motion and returning after a quite long period of time, will notice a very significant age difference. [...] the principle of relativity always requires the ''perfect reciprocity'' of the phenomenons between two systems that possess relative motion. When, in the cited example, the two persons of same age are separated by some relative velocity only to meet again later, the finding of an age difference will be perfectly mutual: A will positively say that B stayed behind in its development, and B will assert with same right that it was A who has not developed fast enough. By that, the absolute relativity principle shows its most extreme consequences and it is clear, that the introduction of the aether is no longer able to resolve this irreducible and inconceivable contradiction.

|-

|colspan=2|{{Lorentzbox|Text=Gruner was probably the first to claim that combining the round-trip experiment with the symmetry of time dilation leads to the contradictory situation, that both must attribute younger age to one another at reunion. At the end of his paper, we also find the expression &quot;clock paradox&quot; (French: paradoxe des horloges). See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem.}}

|}

==Einstein's contributions==

1905:&lt;ref name=einstein05 /&gt; Introduction of the &quot;peculiar&quot; (German: eigentümlich) round-trip experiment with clocks, describing a polygonal path, an continuously curved path, and an experiment comparing a clock at the pole with one at the equator.

January 1911 (published November):&lt;ref name=einstein11a /&gt; In a lecture from January, Einstein extended the &quot;funny&quot; (German: drollig) round-trip experiment to living organisms. According to [[w:Rudolf Lämmel]] in April 1911&lt;ref name=lammel /&gt; and [[w:Fritz Müller-Partenkirchen|Fritz Müller]] in October 1911,&lt;ref name=muller /&gt; Einstein spoke of human beings as well.

January 1911 (published January 1912):&lt;ref name=einstein3 /&gt; During a discussion with Einstein directly after the previous lecture, [[w:Fritz Müller-Partenkirchen|Fritz Müller]] claimed that any time difference during the round-trip should vanish at reunion, in analogy to the fact that the Lorentz contraction of a moving rod vanishes when it is at rest again. Einstein showed that the analogy is incorrect: While the clock rates are indeed the same again when they are mutually at rest, the clocks do not indicate the same time at reunion because &quot;clocks are carriers of the time differential&quot;; he went on to show that any possible influence of acceleration during the turnaround can be made negligible by elongating the constant velocity periods. 

1912:&lt;ref name=einst12manu /&gt; In an unpublished manuscript on special relativity, Einstein showed that if system &lt;math&gt;\Sigma'&lt;/math&gt; makes a round-trip along a polygon, then its inner processes will be retarded with respect to resting system &lt;math&gt;\Sigma &lt;/math&gt; at reunion. He pointed out that any influence of acceleration can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygon. 

April 1914:&lt;ref name=einstpetz /&gt; [[w:Joseph Petzoldt]] criticized asymmetric clock aging in the round-trip experiment as a &quot;fallback into absolutist way of thinking&quot;, claiming that special relativity requires that any difference between the clocks vanishes when their relative velocity is zero again, even though he added that any treatment of the clock paradox in special relativity is unrealistic anyway, since the theory only concerns uniform motions and therefore cannot handle velocity changes, so one has to modify the theory. Einstein responded by letter in which he praised Petzoldt's philosophical take on relativity, yet he rejected Petzoldt's conclusions concerning the clock paradox by showing that any finite acceleration at turnaround during the round-trip can only influence the clock in a finite way and therefore can be neglected by minimizing the time of acceleration with respect to the time of uniform translation, so it &quot;must be concluded&quot; that the clock traveling on a polygonal path is retarded at reunion.

May 1914:&lt;ref name=rowe group=S /&gt; During a conversation with [[w:Ernst Gehrcke]] who claimed that the clock paradox contradicts the relativity principle, Einstein replied that clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are irrelevant for the amount of the time difference, their presence nevertheless cause B to fall behind (&quot;accelerated motions are absolute in the theory of relativity&quot;).

1916:&lt;ref name=einstein16 /&gt; In a lecture of which only an abstract was published, Einstein spoke about the &quot;clock paradox of special relativity from the standpoint of general relativity.&quot;

September 1918:&lt;ref name=einadl /&gt; In a letter to Einstein, [[w:Friedrich Adler (politician)|Friedrich Adler]] (while in prison for the [[w:Assassination of Karl von Stürgkh]]) claimed that the clock paradox which he described on a circular round-trip contradicts the special relativity principle, and also referred to the similar opinions of [[w:#a|Berg and Petzoldt]]. Einstein responded by letter and explained that there is no contradiction as one of them accelerates; he then showed that general relativity makes both inertial frame K and accelerated frame K' equally justified, explaining the time difference in K' by combining the influence of velocity and gravitational potential, concluding that &quot;Berg and Petzoldt were wrong&quot;.

November 1918:&lt;ref name=einstein18 /&gt; In a fictitious dialogue between a relativity critic and a relativity apologist written by Einstein, the &quot;critic&quot; said that special relativity must predict differential clock aging in round-trip experiments, which was confirmed by the &quot;relativist&quot; who regretfully noted that even some pro-relativity authors tried to &quot;circumvent this unavoidable result&quot;. Yet the critic claimed that this leads to a contradiction: From the viewpoint of K, clock U1 is at rest while the clock U2 was in motion and therefore returns being retarded with respect to U1, but from the viewpoint of K', clock U2 is at rest while clock U1 was in motion and therefore returns being retarded with respect to U2, which was rebutted by the relativist by pointing out the acceleration of U2. Then the critic claimed that this problem &quot;rises again from the dead&quot; in general relativity which allows to symmetrically treat both K and K', which was rebutted by the relativist using the equivalence principle: In K', the rate increase of U1 during turnaround period 3) is &quot;the double&quot; of its velocity time dilation in the inertial periods 2) and 4). 

December 1918:&lt;ref name=einstein18b /&gt; In a letter to Einstein, [[w:Max Jakob]] doubted the result from Einstein's dialogue, according to which the advance of U1 in period 3) is the double of its retardation during periods 2) and 4). Einstein responded by letter, in which he used the gravitational time dilation factor &lt;math&gt;1+\Phi/c^{2}&lt;/math&gt; in K' in order to show that U1 at distance &lt;math&gt;l&lt;/math&gt;  is advancing by &lt;math&gt;\Phi/c^{2}=2vl/c^{2}&lt;/math&gt; in period 3), which is indeed the double of approximated delay &lt;math&gt;vl/c^{2}&lt;/math&gt; caused by velocity time dilation during periods 2) and 4).

1920:&lt;ref name=einstein20 /&gt; In conversations with [[w:Alexander Moszkowski]] between 1919 and 1920, Einstein argued that differential aging of the twins is rather a paradox of feeling, not a paradox of thought, because the latter would only arise if there were no reason for the asymmetric aging. The reason in special relativity lies in the fact that one of them suffered accelerations, while a deeper understanding of that question is obtained by using general relativity. Einstein argued that our &quot;common sense&quot; is located in the realm of feeling and analogy drawn from our ordinary experience; since there is no analogy to the example of the twins in our experience, it might appear paradoxical to the common sense, while it appears logical and necessary in light of intensified abstraction of the trained scientific mind.

August 1920:&lt;ref name=rowe group=S /&gt; At an anti-relativity event organized by the right-wing agitator [[w:Paul Weyland]] during which [[w:antisemitic]] leaflets were distributed and [[w:swastika]]s offered at the entrance, a lecture was given by Gehrcke re-iterating his criticism of the twin paradox, claiming that the stationary first organism is old or even dead at reunion while the second organism was in motion and therefore stayed young, but from the standpoint of the second organism he himself is old or even dead while the first organism was in motion and stayed young, thus relativity is either contradictory or it leads to different realities and physical [[w:solipsism]]. Einstein who was present at that event, directly responded in a newspaper article;&lt;ref name=einst20 /&gt; after suspecting antisemitic motives of his critics, he specifically addressed Gehrcke's objections regarding the &quot;well known example of the clocks (or twins)&quot;, remarking that the charge of solipsism will be &quot;greeted by the experts as a joke&quot;, and characterized the claim that relativity requires mutual retardation of two co-located clocks as a &quot;deliberate attempt to misinform the lay public&quot;.

1922:&lt;ref name=morand /&gt;&lt;ref name=nord /&gt; [[w:Paul Painlevé]], Einstein and Langevin discussed the clock paradox at a meeting in Paris. Painlevé imagined a clock on a train that performs a round-trip with constant speed and returns being retarded with respect to the station clock, yet he claimed that the relativity principle also allows to say that the station clock performed the round-trip and returned being retarded with respect to the train clock, in contradiction to the previous result. Einstein replied that the relativity principle cannot be applied since the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two systems having velocities in opposite direction is not an inertial frame; there is no reciprocity between a frame that changes direction and one that doesn't. Langevin consequently gave a detailed analysis in terms of the Lorentz transformation.

1954:&lt;ref name=einstein54 /&gt; In a letter, Einstein explained the &quot;well known clock paradoxon&quot; using two clocks &lt;math&gt;B_{1}&lt;/math&gt; and &lt;math&gt;B_{2}&lt;/math&gt;; clock &lt;math&gt;B_{2}&lt;/math&gt; has to reverse its speed in order to come back, thus it was initially at rest in inertial frame &lt;math&gt;S_{2}&lt;/math&gt; and then at rest in &lt;math&gt;S_{2}^{+}&lt;/math&gt;, whereas &lt;math&gt;B_{1}&lt;/math&gt; constantly remains at rest in &lt;math&gt;S_{1}&lt;/math&gt;, which explains the asymmetry between them.

==Historical references==

&lt;references&gt;

&lt;ref name=einstein05&gt;See p. 904f in: {{Citation |author=Einstein, A. |date=1905 |title=Zur Elektrodynamik bewegter Körper|journal=Annalen der Physik |volume=322 |issue=10 |pages=891–921 |doi=10.1002/andp.19053221004|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 2, Document 23}}. See also: [https://www.fourmilab.ch/etexts/einstein/specrel/www/ English translation at fourmilab].&lt;/ref&gt;

&lt;ref name=einstein11a&gt;See p. 10. in: {{Citation |author=Einstein, A. |title=Die Relativitäts-Theorie|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |issue=1-2|pages=1–14 |date=27 November 1911|orig-date=Lecture 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n11/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 17}}.&lt;br /&gt; The publication date 27 November 1911 can be seen on the [https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n5/mode/2up Title page and TOC of issue 1-2].&lt;/ref&gt;

&lt;ref name=einstein3&gt;Discussion between Einstien, Müller, Lämmel and others after the Zürich lecture: {{Citation |author=Einstein, A.; Müller, F., Lämmel, R.|title=Diskussion zu &quot;Die Relativitäts-Theorie&quot;|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |pages=II-IX |date=January 1912|orig-date=Lecture on 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n587/mode/2up|quote=Reprinted in  ''The Collected Papers of Albert Einstein'', Vol. 3, Document 18, and in the corresponding English translation volume}}&lt;br /&gt; While the discussion already happened on January 1911, the publication followed one year later in January 1912 in the session proceedings (Sitzungsberichte) of the third issue, see [https://www.ngzh.ch/publikationen/vjs/56/3 Full issue Nr. 3] with [http://www.ngzh.ch/archiv/1911_56/56_1-2/56_3.pdf Title page and TOC] and the [http://www.ngzh.ch/archiv/1911_56/56_3/56_30.pdf Sitzungsberichte including Einstein's discussion on pp. II-IX]. &lt;/ref&gt;

&lt;ref name=einst12manu&gt;See p. 46 in: {{Citation |author=Einstein, A. |date=1912 |chapter=Document 1: Einstein's manuscript on the special theory of relativity|title=The collected papers of Albert Einstein|volume=4|pages=3-108|trans-chapter=See also the English translation in the corresponding translation volume}}&lt;/ref&gt;

&lt;ref name=einstlor&gt;{{Citation|author=Einstein, A.|date=1914|title=Review of &quot;Lorentz, H. A. – Das Relativitätsprinzip&quot; |journal=Die Naturwissenschaften|volume=2|pages=1018|url=https://archive.org/details/CAT31421305002/page/1018/mode/2up|quote=Reprinted in  ''The Collected Papers of Albert Einstein'', Vol. 6, Document 11}}&lt;/ref&gt;

&lt;ref name=einstpetz&gt;{{Citation |author=Einstein, A. |date=1914 |chapter=Document 5: Letter from Einstein to Petzoldt|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}&lt;/ref&gt;

&lt;ref name=einstein16&gt;See p. 423f in: {{Citation |author=Einstein, A. |date=1916 |title=Announcement of Einstein's lecture &quot;Über einige anschauliche Überlegungen aus dem Gebiete der Relativitätstheorie&quot;|journal=Berliner Sitzungsberichte|pages=423|volume=1916 (part 1)|url=https://archive.org/details/sitzungsberichte1916deutsch/page/423/mode/2up}}&lt;/ref&gt;

&lt;ref name=einadl&gt;Letter exchange between Einstein and Adler in which the critique on the clock paradox by Berg (1910) and Petzoldt (1914) was mentioned, together with the general relativity solution in terms of the gravitational potential, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Adler's letter in Document 620 and Einstein's reply in Document 628|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}&lt;/ref&gt;

&lt;ref name=einstein18&gt;Einstein discussed in terms of inertial frames (special relativity) on pp. 697f; accelerated frames (general relativity) on pp. 698f.; distant masses (Mach's principle) on pp. 700f. in: {{citation |author=Einstein, A.|title=Dialog über Einwände gegen die Relativitätstheorie|date=November 1918|volume=6|issue=48|journal=Die Naturwissenschaften|pages=697-702|url=https://archive.org/details/sim_naturwissenschaften_1918-11-29_6_48|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 7, Document 13}}; See also English translation [[:s:Translation:Dialog about Objections against the Theory of Relativity|Dialog about Objections against the Theory of Relativity]] on Wikisource.&lt;/ref&gt;

&lt;ref name=einstein18b&gt;Letter exchange between Max Jakob and Einstein from December 1918, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Jakob's letter in Document 661c and Einstein's reply in Document 663a|title=The collected papers of Albert Einstein|volume=10|pages=189-190}}&lt;/ref&gt;

&lt;ref name=einstein20&gt;Interview of Einstein by Moszkowski, see p. 204f. in: {{citation |author=Moszkowski, A.|title=Einstein. Einblicke in seine Gedankenwelt|orig-date=Copyright date 1920 |date=1921|place=Hamburg|url=https://www.archive.org/details/einsteineinblick00moszuoft}}; See also English translation by H. L. Brose (1921): [https://archive.org/details/einsteinsearch00moszrich Einstein, the searcher], p. 206&lt;/ref&gt;

&lt;ref name=einst20&gt;{{Citation|author=Einstein, A.|date=27 August 1920|journal=Berliner Tageblatt|title=Meine Antwort - Ueber die anti-relativitätstheoretische G. m. b. H.|issue=402|pages=1-2|url=https://www.deutsche-digitale-bibliothek.de/newspaper/item/YH65KFT53MOG4SMXDXK4IVUPTR3QRY7Q?issuepage=1|quote=Reprinted in &quot;The Collected Papers of Albert Einstein&quot;, Vol. 7, Document 45}}&lt;/ref&gt;

&lt;ref name=einstein54&gt;Letter from Einstein to N. V. Pope from March 1954; [[w:Albert Einstein Archives]], Object number 27-88 ([https://ein-web.adlibhosting.com/aea/Details/archive/110021626 Online dataset]); Scanned version as [https://groups.google.com/group/npachat/attach/d3121322d37764f2/Einstein%20letter.doc?part=0.1 Word document on Usenet] published by [https://groups.google.com/g/npachat/c/Haoib97d6OA/m/8mR30yITEtMJ Pope himself])&lt;/ref&gt;

&lt;ref name=morand&gt;Discussion between Painlevé, Einstein, and Langevin on p. 316ff in: {{citation |author=Morand, M.|title=Einstein au collège de france|date=April 1922|journal=La Nature|volume=50|issue=2511|pages=315-320|url=http://cnum.cnam.fr/CGI/fpage.cgi?4KY28.102/319/100/620/5/613}}&lt;/ref&gt;

&lt;ref name=lammel&gt;{{Citation|author=Lämmel, R.|date=28 April 1911|title=Die Relativitäts-Lehre|journal=Neue Zürcher Zeitung|volume=117|pages=1|url=https://www.e-newspaperarchives.ch/?a=d&amp;d=NZZ19110428-01.2.4.1}}; English translation of the part concering the twin pardox at [[:v:History of Topics in Special Relativity/Twin paradox#Lämmel 1911-Hum|Wikiversity:Early history of the twin paradox - Lämmel]]&lt;/ref&gt;

&lt;ref name=lammel2&gt;See p. 84ff in: {{Citation|author=Lämmel, R.|date=1921|orig-date=Preface December 1920|title=Die Grundlagen der Relativitätstheorie|place=Berlin|publisher=Springer|url=https://archive.org/details/diegrundlagende00lmgoog}}&lt;/ref&gt;

&lt;ref name=langevin1&gt;He derived differential aging from the proper time integral; pointed out that this demonstrates the &quot;absolute nature of acceleration&quot; with respect to an aether, see: {{citation |author=Langevin, P.|title=[[:s:fr:L’Évolution de l’espace et du temps|L’Évolution de l’espace et du temps]]|journal=Scientia |volume=X |pages=31–54 |date=July 1911|orig-date=Lecture 10 April 1911}}; English translation [[:s:en:Translation:The Evolution of Space and Time|The Evolution of Space and Time]] on Wikisource&lt;/ref&gt;

&lt;ref name=langevin2&gt;See p. 329 in: {{citation |author=Langevin, P. |title=Le temps, l'espace et la causalité dans la physique moderne |journal=Bulletin de la Société française de philosophie |volume=12 |orig-date=Lecture October 1911|date=1912|pages=1-28|url=http://ahp.li/1f7fc22d283fdf0deeca.pdf}}&lt;/ref&gt;

&lt;ref name=wiechert11&gt;See p. 745f. general description and proper time; 757f. space travel; in: {{Citation |author=Wiechert, E. |date=September 1911|orig-date=Lectures March-May 1911, submitted 26 July|title=[[:s:de:Relativitätsprinzip und Äther|Relativitätsprinzip und Äther]]|journal=Physikalische Zeitschrift |volume=12 |issue=17-18 |pages=[https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/741 689-707] published September 1; [https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/789 737–758] published September 15}}&lt;/ref&gt;

&lt;ref name=wiechert15&gt;See p. 46 (Einstein, Langevin, Wiechert) and pp. 51f (Laue versus Wiechert) in: {{citation |author=Wiechert, E.|contribution=Die Mechanik im Rahmen der allgemeinen Physik| title=Die Kultur der Gegenwart: Physik|volume=3.3.1|date=1915 |orig-date=Submitted July 1914|pages=1–78|contribution-url=https://www.archive.org/details/physikunterredak00warbuoft}}&lt;/ref&gt;

&lt;ref name=wiechert20&gt;See p. 46f in: {{citation |author=Wiechert, E.|title=Der Äther im Weltbild der Physik|orig-date=Presented December 1920|date=1921|journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse|pages=29-70|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00250586X}}&lt;/ref&gt;

&lt;ref name=wiechert21&gt;See p. 25ff in: {{citation |author=Wiechert, E.|title=[[:s:de:Prinzipielles über Äther und Relativität|Prinzipielles über Äther und Relativität]]|date=1922|orig-date=Lecture September 1921|journal=Physikalische Zeitschrift|volume=23|pages=25-28}}&lt;/ref&gt;

&lt;ref name=muller&gt;See p. 9 in: {{Citation|author=Müller, F.|date=October 1911|journal=Berliner Tageblatt|title=[[:s:de:Das Zeitproblem (1911)|Das Zeitproblem]]|pages=[https://www.deutsche-digitale-bibliothek.de/newspaper/item/2QKOIOLGNVQILTCEZQOGQPLTRVLPM5PZ?query=zeit&amp;issuepage=9 Part 1 published 16 October 1911] and [https://www.deutsche-digitale-bibliothek.de/newspaper/item/IO44I6QBC4SVV5YUKUDSGXYIPQUXXBN5?query=zeit&amp;issuepage=11 Part 2 published 23 October 1911]}}&lt;/ref&gt;

&lt;ref name=gruner&gt;See p. 253f in: {{Citation |author=Gruner, P. |title=[[:s:fr:Rapport sur la dernière discussion concernant le principe de la relativité et l’éther|Rapport sur la dernière discussion concernant le principe de la relativité et l’éther]] |journal=Archives des sciences physiques et naturelles |volume=33|issue=4 |pages=252-254 |date=March 1912}}&lt;/ref&gt;

&lt;ref name=laue3&gt;See p. 113f in: {{citation |author=Laue, M. v. |title=Das Relativitätsprinzip |journal=Jahrbücher der Philosophie |volume=1 |date=1913 |pages=99–128}}; {{icon|wikisource}} See also English translation of [[:s:Translation:The Principle of Relativity (Laue, Philosophy)|The Principle of Relativity]] on Wikisource&lt;/ref&gt;

&lt;ref name=weyl&gt;See p. 147f. in: {{Citation |author=Weyl, H. |date=March 1918|title=Raum-Zeit-Materie (first edition)|publisher=Berlin: Springer|url=https://archive.org/details/RaumZeitMaterieVolIMeinerFrauGewidmet}}; English translation of the 4th edition by H. Brose (1921): [https://www.gutenberg.org/ebooks/43006 Space—Time—Matter], pp. 278f.&lt;/ref&gt;

&lt;ref name=gbaum&gt;See footnote on p. 507 in: {{Citation|author=Grünbaum, F. |title=Über einige ideelle Versuche zum Relativitätsprinzip|journal=Physikalische Zeitschrift|volume=12|pages=500–509|date=1911|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/540}}&lt;/ref&gt;

&lt;ref name=laue1&gt;Laue introduces the word &quot;paradox&quot;, alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource&lt;/ref&gt;

&lt;ref name=laue2&gt;See p. 42f. for general description; p. 58f. in terms of proper time; in: {{Citation |author=Laue, M. v. |orig-date=Preface December 1912|date=1913 |title=Das Relativitätsprinzip (Second Edition) |publisher=Vieweg |place=Braunschweig|url=https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE4597082}}; See also English translation [[:s:Translation:The Principle of Relativity (Laue 1913)|The Principle of Relativity, Second edition, Part III]] on Wikisource&lt;/ref&gt;

&lt;ref name=laue3&gt;See p. 113f in: {{citation |author=Laue, M. v. |title=Das Relativitätsprinzip |journal=Jahrbücher der Philosophie |volume=1 |date=1913 |pages=99–128}}; {{icon|wikisource}} See also English translation of [[:s:Translation:The Principle of Relativity (Laue, Philosophy)|The Principle of Relativity]] on Wikisource&lt;/ref&gt;

&lt;ref name=berg&gt;See p. 369f in: {{Citation |author=Berg, O. |date=1910 |title=Das Relativitätsprinzip der Elektrodynamik |journal=Abhandlungen der Fries'schen Schule |volume=3 |issue=2|pages=333-382 |url=http://hdl.handle.net/2027/hvd.hnuynk?urlappend=%3Bseq=351}}&lt;/ref&gt;

&lt;ref name=camp&gt;See p. 123f in: {{Citation |author=Campbell, N. |title=Relativitätsprinzip und Äther: Eine Entgegnung an Herrn Wiechert |journal=Physikalische Zeitschrift |volume=13 |pages=120-128 |issue=3|orig-date=Submitted December 1911|date=February 1912|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/150}}. The is based on an English manuscript translated by Max Iklé, and Campbell's first name was Germanised as &quot;Normann&quot;.&lt;/ref&gt;

&lt;ref name=seel&gt;{{Citation|author=Seeliger, R.|title=Review of &quot;P. Gruner – Rapport sur la dernière discussion concernant le principe de la relativité et l'éther&quot;|journal=Die Fortschritte der Physik|volume=68|issue=2|pages=336|date=1913|url=https://books.google.com/books?id=fSJGAQAAMAAJ&amp;pg=PA336}}&lt;/ref&gt;

&lt;ref name=study&gt;See footnote on p. 111 in: {{citation |author=Study, E. |title=Vorlesungen über ausgewählte Gegenstände der Geometrie |date=June 1911|url=https://archive.org/details/vorlesungenber00studuoft|publisher=B.G. Teubner|place=Leipzig}} &lt;/ref&gt;

&lt;ref name=robb1&gt;See pp. 356ff. in: {{Citation|author=Robb, A.|date=1914|title=A theory of time and space|place=Cambridge|publisher=University Press|url=https://archive.org/details/theoryoftimespac00robbrich}} &lt;/ref&gt;

&lt;ref name=robb2&gt;See §12 in: {{citation |author=Robb, A. A.|title=The Straight Path|date=1920 |journal=Nature|pages=599|volume=104|issue=2623|url=https://archive.org/details/sim_nature-uk_1920-02-05_104_2623/page/598/mode/2up}}&lt;/ref&gt;

&lt;ref name=edding2&gt;See p. 22 in: {{Citation |author=Eddington, A. S. |date=1922 |title=The theory of relativity, and its influence on scientific thought |publisher=Oxford Clarendon Press |url=https://archive.org/details/cu31924005748573}}&lt;/ref&gt;

&lt;ref name=rogers&gt;{{citation |author=Rogers, R. A. P.|title=The Time-Triangle and Time-Triad in Special Relativity|date=November 1922|journal=Nature|volume=110|issue=2769|pages=698–699|url=https://archive.org/details/sim_nature-uk_1922-11-25_110_2769/page/698/mode/2up}}&lt;/ref&gt;

&lt;ref name=lorentz1&gt;See pp. 37f, 55ff in: {{citation |author=Lorentz, H. A.|date=1913|title=Het relativiteitsbeginsel : drie voordrachten gehouden in Teyler's stichting|publisher=De Erven Loosjes |place=Haarlem|url=https://resolver.kb.nl/resolve?urn=MMKB24:063387000:00005}}; German translation on pp. 31f, 47f in: {{citation |author=Lorentz, H. A.|date=1914| title=Das Relativitätsprinzip. Drei Vorlesungen gehalten in Teylers Stiftung zu Haarlem|publisher=B.G. Teubner |place=Leipzig and Berlin|url=https://archive.org/details/bub_gb_89PPAAAAMAAJ}}; See also the transcription [[:s:de:Das Relativitätsprinzip (Lorentz)|Das Relativitätsprinzip]] on German Wikisource and the English translation [[:s:Translation:The Principle of Relativity (Lorentz)|The Principle of Relativity]] on English Wikisource&lt;/ref&gt;

&lt;ref name=lorentz3&gt;See §12 in: {{citation |author=Lorentz, H. A.|title=Considérations élémentaires sur le principe de relativité|date=1914 |journal=Revue générale des sciences pures et appliquées|pages=179-186|url=https://archive.org/details/revuegnraled25pari/page/178/mode/2up}}&lt;/ref&gt;

&lt;ref name=bloch&gt;See pp. 67 ff. in: {{Citation | author=Bloch, W.| date=September 1918|title=Einführung in die Relativitätstheorie| publisher=B. G. Teubner |url=https://hdl.handle.net/2027/njp.32101040276907}}&lt;/ref&gt;

&lt;ref name=bloch2&gt;See pp. 69ff. (special relativity) and 102ff. (general relativity) in: {{Citation | author=Bloch, W.| date=1920 |title=Einführung in die Relativitätstheorie (second edition)| publisher=B. G. Teubner |url=https://www.archive.org/details/einfhrungindier00blocgoog}}&lt;/ref&gt;

&lt;ref name=bollert1&gt;See p. 6 (special relativity), pp. 24-26 (EP) in: {{citation |author=Bollert, K.|title=Einstein’s Relativitätstheorie und ihre Stellung im System der Gesamterfahrung |date=April 1921|publisher=Steinkopff|url=https://archive.org/details/dbc.wroc.pl.001504}}&lt;/ref&gt;

&lt;ref name=born&gt;See pp. 190f. (special relativity), 250f (EP) in: {{Citation | author=Born, M.| date=1921 |title=Die Relativitätstheorie Einsteins und ihre physikalischen Grundlagen (Second edition)| publisher=Springer | place=Berlin|url=https://hdl.handle.net/2027/mdp.39015017387310}}; The [https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE5426498 first edition (1920)] of Born's book didn't include the twin paradox. English translation of the third edition by H. Brose (1924): [https://archive.org/details/einsteinstheoryo00born Einstein's theory of relativity]&lt;/ref&gt;

&lt;ref name=pauli&gt;See p. 558f (general description); p. 624f (proper time); p. 713f (accelerated frames); in: {{Citation |author=Pauli, W. |date=1921 |journal=Encyclopädie der Mathematischen Wissenschaften|title=Die Relativitätstheorie|pages=539–776|volume=5|issue=2 |url=http://resolver.sub.uni-goettingen.de/purl?PPN360709672}}; English translation by G. Field (1958): [https://books.google.com/books?id=rc3DAgAAQBAJ Theory of Relativity]&lt;/ref&gt;

&lt;ref name=thirring&gt;See p. 209ff in: {{citation |author=Thirring, H.|title=Über das Uhrenparadoxon in der Relativitätstheorie|date=April 1921|journal=Naturwissenschaften|volume=9|issue=18|pages=209-212|url=https://archive.org/details/sim_naturwissenschaften_1921-04-01_9_13/mode/2up}}&lt;/ref&gt;

&lt;ref name=sommerfeld&gt;See p. 71 in: {{citation |author=Sommerfeld, A. |date=May 1913|chapter=Remarks on Minkowski's &quot;Space and Time&quot;|title=Das Relativitätsprinzip|editor=Otto Blumenthal|pages=69-73|url=https://www.archive.org/details/dasrelativittsp00minkgoog}}&lt;/ref&gt;

&lt;ref name=kopff&gt;See pp. 45ff (special relativity and proper time); pp. 117ff (EP); pp. 189ff (Mach's principle), in: {{citation |author=Kopff, A.|title=Grundzüge der Einsteinschen Relativitätstheorie |date=February 1921|publisher=S. Hirzel|place=Leipzig|url=https://www.archive.org/details/grundzgedereins00kopfgoog}}; English translation by H. Levy (1923): [https://hdl.handle.net/2027/mdp.39015017188817 The mathematical theory of relativity].&lt;/ref&gt;

&lt;ref name=becqu1&gt;See p. 48ff (proper time), p. 240f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Le Principe de relativité et la théorie de la gravitation|Le Principe de relativité et la théorie de la gravitation]] |date=1922 |publisher=Gauthier-Villars|place=Paris}}; See also p. 57ff (proper time), p. 177f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Exposé élémentaire de la théorie d’Einstein et de sa généralisation|Exposé élémentaire de la théorie d’Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}&lt;/ref&gt;

&lt;ref name=nord&gt;Discussion between Painlevé, Einstein, and Langevin on pp. 146ff in: {{citation |author=Nordmann, C.|title=[[s:fr:Einstein expose et discute sa théorie|Einstein expose et discute sa théorie]]|date=May 1922|journal=Revue des deux mondes|volume=IX|pages=129-166}}&lt;/ref&gt;

&lt;/references&gt;

==Secondary sources==

&lt;references group=S&gt;

&lt;ref name=miller&gt;{{Citation |author=Miller, A. I. |date=1981 |title=Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911) |place=Reading |publisher=Addison–Wesley |isbn=978-0-201-04679-3}}; See section 7.4.13 (Langevin, Wiechert, Laue, Einstein), footnotes 29-34 of chapter 7 (Petzoldt, Sommerfeld, Bergson, Einstein)&lt;/ref&gt;

&lt;ref name=lange&gt;{{Citation|author=Lange, L.|date=1927|title=The clock paradox of the theory of relativity|journal=The American Mathematical Monthly|volume=34|issue=1|pages=22-30|jstor=2299914}}&lt;/ref&gt;

&lt;ref name=pes&gt;{{Citation |author=Pesic, P. |date=2003 |title=Einstein and the twin paradox |journal=European Journal of Physics |volume=24 |issue=6 |pages=585–590 |doi=10.1088/0143-0807/24/6/004}}&lt;/ref&gt;

&lt;ref name=during&gt;{{Citation |author=During, É. |date=2014 |title=Langevin ou le paradoxe introuvable |journal=Revue de métaphysique et de morale |volume=84 |pages=513-527 |doi=10.3917/rmm.144.0513|doi-access=free}}; See pp. 515f (Langevin), 520f. (Einstein, Laue, Weyl, Painlevé).&lt;/ref&gt;

&lt;ref name=debs&gt;{{Citation |author=Debs, T. A., &amp; Redhead, M. L. |title=The twin paradox and the conventionality of simultaneity |date=1996 |journal=American Journal of Physics |volume=64|issue=1| pages=384-392 |doi=10.1119/1.18252}}&lt;/ref&gt;

&lt;ref name=alizzi&gt;{{Citation |author=Alizzi, A., Sen, A., &amp; Silagadze, Z. K.|title=Do moving clocks slow down? |year=2022 |journal=European Journal of Physics |volume=43|issue=6|pages=065601 |doi=10.1088/1361-6404/ac93ca|arxiv=2209.12654}}; Appendix B with reference to Lange and Halsbury&lt;/ref&gt;

&lt;ref name=beng&gt;{{Citation |author=Benguigui, L. G. |date=2020 |title=A Tale Of Two Twins: The Langevin Experiment Of A Traveler To A Star |publisher=World Scientific|isbn=9789811219115}}; See early solutions (Einstein, Langevin, Lorentz, Born/Kopff) and the Bergson controversy. A shorter version appeared in {{arxiv|1212.4414}}.&lt;/ref&gt;

&lt;ref name=rowe&gt;{{Citation|author=Rowe, D. E.|date=2006|title=Einstein's allies and enemies: Debating relativity in Germany 1916–1920|journal=Interactions: Mathematics, Physics and Philosophy|pages=231-280|publisher=Springer|doi=10.1007/978-1-4020-5195-1_8}}; Covering the criticism of Gehrcke starting with 1912; discussion between Einstein and Gehrcke in 1914; Einstein's dialogue (1918) as response to antirelativists; the Weyland event in 1920 and Einstein's response.&lt;/ref&gt;

&lt;ref name=weiss&gt;Weiss, W. (Physics FAQ): [https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html The Twin Paradox: The Equivalence Principle Analysis]&lt;/ref&gt;

&lt;ref name=cuvaj&gt;{{Citation |author=Cuvaj, C. |date=1971 |title=Paul Langevin and the theory of relativity|journal=Japanese studies in the history of science|volume=10| pages=113-142|url=http://www.isc.meiji.ac.jp/~sano/hssj/pdf/Cuvaj_C-1972-Langevin_Relativity-JSHS-No_10-pp113-142.pdf}}&lt;/ref&gt;

&lt;ref name=koks&gt;Koks, D. (2018): [https://math.ucr.edu/home/baez/physics/Relativity/SR/sr-gr.html Physics FAQ: Where is the Boundary between Special and General Relativity?]&lt;/ref&gt;

&lt;/references&gt;

[[Category:History of special relativity]]

[[Category:Paradoxes]]</text>
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  <page>
    <title>Coordinates Last: Vector Analysis Done Fast</title>
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        <username>Gavin R Putland</username>
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      <comment>Afterthoughts: some clarifications [including some deletions!], and a correction [dS replaced by &amp;delta;S in two places after eq.(96)].</comment>
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      <text bytes="343975" sha1="qr72lo60jjui2ta7aw6qv2zz37og3o6" xml:space="preserve">{{Author|Gavin R Putland}}{{tertiary}}{{mathematics}}{{physics}}{{engineering}}{{testing}}

== Preface ==

This learning resource (which I call a &quot;paper&quot;, although it's a long one) is an attempt to reduce vector analysis from a second-year undergraduate subject to a ''first''-year undergraduate subject. Its strategy is to delay the use of coordinate systems until their use is required by upcoming topics&amp;mdash;and, behold, assisted by previous topics. It's about ''vector'' analysis as distinct from tensor analysis: it does not deal with dyadics or higher-order tensors, except by way of occasional hints; but, along its unusual path, it ''does'' treat some topics that one might not expect in a &quot;first&quot; course.

[[w:Sheldon Axler|Sheldon Axler]], in his essay &quot;Down with determinants!&quot; ([[#axler-95|1995]]) and his ensuing book ''Linear Algebra Done Right'' (4th Ed., [[#axler-23-|2023–]]), does not eliminate determinants, but introduces them as late as possible, and then exploits them for what he calls their &quot;main reasonable use in undergraduate mathematics&quot;, namely the change-of-variables formula for multiple integrals.&lt;ref&gt;[[#axler-95|Axler, 1995]], &amp;sect;9. The relegation of determinants was anticipated by C.G.&amp;#8239;Broyden ([[#broyden-75|1975]]). But Broyden's approach is less radical: he does not deal with abstract vector spaces or abstract linear transformations, and his eventual definition of the determinant, unlike Axler's, is traditional&amp;mdash;not a product of the preceding narrative.&lt;/ref&gt; Here I treat coordinates in vector analysis somewhat as Axler treats determinants in linear algebra: I introduce coordinate systems as late as possible, and then exploit them in unconventionally ''rigorous'' derivations of vector-analytic identities from (e.g.) vector-algebraic identities. But I contrast with Axler in at least two ways. First, I have no intention of expanding this &quot;paper&quot; into a book. Brevity is of the essence. Second, while one may well avoid determinants in ''numerical''&amp;#8202; linear algebra,&lt;ref&gt;[[#axler-95|Axler, 1995]], &amp;sect;1. But it is Broyden ([[#broyden-75|1975]]), not Axler, who discusses numerical methods at length.&lt;/ref&gt; one can hardly avoid coordinates in ''numerical'' vector analysis! So I cannot offer a coordinate-free path into computation. But I can prepare for computation by expressing the operators of vector analysis in general coordinates and orthogonal coordinates: indeed, readers who stay with me to the end will get a more general treatment of coordinates than is offered by a typical ''book''-length introduction to vector analysis. [''Continued&amp;#8239;&amp;hellip;'']

{{cot|&amp;hellip; Extended content (show or hide)}}

In the meantime, however, coordinates don't get in the way. Familiar coordinates may be mentioned in passing for purposes of illustration; but, until &quot;Cartesian coordinates&quot; are announced under their own heading, I work from ''conceptual'' definitions rather than coordinate-based definitions. This, I submit, keeps the exposition direct and accessible, and facilitates treating related concepts in parallel&amp;mdash;saving time and words, and highlighting similarities and differences.

Something else that doesn't get in the way is an exaggerated pretense of rigor. In the branch of pure mathematics known as ''analysis'', there is a thing called a ''limit'', whereby for every positive ''&amp;epsiv;''&amp;#8201; there exists a positive ''&amp;delta;'' such that if some increment is less than ''&amp;delta;'', some error is less than ''&amp;epsiv;''. In the branch of applied mathematics known as ''[[w:continuum mechanics|continuum mechanics]]'', there is a thing called reality, whereby if the increment is less than some positive ''&amp;delta;'', the assumption of a continuum becomes ridiculous, so that the error cannot be made less than an ''arbitrary &amp;epsiv;''. Yet vector &quot;analysis&quot; (or a superset thereof) is typically studied with the intention of applying it to some form of &quot;continuum&quot; mechanics&amp;mdash;such as the modeling of elasticity, plasticity, fluid flow, or (widening the net) electrodynamics of ordinary matter&amp;mdash;conveniently forgetting that, on a sufficiently small scale, matter is lumpy. (Even if we claim that &quot;particles&quot; of matter are wave functions and therefore continuous, these wave functions are still lumpy on a scale not normally contemplated by continuum mechanics.) One might therefore submit that to express the principles of vector analysis in the language of limits is to strain at a gnat and swallow a camel. Here I avoid that camel by referring to '''elements''' of length or area or volume, each of which is ''small'' enough to allow some quantity or quantities to be considered uniform within it, but, for the same reason, ''large'' enough to allow such local averaging of the said quantity or quantities as is necessary to tune out the lumpiness. We shall see bigger camels, where well-known authors define or misdefine a vector ''operator'' and then derive identities by treating it like an ordinary vector ''quantity''. These I also avoid.

A rough and ready premise is more rigorous than an absurd or meaningless one. Discarding the machinery of limits causes a small lapse in rigor where limits are applicable, but avoids a big lapse where they are not. Maintaining the distinction between operators and quantities cannot cause a loss of rigor, but avoids one wherever the alleged &quot;algebraic&quot; properties of operators get confusing. The resulting standard of rigor is economical but consistent.

This paper is a new arrangement of old knowledge. It does not pretend to offer any new mathematical results, and in that sense does not pretend to be [[original research]]. But, pursuant to its goals as a [[learning resource]], it ''does'' contain independent derivations and independent scholarship.

Much of that scholarship builds on the earlier scholarship of Professor Chen-To Tai, {{serif|FIEEE}}, who died in 2004, and who first came to my attention in 2018 through his invited paper &quot;On the presentation of Maxwell's theory&quot; [''Proc.&amp;#8239;{{serif|IEEE}}'', '''60'''(8):&amp;#8239;936–45, 1972]. In nearly every place where I mention him here, even if I do not accept his conclusion, I am entirely indebted to his works for drawing my attention to the issue raised. In particular, it was through Tai that I became aware of Gibbs's original definitions of the divergence and curl and their suitability for expression in indicial notation ([[#tai-95|Tai, 1995]], pp.&amp;#8239;17,&amp;#8239;21). And although he might not have been pleased, it was through Tai that I first knew with certainty that, if we allow for the variability of the basis vectors, the del-dot and del-cross notations are valid in general coordinates (''ibid.'', pp.&amp;#8239;64–5). Accordingly, this paper is dedicated to him.

{{right|&amp;mdash;&amp;#8201;[[w:User:Gavin R Putland|Gavin R.&amp;#8201;Putland]].}}

{{cob}}

== Overview (for instructors) ==

{{cot}}

The gradient, the curl, the divergence, and the Laplacian are initially defined, without coordinates, as closed-surface integrals per unit volume&amp;mdash;the definition of the Laplacian being indifferent to whether the operand is a scalar field or a vector field. Four integral theorems&amp;mdash;including the divergence theorem&amp;mdash;follow almost immediately, provided that the initial definitions are unambiguous. Their unambiguity, together with some examples of their usefulness, is established as follows, at a level suitable for beginners:
* The gradient is related to an acceleration through an equation of motion;
* The divergence is related to two time-derivatives of density (the partial derivative and the material derivative) through two forms of an equation of continuity;
* The component of the curl in a general direction is expressed as a divergence (now known to be unambiguous);
* The same is done for the general component of the gradient, yielding not only a second proof of unambiguity of the gradient, but also the relation between the gradient and the directional derivative; this together with the original definition of the Laplacian shows that the Laplacian of a ''scalar'' field is the divergence of the gradient and therefore unambiguous. The unambiguity of the Laplacian of a ''vector'' field then follows from a component argument (as for the curl) or a linearity argument.
The derivation of the relation between the gradient and the directional derivative yields a coordinate-free definition of the dot-del operator for a scalar right-hand operand. But, as the directional derivative is also defined for a non-scalar operand, the same relation offers a method of generalizing the dot-del operator, so that the definition of the Laplacian of a general field can be rewritten with that operator. The advection operator&amp;mdash;derived without coordinates, for both scalar and vector properties&amp;mdash;is likewise rewritten.

Meanwhile comparison between the definitions of the various operators leads to coordinate-free definitions of the del-cross, del-dot, and del-squared operators. These together with the dot-del operator allow the four integral theorems to be condensed into a single generalized volume-integral theorem.

If the volume of integration is reduced to a thin curved slab of uniform thickness, with an edge-face perpendicular to the broad faces, the four integral theorems are reduced to their two-dimensional forms, each of which relates an integral over a surface segment to an integral around its enclosing curve, provided that the original ''closed''-surface integral has no contribution from the broad faces of the slab. This proviso can be satisfied by construction in two of the four cases, yielding two general theorems, one of which is the Kelvin&amp;ndash;Stokes theorem. By applying these two theorems to a segment of a closed surface, and expanding the segment to cover the entire surface, it is shown that the gradient is irrotational and the curl is solenoidal.

The next part of the exposition is more conventional, but still coordinate-free. The gradient theorem is derived from the relation between the gradient and the directional derivative. An irrotational field is shown to have a scalar potential. The 1/''r''&amp;#8202; scalar field is shown to be the field whose negative gradient is the inverse-square vector field, whose divergence is a delta function, which is therefore also the negative Laplacian of the 1/''r''&amp;#8202; scalar field. These results enable the construction of a field with a given divergence or a given Laplacian. The wave equation is derived from small-amplitude sound waves in a non-viscous fluid, and shown to be satisfied by a spherical-wave field with a 1/''r''&amp;#8202; amplitude, whose D'Alembertian is a delta function, enabling the construction of a wave function with a given D'Alembertian. But further progress, including the construction of a field with a given ''curl'', seems to require the invocation of a coordinate system.

With the aid of identities already found, expressions are easily obtained for the gradient, curl, divergence, Laplacian, and advection operators in Cartesian coordinates&amp;mdash;with indicial notation and implicit summation, for brevity. While the resulting expressions for the curl and divergence may look unfamiliar, they match the initial definitions given by J.&amp;#8239;Willard Gibbs. The Cartesian expressions are found convenient for deriving further identities: a comprehensive collection (including a multivariate chain rule) is derived, leading to the construction of a field with a given curl in a star-shaped region and, as a by-product, a demonstration that the curl of the velocity field of a rigid body is twice the angular velocity. The curl-of-the-curl identity leads to a second definition of the Laplacian of a vector, the Helmholtz decomposition, and the prediction of electromagnetic waves.

The time-honored method of deriving vector-analytic identities&amp;mdash;treating the divergence and curl as &quot;formal products&quot; with the del operator, varying one field at a time, and adding the results&amp;mdash;is found to be less than rigorous, sometimes less than clear, and hard to justify in view of the ease with which the same thing can be done with Cartesian coordinates, indicial notation, and implicit summation.

The introduction of ''general'' coordinates proceeds through (non-normalized) natural and dual basis vectors, reciprocity, the Kronecker delta, covariance of the natural basis, contravariance of the dual basis, contravariant and covariant components, local bases, contravariance of coordinates, covariance of derivatives w.r.t. coordinates, the Jacobian, and handedness. Reciprocity leads to the dot-product of two vector fields and, via the permutation symbol, to the cross-products of the basis vectors, the definition of one basis in terms of the other, the cross-product of two vector fields, and reciprocity of the covariant and contravariant Jacobians. Thus the stage is set for expressing operators in general coordinates.

The multivariate chain rule leads to expressions for the directional derivative (in terms of the contravariant basis), hence the gradient (del) and advection operators. The identity for the curl of the product of a scalar and a vector leads to an expression for the curl in terms of covariant components. Expressions for the curl and divergence ''operators'' are obtained from the original volume-based definitions, and are found to agree with del-cross and del-dot respectively, with del expressed in the same general coordinates. The volume-based definition of the divergence leads, by a simpler path, to an expression in terms of contravariant components, which in turn yields an expression for the Laplacian.

Affine coordinates are briefly described before proceeding to orthogonal coordinates. In the latter, the Jacobian is simplified and we can choose an orthonormal basis, which is its own reciprocal, so that vectors can be specified in components w.r.t. a single basis. By expressing the old basis vectors and components in terms of the new ones, we can re-express dot-products, cross-products, and differential operators in terms of orthogonal coordinates with an orthonormal basis.

In an appendix, Huygens' principle is mathematized by deriving Green's identities and thence Kirchhoff's integral theorem (''without''&amp;#8202; assuming sinusoidal time-dependence), and then interpreting Kirchhoff's integrand as a distribution of secondary sources.

Some technicalities are relegated to the &quot;Notes&quot;, which are followed by the &quot;Citations&quot;, the &quot;References&quot; cited, and finally&amp;mdash;to compensate for the absence of a &quot;History&quot; section&amp;mdash;some suggested &quot;Further reading&quot;.

=== To-do list ===

Although this resource should be usable already, some improvements are envisaged, namely:
* More illustrations;
* A note on the metric tensor and its determinant.

{{cob}}

== Introduction ==

=== Scalars, vectors, tensors, and coordinates ===

{{cot}}

Elementary calculus concerns differentiation and integration with respect to a ''real'' variable. Vector analysis, or &quot;vector calculus&quot;, concerns what we might call differentiation and integration w.r.t. a ''vector'' variable&amp;mdash;usually the position vector. The function &quot;differentiated&quot; or &quot;integrated&quot; w.r.t. that vector may also be a vector.{{efn|Some authors treat &quot;vector analysis&quot; and &quot;vector calculus&quot; as synonymous. Others, apparently influenced by the difference between elementary &quot;calculus&quot; and real &quot;analysis&quot;, would say that &quot;vector analysis&quot; is more general, more theoretical, and more rigorous than &quot;vector calculus&quot;. That distinction might have surprised the inventors of &quot;vector analysis&quot;, as it was originally called; their motives were specific and practical, and their methods were ad-hoc.}}

But what exactly is a '''vector'''? Mathematicians define a &quot;vector&quot; as a member of a ''[[w:vector space|vector space]]'', which is a [[w:set (mathematics)|set]] whose members satisfy certain basic rules of algebra (called the ''vector-space axioms'') in relation to another set called a ''[[w:field (mathematics)|field]]'' (e.g., the real numbers), which has its own basic rules of algebra (the ''field axioms''), and whose members are called &quot;scalars&quot;. Physicists are more fussy. They typically want a &quot;vector&quot; to be not only a member of a vector space, but also a '''first-order tensor'''&amp;#8239;: a &quot;tensor&quot;, meaning that it exists independently of any coordinate system with which it might be specified; and &quot;first-order&quot; (or &quot;first-degree&quot;, or &quot;first-rank&quot;), meaning that it is specified by a ''one''-dimensional array of numbers. Similarly, a 2nd-order tensor is specified by a 2-dimensional array (a matrix), and a 3rd-order by a 3-dimensional array, and so on. Hence they want a &quot;scalar&quot;, which is specified by a single number (a zero-dimensional array), to be a ''zero-order tensor''. In &quot;vector analysis&quot;, we are greatly interested in applications to physical situations, and accordingly take the physicists' view on what constitutes a vector or a scalar.

So, for our purposes, defining a quantity by three components in (say) a Cartesian coordinate system is not enough to make it a vector, and defining a quantity as a real function of a list of coordinates is not enough to make it a scalar, because we still need to show that the quantity has an independent existence. One method of doing this (''not''&amp;#8202; the method we shall use here!) is to show that the coordinate representation behaves appropriately when the coordinate system is changed. Independent existence of a ''quantity'' means that its coordinate representation changes so as to compensate for the change in the coordinate system.&lt;ref&gt;E.g., Feynman ([[#feynman-63|1963]], vol.&amp;#8239;1, &amp;sect;&amp;#8202;11-5), having defined velocity from displacement in Cartesian coordinates, shows that velocity is a vector by showing that its coordinate representation contra-rotates (like that of displacement) if the coordinate system rotates.&lt;/ref&gt; But independent existence of an ''operator'' means that its expression in one coordinate system (with the operand[s] and the result ''in that system'') gives the same result as the corresponding expression in another coordinate system.&lt;ref&gt;E.g., Feynman ([[#feynman-63|1963]], vol.&amp;#8239;1, &amp;sect;&amp;#8202;11-7), having defined the magnitude and dot-product in Cartesian coordinates, proves that they are scalar functions by showing that the corresponding expressions in rotated (&quot;primed&quot;) coordinates give the same values as the original expressions (in &quot;unprimed&quot; coordinates). And Tai ([[#tai-95|1995]], pp.&amp;#8239;66–7), having found an expression for the &quot;gradient&quot; operator in a general coordinate system (the &quot;unprimed&quot; system), proves the &quot;invariance&quot; of the operator (its vector character in this case) by showing that the corresponding expression in any other general coordinate system (the &quot;primed&quot; system) has the same effect.&lt;/ref&gt;

Here we shall circumvent these complications by the most obvious route: by initially ''defining things without coordinates''. If, having defined something without coordinates, we then need to represent it ''with'' coordinates, we can choose the coordinate system for convenience rather than generality.

For example, without using coordinates, we can define displacements in three-dimensional space by their '''magnitudes''' and '''directions''' and show that they satisfy the vector-space rules, so that they are vectors in the mathematicians' sense, and therefore (because we have defined them without coordinates) in the physicists' sense. Then, by the same rules, we can show that the derivatives of these vectors w.r.t. time are vectors, and that products of these vectors with a scalar (such as mass) are vectors, with the result that not only displacement but also velocity, acceleration, momentum, and force are vectors. Having thus established that these things exist independently of any coordinate system, we can choose convenient coordinates.

{{cob}}

=== Prerequisites ===

{{cot}}

I assume that the reader is familiar with the algebra and geometry of vectors in 3D space, including the dot-product, the cross-product, and the scalar triple product, their geometric meanings, their expressions in Cartesian coordinates, and the identity
:{{big|{{math|'''a''' &amp;times; ('''b''' &amp;times; '''c''')  {{=}}  '''a&amp;sdot;&amp;#8202;c b''' &amp;minus; '''a&amp;sdot;&amp;#8202;b c''' ,}}}}
which we call the &quot;expansion&quot; of the vector triple product.&lt;ref&gt;There are many proofs and interpretations of this identity. My own effort, for what it's worth, is &quot;Trigonometric proof of vector triple product expansion&quot;, ''Mathematics Stack Exchange'', [https://math.stackexchange.com/a/4839213/307861 t.co/NM2v4DJJGo], 2024. The classic is [[#gibbs-1881-4|Gibbs, 1881]], &amp;sect;&amp;sect;&amp;#8239;26–7.&lt;/ref&gt; I further assume that the reader can generalize the concept of a derivative, so as to differentiate a vector with respect to a scalar, e.g.
:&lt;math&gt;\mathbf{r}'(t) = \frac{d\mathbf{r}}{dt}
  =\, \lim_{h\to 0} \frac{\mathbf{r}(t+h) - \mathbf{r}(t)}{h} \,,&lt;/math&gt;
or so as to differentiate a function of several independent variables &quot;partially&quot; w.r.t. one of them while the others are held constant, e.g.
:&lt;math&gt;\tfrac{\part}{\part y} \psi\big(x,y,z\big)
  =\, \lim_{h\to 0} \frac{\psi(x,y{+}h~\!,z) - \psi(x,y,z)}{h} \,.&lt;/math&gt;

But, in view of the limited applicability of limits (see the [[#Preface|Preface]]), I also expect the reader to be tolerant of an argument like this: In a short time{{mvar| dt}}, let the vectors {{math|'''r'''}} and{{math| '''p'''}} change by {{math|''d'''''r'''}} and{{math| ''d'''''p'''}} respectively. Then
:&lt;math&gt;\begin{align}
  \tfrac{d}{dt}\big(\mathbf{r}\!\times\!\mathbf{p}\big)
  &amp;= \frac{(\mathbf{r}+d\mathbf{r})\times(\mathbf{p}+d\mathbf{p})
            \,-\, \mathbf{r}\times\mathbf{p}}{dt}\\[1ex]
  &amp;= \frac{\mathbf{r}\!\times\!d\mathbf{p}+d\mathbf{r}\!\times\!\mathbf{p}}{dt}
       ~~\quad [\mathsf{neglecting}~d\mathbf{r}\!\times\!d\mathbf{p}]\\[1ex]
  &amp;=\, \mathbf{r}\times\!\tfrac{d\mathbf{p}}{dt}
       + \tfrac{d\mathbf{r}}{dt}\!\times\mathbf{p}\\[1ex]
  &amp;=\, \mathbf{r}\times\mathbf{\dot{p}}\,+\,\mathbf{\dot{r}}\times\mathbf{p}\,,
\end{align}&lt;/math&gt;
where, as always, the orders of the cross-products matter.{{efn|If {{math|'''r'''}} is the position of a particle and {{math|'''p'''}} is its momentum, the last term vanishes. If the force is toward the origin, the previous term also vanishes, and we are left with ''conservation of angular momentum'' about the origin.}} Differentiation of a ''dot''-product behaves similarly, except that the orders ''don't'' matter; and if&amp;#8239;{{math| '''p'''&amp;#8201;{{=}}&amp;#8201;''m'''''v'''}}, where {{mvar|m}} is a scalar and {{math|'''v'''}} is a vector, then
:&lt;math&gt;~~\mathbf{\dot{p}} = m\mathbf{\dot{v}} + \dot{m}\mathbf{v} \,.&lt;/math&gt;

Or an argument like this:  If&lt;math&gt;~z\!=\!f(x,y)&lt;/math&gt;, then
:&lt;math&gt;\begin{align}
  \frac{\part^2 z}{\part x\,\part y}
  &amp;= \tfrac{\part}{\part x}\,\tfrac{\part}{\part y} f\big(x,y\big)\\
  &amp;= \frac{\part}{\part x}\,\frac{f(x,y{+}dy)-f(x,y)}{dy}\\[1ex]
  &amp;= \frac{\,\frac{f(x{+}dx~\!,\,y{+}dy)\,-\,f(x{+}dx~\!,\,y)}{dy}
           - \frac{f(x,\,y{+}dy)\,-\,f(x,y)}{dy}\,} {dx}\\[2ex]
  &amp;= \frac{\,\frac{f(x{+}dx~\!,\,y{+}dy)\,-\,f(x,\,y{+}dy)}{dx}
           - \frac{f(x{+}dx~\!,\,y)\,-\,f(x,y)}{dx}\,} {dy}\\[1ex]
  &amp;= \frac{\part}{\part y}\,\frac{f(x{+}dx~\!,~\!y)-f(x,y)}{dx}\\[1ex]
  &amp;= \tfrac{\part}{\part y}\,\tfrac{\part}{\part x} f\big(x,y\big)
   = \frac{\part^2 z}{\part y\,\part x} \,;
\end{align}&lt;/math&gt;
that is, we can switch the order of differentiation in a &quot;mixed&quot; partial derivative. If{{mvar| &amp;part;&lt;sub&gt;x&lt;/sub&gt;}} is an abbreviation for {{mvar|{{sfrac|&amp;part;|&amp;part;x}}&amp;#8202;}}, etc., this rule can be written in '''operational''' terms as
:{{big|{{mvar|&amp;part;&lt;sub&gt;x &lt;/sub&gt;&amp;part;&lt;sub&gt;y&lt;/sub&gt; {{=}} &amp;part;&lt;sub&gt;y &lt;/sub&gt;&amp;part;&lt;sub&gt;x &lt;/sub&gt;.}}}}
More generally, if {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}} is an abbreviation for {{mvar|{{sfrac|&amp;part;|&amp;part;x&lt;sub&gt;i&lt;/sub&gt;}}}}&amp;#8202; where&amp;#8202; {{math|''i''&amp;#8239;&amp;#8714;&amp;#8202;&amp;lcub;1,&amp;#8202;2,&amp;hellip;&amp;rcub;,}}&amp;#8202; the rule becomes
:{{big|{{mvar|&amp;part;&lt;sub&gt;i &lt;/sub&gt;&amp;part;&lt;sub&gt;j&lt;/sub&gt; {{=}} &amp;part;&lt;sub&gt;j &lt;/sub&gt;&amp;part;&lt;sub&gt;i &lt;/sub&gt;.}}}}

The above generalizations of differentiation, however, do not go beyond differentiation w.r.t. ''real'' variables, some of which are scalars, and some of which are coordinates. It is now time to consider various kinds of &quot;differentiation&quot; w.r.t. the position vector.

{{cob}}

== Closed-surface integrals per unit volume ==

{{cot}}

The term ''field'', mentioned above in the context of algebraic axioms, has another meaning, which will be its usual meaning from now on: if {{math|'''r'''}} is the position vector, a '''scalar field''' is a scalar-valued function of{{math| '''r''',}} and a '''vector field''' is a vector-valued function of{{math| '''r'''}}; both may also depend on time. These are the functions of which we want &quot;derivatives&quot; w.r.t. the vector{{math| '''r'''}}.

In this section I introduce four such derivatives&amp;mdash;the ''gradient'', the ''curl'', the ''divergence'', and the ''Laplacian''&amp;#8202;&amp;mdash;in a way that will seem unremarkable to those readers who aren't already familiar with them, but idiosyncratic to those who are. The gradient is commonly introduced in connection with a curve and its endpoints, the curl in connection with a surface segment and its enclosing curve, the divergence in connection with a volume and its enclosing surface, and the Laplacian as a composite of two of the above, initially applicable only to a scalar field. Here I introduce all four in connection with a volume and its enclosing surface, and I introduce the Laplacian as a concept in its own right, equally applicable to a scalar ''or vector''&amp;#8202; field; only later do I express the Laplacian in terms of other &quot;derivatives&quot;. My initial definitions of the gradient, the curl, and the Laplacian, although not novel, are usually thought to be more advanced than the common ones&amp;mdash;in spite of being conceptually simpler, and in spite of being obvious variations on the same theme.

{{cob}}

=== Instant integral theorems (with a caveat) ===

{{cot}}

Let {{mvar|V}} be a volume (3D region) enclosed by a surface {{mvar|S}} (a mathematical surface, ''not'' generally a physical barrier). Let &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; be the unit normal vector at a general point on {{mvar|S}}, pointing ''out'' of{{mvar| V}}. Let {{mvar|n}} be the distance from {{mvar|S}} in the direction of&lt;math&gt;~\mathbf{\hat{n}}&lt;/math&gt; (positive outside {{mvar|V}}, negative inside), and let {{mvar|&amp;part;&lt;sub&gt;n&lt;/sub&gt;}} be an abbreviation for{{mvar| {{sfrac|&amp;part;|&amp;part;n}}&amp;#8202;}}, where the derivative&amp;mdash;commonly called the '''normal derivative'''&amp;mdash;is tacitly assumed to exist.

In {{mvar|V}}, and on {{mvar|S}}, let {{mvar|p}} be a scalar field (e.g., pressure in a fluid, or temperature), and let {{math|'''q'''}} be a vector field (e.g., flow velocity, or heat-flow density), and let {{mvar|&amp;psi;}} be a generic field which may be a scalar or a vector. Let a general ''element'' (small segment) of the surface {{mvar|S}} have area {{mvar|dS}}, and let it be small enough to allow &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt;, {{mvar|p}}, {{math|'''q'''}}, and {{mvar|&amp;part;&lt;sub&gt;n&lt;/sub&gt;&amp;#8202;&amp;psi;}} to be considered uniform over the element (making a tacit assumption of local continuity). Then, for every element, the following four products are well defined:
{{NumBlk|:|&lt;math&gt;\mathbf{\hat{n}}            ~\!p\,dS ~,\qquad
                 \mathbf{\hat{n}}\times\mathbf{q}\,dS ~,\qquad
                 \mathbf{\hat{n}         \cdot q}\,dS ~,\qquad
                 \part_n                     \psi\;dS \,.
          &lt;/math&gt;|{{EquationRef|1}}}}
If {{mvar|p}} is pressure in a non-viscous fluid, the first of these products is the force exerted by the fluid in {{mvar|V}}&amp;#8202; through the area {{mvar|dS}}. The second product does not have such an obvious physical interpretation; but if{{math| '''q'''}} is ''circulating'' clockwise about an axis directed through{{mvar| V}}, the cross-product will be exactly tangential to{{mvar| S}} and will tend to have a component in the direction of that axis. The third product is the ''flux'' of{{math| '''q'''}} through the surface element; if{{math| '''q'''}} is flow velocity, the third product is the volumetric flow rate (volume per unit time) ''out'' of{{mvar| V}}&amp;#8202; through{{mvar| dS&amp;#8202;}}; or if {{math|'''q'''}} is heat-flow density, the third product is the heat transfer rate (energy per unit time) ''out'' of{{mvar| V}}&amp;#8202; through{{mvar| dS}}. The fourth product, by analogy with the third, might be called the flux of the normal derivative of{{mvar| &amp;psi;}} through the surface element, but is equally well defined whether {{mvar|&amp;psi;}} is a scalar or a vector&amp;mdash;or, for that matter, a matrix, or a tensor of any order, or anything else that we can differentiate w.r.t.{{mvar| n}}.

If we add up each of the four products over all the elements of the surface {{mvar|S}}, we obtain, respectively, the four '''surface integrals'''
{{NumBlk|:|&lt;math&gt;\iint_S \!\mathbf{\hat{n}}            ~\!p\,dS \,,~
                 \iint_S \!\mathbf{\hat{n}}\times\mathbf{q}\,dS \,,~
                 \iint_S \!\mathbf{\hat{n}         \cdot q}\,dS \,,~
                 \iint_S \!\part_n                     \psi\;dS \,,
          &lt;/math&gt;|{{EquationRef|2}}}}
in which the double integral sign indicates that the range of integration is two-dimensional. The first surface integral takes a scalar field and yields a vector; the second takes a vector field and yields a vector; the third takes a vector field and yields a scalar; and the fourth takes (e.g.) a scalar field yielding a scalar, or a vector field yielding a vector. If{{mvar| p}} is pressure in a non-viscous fluid, the first integral is the force exerted by the fluid in {{mvar|V}}&amp;#8202; on the fluid outside {{mvar|V}}. The second integral may be called the ''skew'' surface integral of{{math| '''q'''}} over {{mvar|S&amp;#8202;}},&lt;ref&gt;[[#gibbs-1881-4|Gibbs, 1881]], &amp;sect;&amp;#8239;56.&lt;/ref&gt; or, for the reason hinted above, the ''circulation'' of{{math| '''q'''}} over {{mvar|S}}.&amp;#8201; The third integral, commonly called the ''flux integral'' (or simply the surface integral) of{{math| '''q'''}} over {{mvar|S}}, is the total ''flux'' of{{math| '''q'''}} out of{{mvar| V}}. And the fourth integral is the surface integral of the outward normal derivative of{{mvar| &amp;psi;}}.

Let the volume {{mvar|V}}&amp;#8202; be divided into elements. Let a general volume element have the volume {{mvar|dV}} and be enclosed by the surface {{mvar|&amp;delta;S}}&amp;#8201;&amp;mdash;not to be confused with the area {{mvar|dS}} of a surface ''element'', which may be an element of{{mvar| S}} or of{{mvar| &amp;delta;S}}. Then consider what happens if, instead of evaluating each of the above surface integrals over {{mvar|S}}, we evaluate it over each {{mvar|&amp;delta;S}} and add up the results for all the volume elements. In the ''interior'' of{{mvar| V}}, each surface element of area {{mvar|dS}} is on the boundary between two volume elements, for which the unit normals &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; at {{mvar|dS}}, and the respective values of{{mvar| &amp;part;&lt;sub&gt;n&lt;/sub&gt;&amp;#8202;&amp;psi;}}, are equal and opposite. Hence when we add up the integrals over the surfaces {{mvar|&amp;delta;S}}, the contributions from the elements {{mvar|dS}} cancel in pairs, except on the original surface {{mvar|S}}, so that we are left with the original integral over {{mvar|S}}. So, for the four surface integrals in ({{EquationNote|2}}), we have respectively
{{NumBlk|:|&lt;math&gt;\begin{align}
    \iint_S \mathbf{\hat{n}}~\!p \,dS
    &amp; \,= \sum_V\iint_{\delta S} \mathbf{\hat{n}}~\!p \,dS \,, \\
    \iint_S \mathbf{\hat{n}}\times\mathbf{q} \,dS
    &amp; \,= \sum_V\iint_{\delta S} \mathbf{\hat{n}}\times\mathbf{q} \,dS \,, \\
    \iint_S \mathbf{\hat{n}\cdot q} \,dS
    &amp; \,= \sum_V\iint_{\delta S} \mathbf{\hat{n}\cdot q} \,dS \,, \\
    \iint_S \part_n \psi \;dS
    &amp; \,= \sum_V\iint_{\delta S} \part_n \psi \;dS \,.
  \end{align}&lt;/math&gt;|{{EquationRef|3}}}}

Now comes a big &quot;if&quot;:&amp;#8201; ''if''&amp;#8202; we define the '''gradient''' of{{mvar| p}} (pronounced &quot;grad {{mvar|p}}&quot;) inside {{mvar|dV}}&amp;#8202; as
{{NumBlk|:|&lt;math&gt;\nabla p
    \,:=\, \frac{1}{dV}\iint_{\delta S} \mathbf{\hat{n}}~\!p \,dS
  &lt;/math&gt;|{{EquationRef|4g}}}}
and the '''curl''' of {{math|'''q'''}} inside {{mvar|dV}}&amp;#8202; as
{{NumBlk|:|&lt;math&gt;\operatorname{curl}\mathbf{q}
    \,:=\, \frac{1}{dV}\iint_{\delta S} \mathbf{\hat{n}}\times\mathbf{q} \,dS
  &lt;/math&gt;|{{EquationRef|4c}}}}
and the '''divergence''' of {{math|'''q'''}} inside {{mvar|dV}}&amp;#8202; as
{{NumBlk|:|&lt;math&gt;\operatorname{div}\mathbf{q}
    \,:=\, \frac{1}{dV}\iint_{\delta S} \mathbf{\hat{n}\cdot q} \,dS
  &lt;/math&gt;|{{EquationRef|4d}}}}
and the '''Laplacian''' of {{mvar|&amp;psi;}} inside {{mvar|dV}}&amp;#8202; as
{{NumBlk|:|&lt;math&gt;\triangle\psi
    \,:=\, \frac{1}{dV}\iint_{\delta S} \part_n \psi \;dS
  &lt;/math&gt;|{{EquationRef|4L}}}}
(where the letters after the equation number stand for ''gradient'', ''curl'', ''divergence'', and ''Laplacian'', respectively), then equations ({{EquationNote|3}}) can be rewritten
:&lt;math&gt;\begin{align}
    \iint_S \mathbf{\hat{n}}~\!p \,dS
    &amp; \,= \sum_V \nabla p ~dV \,, \\
    \iint_S \mathbf{\hat{n}}\times\mathbf{q} \,dS
    &amp; \,= \sum_V \operatorname{curl}\mathbf{q} ~dV \,, \\
    \iint_S \mathbf{\hat{n}\cdot q} \,dS
    &amp; \,= \sum_V \operatorname{div}\mathbf{q} ~dV \,, \\
    \iint_S \part_n \psi \;dS
    &amp; \,= \sum_V \triangle\psi ~dV \,.
  \end{align}&lt;/math&gt;
(For the Laplacian operator, we have used the broad triangle symbol{{math| (&amp;#9651;)}} rather than the narrower Greek Delta{{math| (&amp;Delta;)}}; the latter would more readily be misinterpreted as &quot;change in&amp;hellip;&quot;)&amp;#8201; But because each term in each sum above has a factor {{mvar|dV}}, we call the sum an integral; and because the range of integration is three-dimensional, we use a triple integral sign. Thus we obtain the following four theorems relating integrals over an enclosing surface {{mvar|S}}&amp;#8202; to integrals over the enclosed volume {{mvar|V&amp;#8202;}}:
{{NumBlk|:|&lt;math&gt;~~~~~\!\iint_S \mathbf{\hat{n}}~\!p \,dS
    \,= \iiint_V \nabla p ~dV \,;
  &lt;/math&gt;|{{EquationRef|5g}}}}
{{NumBlk|:|&lt;math&gt;\iint_S \mathbf{\hat{n}}\times\mathbf{q} \,dS
    \,= \iiint_V \operatorname{curl}\mathbf{q} ~dV \,;
  &lt;/math&gt;|{{EquationRef|5c}}}}
{{NumBlk|:|&lt;math&gt;~~\!\iint_S \mathbf{\hat{n}\cdot q} \,dS
    \,= \iiint_V \operatorname{div}\mathbf{q} ~dV \,;
  &lt;/math&gt;|{{EquationRef|5d}}}}
{{NumBlk|:|&lt;math&gt;~~\iint_S \part_n \psi \;dS
    \,= \iiint_V \triangle\psi ~dV \,.
  &lt;/math&gt;|{{EquationRef|5L}}}}
Of the above four results, only the third ({{EquationNote|5d}}) seems to have a standard name; it is called the '''divergence theorem''' (or ''Gauss's theorem'' or, more properly, ''[[w:Mikhail Ostrogradsky|Ostrogradsky]]'s theorem''&lt;ref&gt;[[#katz-79|Katz, 1979]], pp.&amp;#8239;146–9.&lt;/ref&gt;), and is indeed the best known of the four&amp;mdash;although the other three, having been derived in parallel with it, may be said to be equally fundamental.

As each of the operators {{math|&amp;nabla;,}} {{math|curl,}} and {{math|div}} calls for an integration w.r.t. area and then a division by volume, the ''dimension'' (or unit of measurement) of the result is the dimension of the operand divided by the dimension of length, as if the operation were some sort of differentiation w.r.t. position. Moreover, in each of equations ({{EquationNote|5g}}) to ({{EquationNote|5d}}), there is a triple integral on the right but only a double integral on the left, so that each of the operators {{math|&amp;nabla;,}} {{math|curl,}} and {{math|div}} appears to compensate for a single integration. For these reasons, and for convenience, we shall describe them as '''differential operators'''. By comparison, the {{math|&amp;#9651; }}operator in ({{EquationNote|4L}}) or ({{EquationNote|5L}}) calls for a further differentiation w.r.t.{{mvar| n&amp;#8202;}}; we shall therefore describe {{math|&amp;#9651;}} as a ''2nd-order'' differential operator. (An additional reason for these descriptions will emerge later.) As promised, the four definitions ({{EquationNote|4g}}) to ({{EquationNote|4L}}) are &quot;obvious variations on the same theme&quot; (although the fourth is somewhat less obvious than the others).

But remember the &quot;if&quot;: Theorems ({{EquationNote|5g}}) to ({{EquationNote|5L}}) depend on definitions ({{EquationNote|4g}}) to ({{EquationNote|4L}}) and are therefore only as definite as those definitions! Equations ({{EquationNote|3}}), without assuming anything about the shapes and relative sizes of the closed surfaces {{mvar|&amp;delta;S}} (except, tacitly, that&amp;#8202; &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is piecewise well-defined), indicate that the surface integrals are ''additive with respect to volume''. But this additivity, by itself, does not guarantee that the surface integrals are shared among neighboring volume elements ''in proportion'' to their volumes, as envisaged by &quot;definitions&quot; ({{EquationNote|4g}}) to ({{EquationNote|4L}}). Each of these &quot;definitions&quot; is unambiguous if, and only if, the ratio of the surface integral to{{mvar| dV}}&amp;#8202; is insensitive to the shape and size of{{mvar| &amp;delta;S}}&amp;#8202; for a sufficiently small {{mvar|&amp;delta;S}}. Notice that the issue here is ''not'' whether the ratios specified in equations ({{EquationNote|4g}}) to ({{EquationNote|4L}}) are true vectors or scalars, independent of the coordinates; all of the operations needed in those equations have coordinate-free definitions. Rather, the issue is whether the resulting ratios are unambiguous ''notwithstanding the ambiguity of'' {{mvar|&amp;delta;S}}, provided only that {{mvar|&amp;delta;S}} is sufficiently small. That is the advertised &quot;caveat&quot;, which must now be addressed.

Our proofs of the unambiguity of the differential operators will rest on a few [[w:thought experiment|thought experiments]], each of which applies an operator to a physical field, say{{mvar| f}}, and obtains another physical field whose unambiguity is beyond dispute, provided only that it can be considered uniform over the (small) volume element. The conclusion of the thought experiment is then applicable to any operand field whose ''mathematical'' properties are consistent with the physical interpretation; the loss of generality, if any, is only what is incurred by that interpretation.

{{cob}}

=== Unambiguity of the gradient ===

{{cot}}

Suppose that a fluid with density {{mvar|&amp;rho;}} (a scalar field) flows with velocity{{math| '''v'''}} (a vector field) under the influence of the internal pressure {{mvar|p}} (a scalar field). Then the integral in ({{EquationNote|4g}}) is the force exerted by the pressure of the fluid inside {{mvar|&amp;delta;S}} on the fluid outside, so that ''minus'' the integral is the force exerted ''on'' the fluid inside{{mvar| &amp;delta;S}}&amp;#8202; by the pressure of the fluid outside. Dividing by {{mvar|dV}}, we find that {{math|&amp;minus;&amp;nabla;''p''}}, as defined by ({{EquationNote|4g}}), is the force per unit volume, due to the pressure outside the volume.&lt;ref&gt;In [[#feynman-63|Feynman, 1963]],&amp;#8201; {{math|&amp;minus;&amp;nabla;''p'' }}as the &quot;pressure force per unit volume&quot; eventually appears in the 3rd-last lecture of Volume 2 (&amp;sect;40-1).&lt;/ref&gt; If this is the ''only'' force per unit volume acting ''on'' the volume (e.g., because the fluid is non-viscous and in a weightless environment, and the volume element is not in contact with the container), then it is equal to the acceleration times the mass per unit volume; that is,
{{NumBlk|:|&lt;math&gt;
    \rho\,\frac{d\mathbf{v}}{dt} = -\nabla p \,.
  &lt;/math&gt;|{{EquationRef|6g}}}}
Now provided that the left-hand expression can be considered uniform inside the small {{mvar|&amp;delta;S}}, it is unambiguous, whence&amp;#8202; {{math|&amp;nabla;''p'' }}''is also unambiguous''. If there are additional forces on the fluid element, e.g. due to gravity and&amp;#10744;or viscosity, then {{math|&amp;minus;&amp;nabla;''p''}} is not the sole contribution to density-times-acceleration, but is still the contribution due to pressure, which is still unambiguous.

By showing the unambiguity of definition ({{EquationNote|4g}}), we have confirmed theorem ({{EquationNote|5g}}). In the process we have seen that the volume-based definition of the gradient is useful for the modeling of fluids, and intuitive in that it formalizes the common notion that a pressure &quot;gradient&quot; gives rise to a force.

{{cob}}

=== Unambiguity of the divergence ===

{{cot}}

In the aforesaid fluid, in a short time{{mvar| dt}}, the volume that flows out of fixed closed surface {{mvar|&amp;delta;S}}&amp;#8202; through a fixed surface element of area {{mvar|dS}}&amp;#8202; is &lt;math&gt;\mathbf{v}~\!dt\!\cdot\!\mathbf{\hat{n}}\,dS&lt;/math&gt;&amp;#8239; (i.e., the displacement normal to the surface element, times the area).&amp;#8201; Multiplying this by density and integrating over {{mvar|&amp;delta;S}}, we find that the mass flowing out of{{mvar| &amp;delta;S}}&amp;#8202; in time{{mvar| dt}} is&amp;#8201; &lt;math&gt;\textstyle\iint_{\delta S}\rho\mathbf{v}~\!dt\cdot\mathbf{\hat{n}}\,dS&lt;/math&gt;.  Dividing this by {{mvar|dV}}, and then by {{mvar|dt}}, we get the rate of reduction of density inside {{mvar|&amp;delta;S&amp;#8202;}}; that is,
:&lt;math&gt;\tfrac{1}{dV}\!\iint_{\delta S}\mathbf{\hat{n}}\cdot\rho\mathbf{v}\,dS
       \,= -\frac{\part\rho}{\part t} \,,&lt;/math&gt;
where the derivative w.r.t. time is evaluated at a fixed location (because {{mvar|&amp;delta;S}} is fixed), and is therefore written as a ''partial'' derivative (because other variables on which {{mvar|&amp;rho;}} might depend&amp;mdash;namely spatial coordinates&amp;mdash;are held constant). Provided that the right-hand side can be considered uniform inside {{mvar|&amp;delta;S}}, it is unambiguous, so that the left side is likewise unambiguous. But the left side is simply{{math| div&amp;#8239;''&amp;rho;'''''v'''}}&amp;#8201; as defined by ({{EquationNote|4d}}),{{efn|There is no need for parentheses around{{math| ''&amp;rho;'''''v'''&amp;#8202;,}} because {{math|div&amp;#8239;''&amp;rho;'''''v'''&amp;#8202;}} cannot mean {{math|(div&amp;#8239;''&amp;rho;'')'''v'''&amp;#8202;,}} because the divergence of a scalar field is not defined.}} which is therefore also unambiguous,&lt;ref&gt;A demonstration like the foregoing is outlined by Gibbs ([[#gibbs-1881-4|1881]], &amp;sect;&amp;#8239;55).&lt;/ref&gt; confirming theorem ({{EquationNote|5d}}). In short, the divergence operator is that which maps {{math|''&amp;rho;'''''v'''}} to the rate of reduction of density at a fixed point:
{{NumBlk|:|&lt;math&gt;
    \operatorname{div}\rho\mathbf{v} = -\frac{\part\rho}{\part t} \,.
  &lt;/math&gt;|{{EquationRef|7d}}}}
This result, which expresses ''conservation of mass'', is a form of the so-called '''equation of continuity'''.

The partial derivative {{mvar|{{sfrac|&amp;part;&amp;rho;|&amp;part;t}}&amp;#8202;}} in ({{EquationNote|7d}}) must be distinguished from the '''material derivative''' {{mvar|{{sfrac|d&amp;rho;|dt}}&amp;#8202;}}, which is evaluated at a point that moves ''with the fluid''.{{efn|The material derivative operator {{mvar|{{sfrac|d|dt}}}} is also called the ''substantive'' derivative, and is sometimes written {{mvar|{{sfrac|D|Dt}}}} if the result is meant to be understood as a field rather than simply a function of time ([[#kemmer-77|Kemmer, 1977]], pp.&amp;#8239;184–5).}} [Similarly, {{math|{{sfrac|''d''&amp;#8202;'''v'''|''dt''}}}} in ({{EquationNote|6g}}) is the ''material'' acceleration, because it is the acceleration of the mobile mass&amp;mdash;not of a fixed point!&amp;#8239;]&amp;#8239; To re-derive the equation of continuity in terms of the ''material'' derivative, the volume &lt;math&gt;\mathbf{v}~\!dt\!\cdot\!\mathbf{\hat{n}}\,dS~\!,&lt;/math&gt; which flows out through{{mvar| dS}} in time{{mvar| dt}} (as above), is integrated over {{mvar|&amp;delta;S}} to obtain the increase in volume of the mass ''initially'' contained in {{mvar|dV}}. Dividing this by the mass, {{mvar|&amp;rho;&amp;#8239;dV}}, gives the increase in ''[[w:specific volume|specific volume]]'' {{math|(1&amp;#10744;''&amp;rho;'')}} of that mass, and then dividing by {{mvar|dt}} gives the rate of change of specific volume; that is,
:&lt;math&gt;\tfrac{1}{\rho~\!dV}\!\iint_{\delta S}\mathbf{\hat{n}}\cdot\mathbf{v}\,dS
       \,= \tfrac{d}{dt}\big(\rho^{-1}\big)
       = -\rho^{-2\,}\tfrac{d\rho}{dt} \,.&lt;/math&gt;
Multiplying by {{math|''&amp;rho;''&amp;sup2;}} and comparing the left side with ({{EquationNote|4d}}), we obtain
{{NumBlk|:|&lt;math&gt;
    \rho\operatorname{div}\mathbf{v} = -\frac{d\rho}{dt} \,.
  &lt;/math&gt;|{{EquationRef|7d'}}}}
Whereas ({{EquationNote|7d}}) shows that {{math|div&amp;#8239;''&amp;rho;'''''v'''&amp;#8202;}} is unambiguous, ({{EquationNote|7d'}}) shows that {{math|div&amp;#8239;'''v'''&amp;#8202;}} is unambiguous (provided that other things are locally continuous). In accordance with the everyday meaning of &quot;divergence&quot;, ({{EquationNote|7d'}}) also shows that {{math|div&amp;#8239;'''v'''&amp;#8202;}} is positive if the fluid is expanding ({{mvar|&amp;rho; }}decreasing), negative if it is contracting ({{mvar|&amp;rho; }}increasing), and zero if it is incompressible. In the last case, the equation of continuity reduces to
{{NumBlk|:|&lt;math&gt;
    \operatorname{div}\mathbf{v} = 0
  \qquad&lt;/math&gt;[&amp;#8202;for an incompressible fluid&amp;#8202;].|{{EquationRef|7i}}}}

For incompressible flow, any tubular surface tangential to the flow velocity, and consequently with no flow in or out of the &quot;tube&quot;, has the same volumetric flow rate across all cross-sections of the &quot;tube&quot;, as if the surface were the wall of a pipe full of liquid (except that the surface is not necessarily stationary). Accordingly, ''a vector field with zero divergence is described as '''solenoidal''''' (from the Greek word for &quot;pipe&quot;). More generally, a solenoidal vector field has the property that for any tubular surface tangential to the field, the flux integrals across any two cross-sections of the &quot;tube&quot; are the same&amp;mdash;because otherwise there would be a net flux integral out of the closed surface comprising the two cross-sections and any segment of tube between them, in which case, by the divergence theorem ({{EquationNote|5d}}), the divergence would have to be non-zero somewhere inside, contrary to ({{EquationNote|7i}}).

{{cob}}

=== Unambiguity of the curl (and gradient) ===

{{cot}}

The unambiguity of the curl ({{EquationNote|4c}}) follows from the unambiguity of the divergence. Let {{math|'''b'''}} be any ''uniform'' vector (i.e., any vector that is independent of location&amp;mdash;e.g. a uniform vector field, possibly time-dependent). Taking dot-products of ({{EquationNote|4c}}) with{{math| '''b''',}} we get
:&lt;math&gt;\begin{align}
    \mathbf{b}\cdot\operatorname{curl}\mathbf{q}\,
    &amp;=\, \mathbf{b}\cdot\tfrac{1}{dV}\!\iint_{\delta S}
           \mathbf{\hat{n}}\times\mathbf{q}
         \,dS \\[.5ex]
    &amp;=\, \tfrac{1}{dV}\!\iint_{\delta S}
           \mathbf{b}\cdot\mathbf{\hat{n}}\!\times\!\mathbf{q}
         \,dS \\[1ex]
    &amp;=\, \tfrac{1}{dV}\!\iint_{\delta S}
           \mathbf{\hat{n}}\cdot\mathbf{q}\!\times\!\mathbf{b}
         \,dS \,;
  \end{align}&lt;/math&gt;
that is, by ({{EquationNote|4d}}),
{{NumBlk|:|&lt;math&gt;\operatorname{curl}\mathbf{q} \cdot \mathbf{b}
    = \operatorname{div}(\mathbf{q}\!\times\!\mathbf{b})
  \qquad&lt;/math&gt;[&amp;#8202;for uniform {{math|'''b'''}}].|{{EquationRef|8c}}}}
(The parentheses around&amp;#8202; {{math|'''q'''&amp;#8202;&amp;times;&amp;#8202;'''b'''}}&amp;#8202; on the right, although helpful because of the spacing, are not strictly necessary, because the alternative binding would be {{math|(div&amp;#8201;'''q''')}}, which is a scalar, whose cross-product with the vector {{math|'''b'''}} is not defined. And the left-hand expression does not need parentheses, because it can only mean the dot-product of a curl with the vector {{math|'''b'''}}; it cannot mean the curl of a dot-product, because the curl of a scalar field is not defined.) Equation ({{EquationNote|8c}}) is an identity for ''uniform''{{math| '''b'''}}. If we make {{math|'''b'''}} a ''unit'' vector in any fixed direction, the left-hand side of the identity is the (scalar) component of {{math|curl&amp;#8239;'''q'''}} in that direction, and the right-hand side is unambiguous. Thus ''the curl is unambiguous because its component in any direction is unambiguous''. This confirms theorem ({{EquationNote|5c}}).

Similarly, the unambiguity of the divergence implies the unambiguity of the gradient. Starting with ({{EquationNote|4g}}), taking dot-products with an arbitrary uniform vector {{math|'''b''',}} and proceeding as above, we obtain
{{NumBlk|:|&lt;math&gt;\nabla p \cdot \mathbf{b}
    = \operatorname{div} p\mathbf{b}
  \qquad&lt;/math&gt;[&amp;#8202;for uniform {{math|'''b'''}}].|{{EquationRef|8g}}}}
(The left-hand side does not need parentheses, because it can only mean the dot-product of a gradient with the vector {{math|'''b'''}}; it cannot mean the gradient of the dot-product of a scalar field with a vector field, because that dot-product would not be defined.) If we make {{math|'''b'''}} a ''unit'' vector, this result ({{EquationNote|8g}}) says that the (scalar) component of{{math| &amp;nabla;''p''}} in the direction of{{math|&amp;#8202; '''b'''}} is given by the right-hand side, which again is unambiguous. So here we have a second explanation of the unambiguity of the gradient: like the curl, it is unambiguous because its component in any direction is unambiguous.

We might well ask what happens if we take ''cross''-products with {{math|'''b'''}} on the left, instead of dot-products. If we start with ({{EquationNote|4g}}), the process is straightforward: in the end we can switch the order of the cross-product on the left, and change the sign on the right, obtaining
{{NumBlk|:|&lt;math&gt;\nabla p \times \mathbf{b}
    = \operatorname{curl} p\mathbf{b}
  \qquad&lt;/math&gt;[&amp;#8202;for uniform {{math|'''b'''}}].|{{EquationRef|8p}}}}
(Again no parentheses are needed.) If we start with ({{EquationNote|4c}}) instead, and take {{math|'''b'''}} inside the integral, we get a vector triple product to expand, which leads to
:&lt;math&gt;\mathbf{b} \times \operatorname{curl}\mathbf{q}
    = \tfrac{1}{dV}\!\iint_{\delta S}\!\mathbf{\hat{n}}\,\mathbf{b{\cdot}q}\,dS
    - \tfrac{1}{dV}\!\iint_{\delta S}\!\mathbf{b{\cdot}\hat{n}\,q}\,dS \,,
  &lt;/math&gt;
in which the first term on the right is simply&amp;#8202; {{math|&amp;nabla;&amp;#8239;'''b&amp;sdot;q'''}}&amp;#8201; (the gradient of the dot-product). The second term is more problematic. ''If''&amp;#8202; we had a scalar {{mvar|p}} instead of the vector {{math|'''q''',}} we could take {{math|'''b'''}} outside the second integral, so that the second term would be (minus) {{math|'''b&amp;#8239;&amp;sdot;'''&amp;#8239;&amp;nabla;''p''}}. This suggests that the actual second term should be (minus) {{math|'''b&amp;#8239;&amp;sdot;'''&amp;#8239;&amp;nabla;'''q'''}}.&amp;#8201; Shall we therefore adopt the second term (without the sign) as the ''definition'' of{{math|&amp;#8202; '''b&amp;sdot;'''&amp;nabla;&amp;#8239;'''q'''}} for a ''vector'' {{math|'''q'''}} (treating {{math|'''b&amp;sdot;'''&amp;nabla;}} as an operator), and write
{{NumBlk|:|&lt;math&gt;\mathbf{b} \times \operatorname{curl}\mathbf{q} ~\!=
    \nabla\,\mathbf{b{\cdot}q} - \mathbf{b}~\!{\cdot}\nabla\,\mathbf{q}
  \qquad&lt;/math&gt;[&amp;#8202;for uniform {{math|'''b'''}}] ?|{{EquationRef|8q}}}}
The proposal would be open to the objection that&amp;#8202; {{math|'''b&amp;sdot;'''&amp;nabla;&amp;#8239;'''q'''}}&amp;#8202; had been defined only for ''uniform''{{math| '''b'''&amp;#8202;,}} whereas&amp;#8202; {{math|'''b&amp;#8239;&amp;sdot;'''&amp;#8239;&amp;nabla;''p''&amp;#8202;}} (for scalar{{mvar| p}}) is defined whether {{math|'''b'''}} is uniform or not.&amp;#8201; So, for the moment, let us put ({{EquationNote|8q}}) aside and run with ({{EquationNote|8c}}), ({{EquationNote|8g}}), and ({{EquationNote|8p}}).

{{cob}}

=== Another meaning of the gradient ===

{{cot}}

Let {{math|'''s&amp;#770;'''}} be a unit vector in a given direction, and let {{mvar|s}} be a parameter measuring distance (arc length) along a path in that direction. By equation ({{EquationNote|8g}}) and definition ({{EquationNote|4d}}), we have
:&lt;math&gt;\nabla p \cdot \mathbf{\hat{s}} =
    \operatorname{div} p\mathbf{\hat{s}} =
    \tfrac{1}{dV}\!\iint_{\delta S}\mathbf{\hat{n}}\cdot p\mathbf{\hat{s}}\,dS\,,
  &lt;/math&gt;
where, by the unambiguity of the divergence, the shape of the closed surface {{mvar|&amp;delta;S}} enclosing {{mvar|dV}}&amp;#8202; can be chosen for convenience. So let {{mvar|&amp;delta;S}} be a right cylinder with cross-sectional area {{mvar|&amp;alpha;}}&amp;#8201; and perpendicular height {{mvar|ds&amp;#8202;,}} with the path passing perpendicularly through the end-faces at parameter-values {{mvar|s}} and {{mvar|s+ds&amp;#8202;,}} where the outward unit normal &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; consequently takes the values {{math|&amp;minus;'''s&amp;#770;'''}} and {{math|'''s&amp;#770;'''&amp;#8202;,}} respectively. And let the cross-sectional dimensions be small compared with {{mvar|ds}}&amp;#8202; so that the values of{{mvar| p}} at the end-faces, say {{mvar|p}} and {{mvar|p+dp}}, can be taken to be the same as where the end-faces cut the path. Then&amp;#8202; {{mvar|dV&amp;#8201;{{=}}&amp;#8201;&amp;alpha;&amp;#8239;ds&amp;#8202;}}, and the surface integral over{{mvar| &amp;delta;S}} includes only the contributions from the end-faces (because &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is perpendicular to {{math|'''s&amp;#770;'''}} elsewhere); those contributions are respectively&amp;#8201; &lt;math&gt;-\mathbf{\hat{s}}\!\cdot\!p\mathbf{\hat{s}}\,\alpha\,&lt;/math&gt; and&amp;#8201; &lt;math&gt;\mathbf{\hat{s}}\!\cdot\!(p\!+\!dp)\mathbf{\hat{s}}\,\alpha~\!,&lt;/math&gt;  i.e.&amp;#8239; &lt;math&gt;-p\alpha\,&lt;/math&gt; and &lt;math&gt;(p\!+\!dp)\alpha&lt;/math&gt;.  With these substitutions the above equation becomes
:&lt;math&gt;\begin{align}
    \nabla p \cdot \mathbf{\hat{s}}
    &amp;= \tfrac{1}{\alpha\,ds}\Big({-}p\alpha + (p\!+\!dp)\alpha\Big) \\[1ex]
    &amp;= \frac{\,p\!+\!dp ~-~ p\,}{ds} = \frac{\part p}{\part s} ~;
  \end{align}&lt;/math&gt;
that is,
{{NumBlk|:|&lt;math&gt;
    \nabla p \cdot \mathbf{\hat{s}} = \part_s p \,,
  &lt;/math&gt;|{{EquationRef|9g}}}}
where the right-hand side, commonly called the '''directional derivative''' of{{mvar| p}} in the {{math|'''s&amp;#770;'''}} direction,&lt;ref&gt;[[#wilson-1901|Wilson, 1901]], pp.&amp;#8239;147–8; [[#borisenko-tarapov-68|Borisenko &amp;amp; Tarapov, 1968]], pp.&amp;#8239;147–8 (again); [[#hsu-84|Hsu, 1984]], p.&amp;#8239;92; [[#kreyszig-62-|Kreyszig, 1988]], pp.&amp;#8239;485–6; [[#wrede-spiegel-10|Wrede &amp;amp; Spiegel, 2010]], p.&amp;#8239;198.&lt;/ref&gt; is the derivative of{{mvar| p}} w.r.t. distance in that direction. Although ({{EquationNote|9g}}) has been obtained by taking that direction as fixed, the equality is evidently maintained if {{mvar|s}} measures arc length along any path ''tangential''&amp;#8202; to{{math| '''s&amp;#770;'''}} at the point of interest.

Equation ({{EquationNote|9g}}) is an alternative definition of the gradient: it says that ''the gradient of&lt;math&gt;~p&lt;/math&gt; is the vector whose scalar component in any direction is the directional derivative of&lt;math&gt;~p&lt;/math&gt; in that direction''. For ''real&lt;math&gt;~p&lt;/math&gt;'', this component has its maximum, namely {{math|{{abs|&amp;nabla;''p''}}&amp;#8202;,}} in the direction of{{math| &amp;nabla;''p''&amp;#8202;}}; thus ''the gradient of&lt;math&gt;~p&lt;/math&gt; is the vector whose direction is that in which the derivative of&lt;math&gt;~p&lt;/math&gt; w.r.t. distance is a maximum, and whose magnitude is that maximum''. This is the usual conceptual definition of the gradient.&lt;ref&gt;Gibbs ([[#gibbs-1881-4|1881]], &amp;sect;&amp;#8239;50) ''introduces'' the gradient with this definition, except that he calls {{math|&amp;nabla;''u''}} simply the ''derivative'' of{{mvar| u}}, and {{mvar|u}} the ''primitive'' of{{math| &amp;nabla;''u''}}. Use of the term ''gradient'' as an alternative to ''derivative'' is reported by Wilson ([[#wilson-1901|1901]], p.&amp;#8239;138).&lt;/ref&gt; Sometimes it is convenient to work directly from this definition. For example, in Cartesian coordinates {{math|(''x'',&amp;#8239;''y'',&amp;#8239;''z''),}} if a scalar field is given by {{mvar|x&amp;#8202;,}} its gradient is obviously the unit vector in the direction of the {{mvar|x }}axis, usually called {{math|'''i'''&amp;#8202;}}; that is, {{math|&amp;nabla;''x''&amp;#8201;{{=}}&amp;#8201;'''i'''}}. Similarly, if&amp;#8201; &lt;math&gt;\mathbf{r}=r\mathbf{\hat{r}}&lt;/math&gt;&amp;#8201; is the position vector, then &lt;math&gt;\nabla r = \mathbf{\hat{r}}&lt;/math&gt;.

If&amp;#8202; {{math|'''s&amp;#770;'''}} is ''tangential''&amp;#8202; to a '''level surface''' of{{mvar| p}} (a surface of constant{{mvar| p}}), then {{mvar|&amp;part;&lt;sub&gt;s&lt;/sub&gt;&amp;#8201;p}}&amp;#8202; in that direction is zero, in which case ({{EquationNote|9g}}) says that {{math|&amp;nabla;''p''}} (if not zero) is orthogonal to{{math| '''s&amp;#770;'''}}.&amp;#8201; So&lt;math&gt;\,\nabla p&lt;/math&gt; ''is orthogonal to the surfaces of constant&lt;math&gt;~p\,&lt;/math&gt;'' (as we would expect, having just shown that the direction of{{math| &amp;nabla;''p''}} is that in which {{mvar|p}} varies most steeply). This result leads to a method of finding a vector normal to a curved surface at a given point: if the equation of the surface is&amp;#8239; {{math|''f''&amp;#8239;('''r''')&amp;#8201;{{=}}&amp;#8201;''C''&amp;#8202;,}}&amp;#8202; where {{math|'''r''' }}is the position vector and {{mvar|C&amp;#8202; }}is a constant (possibly zero), a suitable vector is {{math|&amp;nabla;''f''}}&amp;#8202; evaluated at the given point.

If {{mvar|p}} is ''uniform''&amp;#8202;&amp;mdash;that is, if it has no spatial variation&amp;mdash;then its derivative w.r.t. distance in every direction is zero; that is, the component of{{math| &amp;nabla;''p''}} in every direction is zero, so that {{math|&amp;nabla;''p''}} must be the zero vector. In short, ''the gradient of a uniform scalar field is zero''. Conversely, if {{mvar|p}} is ''not'' uniform, there must be some location and some direction in which its derivative w.r.t. distance, if defined at all, is non-zero, so that its gradient, if defined at all, is also non-zero. Thus ''a scalar field with zero gradient in some region is uniform in that region''.

{{cob}}

=== Unambiguity of the Laplacian ===

{{cot}}

Armed with our new definition of the gradient ({{EquationNote|9g}}), we can revisit our definition of the Laplacian ({{EquationNote|4L}}). If{{mvar| &amp;psi;}} is a ''scalar'' field, then, by ({{EquationNote|9g}}),  &lt;math&gt;\part_n \psi&lt;/math&gt; can be replaced by &lt;math&gt;\nabla\psi\cdot\mathbf{\hat{n}}\,&lt;/math&gt; in ({{EquationNote|4L}}), which then becomes
{{NumBlk|:|&lt;math&gt;\triangle\psi \,=\,
    \tfrac{1}{dV}\!\iint_{\delta S} \mathbf{\hat{n}}\cdot\nabla\psi \;dS \,;
  &lt;/math&gt;|{{EquationRef|9L}}}}
that is, by definition ({{EquationNote|4d}}),
{{NumBlk|:|&lt;math&gt;\triangle\psi
    \,=\, \operatorname{div}\nabla\psi
  \qquad&lt;/math&gt;[&amp;#8202;for scalar {{mvar|&amp;psi;}}].|{{EquationRef|9L'}}}}
So ''the Laplacian of a scalar field is the divergence of the gradient''. This is the usual ''introductory'' definition of the Laplacian&amp;mdash;and on its face is applicable only in the case of a scalar field. The unambiguity of the Laplacian, in this case, follows from the unambiguity of the divergence and the gradient.

If, on the contrary, {{mvar|&amp;psi;}} in definition ({{EquationNote|4L}}) is a ''vector'' field, then we can again take dot-products with a uniform vector {{math|'''b''',}} obtaining
:&lt;math&gt;(\triangle\psi)\cdot\mathbf{b} \,=\,
    \tfrac{1}{dV}\!\iint_{\delta S} \part_n(\psi\!\cdot\!\mathbf{b}) \,dS \,.
  &lt;/math&gt;
If we make {{math|'''b'''}} a ''unit'' vector, this says that ''the scalar component of the Laplacian of a vector field, in any direction, is the Laplacian of the scalar component of that vector field in that direction''. As we have just established that the latter is unambiguous, so is the former.

But the unambiguity of the Laplacian can be generalized further. If
:{{big|{{math|''&amp;psi;'' {{=}} &amp;sum;&lt;sub&gt;''i'' &lt;/sub&gt;''&amp;alpha;&lt;sub&gt;i &lt;/sub&gt;&amp;phi;&lt;sub&gt;i&lt;/sub&gt;''}}}}
where each {{mvar|&amp;phi;&lt;sub&gt;i&lt;/sub&gt;}} is a scalar field, and each {{mvar|&amp;alpha;&lt;sub&gt;i&lt;/sub&gt;}} is a constant, and the counter {{mvar|i}} ranges from (say) 1 to{{mvar| k&amp;#8202;}}, then it is clear from ({{EquationNote|4L}}) that
{{NumBlk|:|{{big|{{math|&amp;#9651;{{big|(}}&amp;sum;&lt;sub&gt;''i'' &lt;/sub&gt;''&amp;alpha;&lt;sub&gt;i &lt;/sub&gt;&amp;phi;&lt;sub&gt;i&lt;/sub&gt;''{{big|)}} {{=}} &amp;sum;&lt;sub&gt;''i'' &lt;/sub&gt;{{big|(}}''&amp;alpha;&lt;sub&gt;i &lt;/sub&gt;''&amp;#9651;''&amp;phi;&lt;sub&gt;i&lt;/sub&gt;''{{big|)}}}} .}}|{{EquationRef|10}}}}
In words, this says that ''the Laplacian of a linear combination of fields is the same linear combination of the Laplacians of the same fields''&amp;mdash;or, more concisely, that ''the Laplacian is '''[[w:linearity|linear]]'''''. I say &quot;it is clear&quot; because the Laplacian as defined by ({{EquationNote|4L}}) is itself a linear combination, so that ({{EquationNote|10}}) merely asserts that we can regroup the terms of a nested linear combination; the gradient, curl, and divergence as defined by ({{EquationNote|4g}}) to ({{EquationNote|4d}}) are likewise linear. It follows from ({{EquationNote|10}}) that ''the Laplacian of a linear combination of fields is unambiguous if the Laplacians of the separate fields are unambiguous''. Now we have supposed that the fields {{mvar|&amp;phi;&lt;sub&gt;i&lt;/sub&gt;}} are scalar and that the coefficients {{mvar|&amp;alpha;&lt;sub&gt;i&lt;/sub&gt;}} are constants. But the same logic applies if the &quot;constants&quot; are uniform basis vectors (e.g.,{{math| '''i''',&amp;#8202;'''j''','''k'''}}), so that the &quot;linear combination&quot; can represent any vector field, whence the Laplacian of any vector field is unambiguous. And the same logic applies if the &quot;constants&quot; are chosen as a &quot;basis&quot; for a space of tensors of any order, so that the Laplacian of any tensor field of that order is unambiguous, and so on. In short, ''the Laplacian of any field that we can express with a uniform basis is unambiguous''.

{{cob}}

=== The dot-del, del-cross, and del-dot operators ===

{{cot}}

The gradient operator {{math|&amp;nabla;}} is also called {{mvar|'''del'''}}.{{efn|Or ''nabla'', because it allegedly looks like the ancient Phoenician harp that the Greeks called by that name.}} If it simply denotes the gradient, we tend to pronounce it &quot;grad&quot; in order to emphasize the result. But it can also appear in combination with other operators to give other results, and in those contexts we tend to pronounce it &quot;del&quot;.

One such combination is &quot;dot del&quot;&amp;mdash;&amp;#8202;as in &quot;{{math|&amp;#8202;'''b&amp;sdot;'''&amp;nabla;&amp;#8202;}}&quot;, which we proposed for ({{EquationNote|8q}}), but did not quite manage to define satisfactorily for a vector operand. With our new definition of the gradient ({{EquationNote|9g}}), we can now make a second attempt. A general vector field {{math|'''q'''}} can be written &lt;math&gt;|\mathbf{q}|\,\mathbf{\hat{q}}~\!,&lt;/math&gt; so that
:&lt;math&gt;\mathbf{q}\cdot\nabla\psi
    \,=\, |\mathbf{q}| \,\mathbf{\hat{q}}\cdot\nabla\psi \,.
  &lt;/math&gt;
If {{mvar|&amp;psi;}} is a ''scalar'' field, we can apply ({{EquationNote|9g}}) to the right-hand side, obtaining
:{{big|&lt;math&gt;\mathbf{q}\cdot\nabla\psi
    ~\!=~\! |\mathbf{q}| \,\part_{s_q} \psi \,,
  &lt;/math&gt;}}
where {{mvar|s&lt;sub&gt;q&lt;/sub&gt;}} is distance in the direction of{{math| '''q'''}}. For ''scalar'' {{mvar|&amp;psi;}}, this result is an identity between previously defined quantities. For ''non-scalar'' {{mvar|&amp;psi;}}, we have not yet defined the left-hand side, but the right-hand side is still well-defined and self-explanatory (provided that we can differentiate {{mvar|&amp;psi;}} w.r.t.{{mvar| s&lt;sub&gt;q&lt;/sub&gt;}}). So we are free to adopt
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,\psi \,:=\, |\mathbf{q}| \,\part_{s_q} \psi
  &lt;/math&gt;}}|{{EquationRef|11}}}}
(where {{mvar|s&lt;sub&gt;q&lt;/sub&gt;}} is distance in the direction of{{math| '''q'''}}) as the general definition of the ''operator'' {{math|'''q&amp;sdot;'''&amp;nabla;&amp;#8202;,}} and to interpret it as defining both a ''unary'' operator&amp;#8202; {{math|'''q&amp;sdot;'''&amp;nabla;}} which operates on a generic field, and a ''binary'' operator&amp;#8202; {{math|'''&amp;sdot;'''&amp;nabla;}} which takes a (possibly uniform) vector field on the left and a generic field on the right.

For any vector field {{math|'''q'''&amp;#8202;,}} it follows from ({{EquationNote|11}}) that ''if&lt;math&gt;~\psi&lt;/math&gt; is a uniform field, then&lt;math&gt;\,\,\mathbf{q}\;\!{\cdot}\nabla~\!\psi=0&lt;/math&gt;''.

For the special case in which {{math|'''q'''}} is a unit vector {{math|'''s&amp;#770;'''&amp;#8202;,}} with {{mvar|s}} measuring distance in the direction of{{math|&amp;#8202; '''s&amp;#770;'''&amp;#8202;,}} definition ({{EquationNote|11}}) reduces to
{{NumBlk|:|&lt;math&gt;\mathbf{\hat{s}}{\cdot}\nabla\,\psi = \part_s \psi \,,
  &lt;/math&gt;|{{EquationRef|12}}}}
which agrees with ({{EquationNote|9g}}) but now holds for a ''generic'' field {{mvar|&amp;psi;}} [whereas ({{EquationNote|9g}}) was for a ''scalar'' field, and was derived as a ''theorem'' based on earlier definitions]. So{{math| '''s&amp;#770;&amp;sdot;'''&amp;nabla;&amp;#8202;,}} with a unit vector {{math|'''s'''&amp;#8202;,}} is the '''directional-derivative operator''' on a generic field; and by ({{EquationNote|11}}),&amp;#8201; {{math|'''q&amp;sdot;'''&amp;nabla;}} is a '''scaled directional derivative''' operator on a generic field.

In particular, if&amp;#8202; {{math|'''s&amp;#770;'''}} is &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt;,&amp;#8201; we have
:&lt;math&gt;\part_n \psi \,=\, \mathbf{\hat{n}}\;\!{\cdot}\nabla\,\psi \,,&lt;/math&gt;
which we may substitute into the original definition of the Laplacian ({{EquationNote|4L}}) to obtain
{{NumBlk|:|&lt;math&gt;\triangle\psi \,=\,
    \tfrac{1}{dV}\!\iint_{\delta S} \mathbf{\hat{n}}{\cdot}\nabla\,\psi \;dS \,,
  &lt;/math&gt;|{{EquationRef|13L}}}}
which is just ({{EquationNote|9L}}) again, except that it now holds for for a ''generic'' field.

If our general definition of the gradient ({{EquationNote|4g}}) is also taken as the general definition of the {{math|&amp;nabla;}} operator,&lt;ref&gt;''Cf''. [[#borisenko-tarapov-68|Borisenko &amp;amp; Tarapov, 1968]], p.&amp;#8239;157, eq.&amp;#8239;(4.43), quoted in [[#tai-95|Tai, 1995]], p.&amp;#8239;33, eq.&amp;#8239;(4.19).&lt;/ref&gt; then, comparing ({{EquationNote|4g}}) with ({{EquationNote|4c}}), ({{EquationNote|4d}}), and ({{EquationNote|13L}}), we see that
:&lt;math&gt;\begin{align}
    \operatorname{curl}\mathbf{q} ~\!&amp;= \nabla(\times\mathbf{q}) \\
    \operatorname{div}\mathbf{q}  ~\!&amp;= \nabla(\cdot\,\mathbf{q}) \\
    \triangle\psi                 ~\!&amp;= \nabla(\cdot\nabla\,\psi) \,,
  \end{align}&lt;/math&gt;
where the parentheses may seem to be required on account of the closing {{mvar|dS}}&amp;#8202; in ({{EquationNote|4g}}).&lt;ref&gt;The first two cases may be compared with Javid &amp;amp; Brown, 1963, cited in [[#tai-94|Tai, 1994]], p.&amp;#8239;15.&lt;/ref&gt; But if we write the factor {{mvar|dS}} ''before'' the integrand, the del operator in ({{EquationNote|4g}}) becomes
:&lt;math&gt;\nabla = \tfrac{1}{dV}\!\iint_{\delta S} dS\,\mathbf{\hat{n}} &lt;/math&gt;
&amp;mdash;''if''&amp;#8202; we insist that it is to be read as a operator looking for an operand, and not as a self-contained expression. Then, if we similarly bring forward the {{mvar|dS}} in ({{EquationNote|4c}}), ({{EquationNote|4d}}), and ({{EquationNote|13L}}), the respective operators become&lt;ref&gt;The first two cases may be compared with Neff, 1991, cited in [[#tai-94|Tai, 1994]], p.&amp;#8239;16.&lt;/ref&gt;
{{NumBlk|:|&lt;math&gt;\begin{align}
    \operatorname{curl} &amp;= \nabla\times \\
    \operatorname{div}  &amp;= \nabla~\!\boldsymbol{\cdot} \\
    \triangle           &amp;= \nabla\boldsymbol{\cdot}\nabla
  \end{align}&lt;/math&gt;|{{EquationRef|14}}}}
(pronounced &quot;del cross&quot;, &quot;del dot&quot;, and &quot;del dot del&quot;), of which the last is usually abbreviated as{{math| &amp;nabla;&lt;sup&gt;2&lt;/sup&gt;}}&amp;#8201; (&quot;del squared&quot;).&lt;ref&gt;But Gibbs ([[#gibbs-1881-4|1881]]) and Wilson ([[#wilson-1901|1901]]) were content to leave it as {{math|&amp;nabla;'''&amp;sdot;'''&amp;nabla;}}.  And they did not call it the ''Laplacian''; they used that term with a different meaning, which has apparently fallen out of fashion.&lt;/ref&gt; These notations are ubiquitous.

Another way to obtain the {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; operators (but ''not''{{math|&amp;#8202; &amp;nabla;&lt;sup&gt;2&lt;/sup&gt;}}), again inspired by ({{EquationNote|4g}}), is to define
{{NumBlk|:|&lt;math&gt;T(\nabla)
    \,:=\, \tfrac{1}{dV}\!\iint_{\delta S} T(\mathbf{\hat{n}}) \,dS \,,
  &lt;/math&gt;|{{EquationRef|14s}}}}
where {{mvar|T}}&amp;#8202; is any well-defined function that takes a vector argument. Setting{{math|&amp;#8202; ''T''&amp;#8202;(&amp;nabla;)}} to {{math|&amp;nabla;''p''&amp;#8202;,}} {{math|&amp;nabla;&amp;#8202;&amp;times;&amp;#8202;'''q'''&amp;#8202;,}} and{{math| &amp;nabla;'''&amp;sdot;&amp;#8239;q'''}}&amp;#8239; in ({{EquationNote|14s}}), we obtain respectively {{math|&amp;nabla;''p''&amp;#8202;,}}&amp;#8202; {{math|curl&amp;#8201;'''q'''&amp;#8202;,}} and&amp;#8202; {{math|div&amp;#8239;'''q'''}}&amp;#8239; as given by ({{EquationNote|4g}}) to ({{EquationNote|4d}}). But this approach has undesirable side-effects&amp;mdash;for example, that {{math|&amp;nabla;''p''}}&amp;#8202; becomes synonymous with{{math| ''p''&amp;nabla;}}.&amp;#8201; Accordingly, Chen-To Tai,&lt;ref&gt;[[#tai-fang-91|Tai &amp;amp; Fang, 1991]], pp.&amp;#8239;168–9.&lt;/ref&gt; on the left of ({{EquationNote|14s}}), replaces{{math| &amp;nabla;}} with his original symbol &lt;math&gt;\nabla\!\!\!\!^{\textstyle_-}~\!\!,\,&lt;/math&gt; which he calls the &quot;symbolic operator&quot; or the &quot;{{nowrap|''S''&amp;#8202;-operator}}&quot; or, later, the &quot;symbolic vector&quot; or the &quot;dummy vector&quot;. Tai in his later works (e.g.,&amp;#8239;[[#tai-94|1994]],&amp;#8239;[[#tai-95|1995]]) does not tolerate cross- or dot-products involving the del operator, but ''does'' tolerate such products involving his symbolic vector ([[#tai-95|1995]], pp.&amp;#8239;50–52).

There is a misconception that the operational equivalences in ({{EquationNote|14}}) apply ''only'' in Cartesian coordinates.&lt;ref&gt;Durney &amp;amp; Johnson, in ''Introduction to Modern Electromagnetics'' (1969, p.&amp;#8239;45, cited in [[#tai-94|Tai, 1994]], p.&amp;#8239;12), make the absurd statement that &quot;a{{math| &amp;nabla;}} operator cannot be defined in the other coordinate systems&amp;hellip;&quot; In the context, they apparently meant to say that&amp;#8202; {{math|div&amp;#8202;'''A'''}} isn't&amp;#8202; {{math|&amp;nabla;'''&amp;sdot;A'''}}&amp;#8202; in other coordinate systems. Robert S. Elliott, in ''Electromagnetics'' (1966, p.&amp;#8239;606, cited in [[#tai-94|Tai, 1994]], p.&amp;#8239;13), says that &quot;only in Cartesian coordinates&amp;hellip; do the gradient and divergence operators turn out to be identical.&quot; Apparently he meant to say that only in Cartesian coordinates do the two operators differ by a dot. But what these authors apparently meant to say is still wrong, as shown with counterexamples by Kemmer (next citation).&lt;/ref&gt; Tai does not accept them even in that case. But, because these equivalences have been derived from ''coordinate-free'' definitions of the operators, they must remain valid in any coordinate system ''provided that they are expressed correctly''&amp;mdash;without (e.g.) inadvertently taking dependent variables inside or outside differentiations.&lt;ref&gt;The perception that they are restricted to Cartesian coordinates arises partly from failure to allow for the variability of the basis vectors in curvilinear coordinate systems; ''cf''. [[#kemmer-77|Kemmer, 1977]], pp.&amp;#8239;163–5, 172–3 (Exs.&amp;#8239;2,&amp;#8239;3,&amp;#8239;5), 230–33 (sol'ns). From the del operator and the derivatives of the basis vectors w.r.t. the coordinates, Kemmer finds the curl and divergence in cylindrical coordinates, notes that we can do the same &quot;with a little greater effort&quot; in spherical coordinates (p.&amp;#8239;230), and finds the Laplacian of a scalar in both coordinate systems (p.&amp;#8239;231). He further reports that the method works for the Laplacian of a vector in cylindrical and spherical coordinates and is relatively convenient for the former (p.&amp;#8239;232), for which &quot;differentiation of the unit vectors is very simple&quot; (p.&amp;#8239;165).&lt;/ref&gt; That does ''not'' mean that they are always convenient, or easily verified, or conducive to the avoidance of error. But they sometimes make useful mnemonics; e.g., they let us rewrite identities ({{EquationNote|8c}}), ({{EquationNote|8g}}), and ({{EquationNote|8p}}) as
{{NumBlk|:|&lt;math&gt;\left.\begin{align}
    \nabla\!\times\!\mathbf{q}\cdot\mathbf{b}
    &amp;\,=\, \nabla\cdot\mathbf{q}\!\times\!\mathbf{b}\\[.5ex]
    \nabla p \cdot \mathbf{b}
    &amp;\,=\, \nabla\cdot\;\! p\mathbf{b}\\[.5ex]
    \nabla p \times \mathbf{b}
    &amp;\,=\, \nabla \times p\mathbf{b}
  \end{align}~\right\}\quad&lt;/math&gt;for uniform {{math|'''b'''}}.
    |{{EquationRef|15}}}}
These would be basic ''algebraic'' vector identities if&amp;#8202; {{math|&amp;nabla;}} were an ordinary vector, and one could try to derive them from the &quot;algebraic&quot; behavior of{{math| &amp;nabla;}}; but they're not, because it isn't, so we didn't&amp;#8239;!  Moreover, these simple &quot;algebraic&quot; rules are for a uniform {{math|'''b''',}} and do not of themselves tell us what to do if&amp;#8202; {{math|'''b'''}} is spatially variable; for example, ({{EquationNote|8g}}) is not applicable to ({{EquationNote|7d}}).

{{cob}}

=== The advection operator ===

{{cot}}

Variation or transportation of a property of a medium due to motion with the medium is called '''advection''' (which, according to its Latin roots, means &quot;carrying to&quot;). Suppose that a medium (possibly a fluid) moves with a velocity field {{math|'''v'''}} in some inertial reference frame. Let {{mvar|&amp;psi;}} be a field (possibly a scalar field or a vector field) expressing some property of the medium (e.g., density, or acceleration, or stress,{{efn|Stress is a second-order tensor, and the origin of the term &quot;tensor&quot;; but, for present purposes, it's just another possible example of a field called{{mvar| &amp;psi;}}.}}&amp;hellip; or even {{math|'''v''' }}itself). We have seen that the time-derivative of{{mvar| &amp;psi;}} may be specified in two different ways: as the ''partial'' derivative {{mvar|{{sfrac|&amp;part;&amp;psi;|&amp;part;t}}&amp;#8202;,}} evaluated at a fixed point (in the chosen reference frame), or as the ''material'' derivative {{mvar|{{sfrac|d&amp;psi;|dt}}&amp;#8202;}}, evaluated at a point moving at velocity {{math|'''v'''}} (i.e., ''with the medium''). The difference&amp;#8202; {{mvar|{{sfrac|d&amp;psi;|dt}}&amp;#8202;&amp;minus;&amp;#8202;{{sfrac|&amp;part;&amp;psi;|&amp;part;t}}&amp;#8202;}} is due to motion with the medium. To find another expression for this difference, let {{mvar|s}} be a parameter measuring distance along the path traveled by a particle of the medium. Then, for points along the path, the surface-plot of the small change in {{mvar|&amp;psi;}} (or any component thereof) as a function of small changes in {{mvar|t}} and {{mvar|s&amp;#8202;}} (plotted on perpendicular axes) can be taken as a plane through the origin, so that
:{{big|&lt;math&gt;d\psi
    = \tfrac{\part\psi}{\part t}~\!dt + \tfrac{\part\psi}{\part s}~\!ds \;;
  &lt;/math&gt;}}
that is, the change in {{mvar|&amp;psi;}} is the sum of the changes due to the change in {{mvar|t}} and the change in {{mvar|s&amp;#8202;}}. Dividing by {{mvar|dt}} gives
:{{big|&lt;math&gt;\begin{align}\tfrac{d\psi}{dt}
  &amp;= \tfrac{\part\psi}{\part t}+\tfrac{\part\psi}{\part s}~\!\tfrac{ds}{dt}\\[1ex]
  &amp;= \tfrac{\part\psi}{\part t}+\tfrac{\part\psi}{\part s}~\!|\mathbf{v}| \,;
  \end{align}&lt;/math&gt;}}
i.e.,
:{{big|&lt;math&gt;\tfrac{d\psi}{dt}
    = \tfrac{\part\psi}{\part t} + |\mathbf{v}|\,\part_s \psi
  &lt;/math&gt;}}
(and the first term on the right could have been written {{mvar|&amp;part;&lt;sub&gt;t&lt;/sub&gt;&amp;#8239;&amp;psi;}}). So the second term on the right is the contribution to the material derivative due to motion with the medium; it is called the '''advective term''', and is non-zero wherever a particle of the medium moves along a path on which {{mvar|&amp;psi;}} varies with location&amp;mdash;even if {{mvar|&amp;psi;}} at ''each'' location is constant over time.&amp;#8201; So the operator&amp;#8202; {{math|{{abs|'''v'''}}&amp;#8201;''&amp;part;&lt;sub&gt;s&lt;/sub&gt;''&amp;#8202;,}} where {{mvar|s}} measures distance along the path, is the ''advection operator''&amp;#8239;: it maps a property of a medium to the advective term in the time-derivative of that property. If{{mvar| &amp;psi;}} is {{math|'''v''' }}itself, the above result becomes
:{{big|&lt;math&gt;\tfrac{d\mathbf{v}}{dt}
    = \tfrac{\part\mathbf{v}}{\part t} + |\mathbf{v}|\,\part_s \mathbf{v} \,,
  &lt;/math&gt;}}
where the left-hand side (the ''material'' acceleration) is as given by Newton's second law, and the first term on the right (which we might call the &quot;partial&quot; acceleration) is the time-derivative of velocity in the chosen reference frame, and the second term on the right (the ''advective'' term) is the correction that must be added to the &quot;partial&quot; acceleration in order to obtain the material acceleration. This term is non-zero wherever velocity is non-zero and varies along a path, even if the velocity at each point on the path is constant over time (as when water speeds up while flowing at a constant volumetric rate into a nozzle). Paradoxically, while the material acceleration and the &quot;partial&quot; acceleration are apparently linear (first-degree) in {{math|'''v''',}} their difference (the advective term) is not. Thus the distinction between {{mvar|{{sfrac|&amp;part;&amp;psi;|&amp;part;t}}}} and {{mvar|{{sfrac|d&amp;psi;|dt}}}}&amp;#8202; has the far-reaching implication that ''fluid dynamics is non-linear''.

Applying ({{EquationNote|11}}) to the last two equations, we obtain respectively
{{NumBlk|:|{{big|&lt;math&gt;\tfrac{d\psi}{dt}
    = \tfrac{\part\psi}{\part t} + \mathbf{v}{\cdot}\nabla\,\psi
  &lt;/math&gt;}}|{{EquationRef|16}}}}
and
{{NumBlk|:|{{big|&lt;math&gt;\tfrac{d\mathbf{v}}{dt}
    = \tfrac{\part\mathbf{v}}{\part t} + \mathbf{v}{\cdot}\nabla\,\mathbf{v} \,,
  &lt;/math&gt;}}|{{EquationRef|16v}}}}
where, in each case, the second term on the right is the advective term. So ''the '''advection operator''' can also be written''&amp;#8239;{{math| '''v&amp;sdot;'''&amp;nabla;&amp;#8202;}}.

When the generic {{mvar|&amp;psi;&amp;#8202;}} in ({{EquationNote|16}}) is replaced by the density {{mvar|&amp;rho;&amp;#8202;}}, we get a relation between {{mvar|{{sfrac|&amp;part;&amp;rho;|&amp;part;t}}&amp;#8202;}} and {{mvar|{{sfrac|d&amp;rho;|dt}}&amp;#8202;}}, both of which we have seen before&amp;mdash;in equations ({{EquationNote|7d}}) and ({{EquationNote|7d'}}) above. Substituting from those equations then gives
{{NumBlk|:|&lt;math&gt;
    \operatorname{div}\rho\mathbf{v}
    \,=\, \rho\operatorname{div}\mathbf{v} \,+\, \mathbf{v}\cdot\nabla\rho \,,
  &lt;/math&gt;|{{EquationRef|17}}}}
where {{math|&amp;nabla;''&amp;rho;''}} can be taken as a gradient since {{mvar|&amp;rho;}} is scalar. This result is in fact an identity&amp;mdash;a ''product rule for the divergence''&amp;mdash;as we shall eventually confirm by another method.

{{cob}}

=== Generalized volume-integral theorem ===

{{cot}}

We can rewrite the fourth integral theorem ({{EquationNote|5L}}) in the &quot;dot del&quot; notation as
{{NumBlk|:|&lt;math&gt;\iint_S \mathbf{\hat{n}}\;\!{\cdot}\nabla\,\psi \;dS
    \,= \iiint_V \triangle\psi ~dV \,.
  &lt;/math&gt;|{{EquationRef|18L}}}}
Then, using notations ({{EquationNote|14}}), we can condense ''all four'' integral theorems ({{EquationNote|5g}}), ({{EquationNote|5c}}), ({{EquationNote|5d}}), and ({{EquationNote|18L}}) into the single equation
{{NumBlk|:|&lt;math&gt;\iint_S \mathbf{\hat{n}} * \psi \;dS
    \,= \iiint_V \nabla * \psi ~dV \,,
  &lt;/math&gt;|{{EquationRef|19}}}}
where the wildcard {{math|&amp;lowast;}} (conveniently pronounced &quot;star&quot;) is a generic binary operator which may be replaced by a null (direct juxtaposition of the operands) for theorem ({{EquationNote|5g}}), or a cross for ({{EquationNote|5c}}), or a dot for ({{EquationNote|5d}}), or&amp;#8202; {{math|'''&amp;sdot;'''&amp;nabla;}} for ({{EquationNote|18L}}); and the operand {{mvar|&amp;psi;}} is of a kind that makes the operator meaningful. This single equation is a ''generalized volume-integral theorem'', relating an integral over a volume to an integral over its enclosing surface.&lt;ref&gt;Kemmer ([[#kemmer-77|1977]], p.&amp;#8239;98, eq.&amp;#8239;4) gives an equivalent result for our first three integral theorems ({{EquationNote|5g}} to {{EquationNote|5d}}) only, and calls it the ''generalized divergence theorem'' because the divergence theorem is its most familiar special case.&lt;/ref&gt;

Theorem ({{EquationNote|19}}) is based on the following definitions, which have been found unambiguous:
* the ''gradient'' of a scalar field {{mvar|p}} is the closed-surface integral of&amp;#8202; &lt;math&gt;\mathbf{\hat{n}}p\,&lt;/math&gt; per unit volume, where &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is the outward unit normal;
* the ''curl'' of a vector field is the skew surface integral per unit volume, also called the surface circulation per unit volume;
* the ''divergence'' of a vector field is the outward flux integral per unit volume; and
* the ''Laplacian'' is the closed-surface integral of the outward normal derivative, per unit volume.
The gradient maps a scalar field to a vector field; the curl maps a vector field to a vector field; the divergence maps a vector field to a scalar field; and the Laplacian maps a scalar field to a scalar field, or a vector field to a vector field, etc.

The ''gradient'' of {{mvar|p}}, as defined above, has been shown to be also
* the vector whose (scalar) component in any direction is the ''directional derivative'' of{{mvar| p}} in that direction (i.e. the derivative of{{mvar| p}} w.r.t. distance in that direction), and
* the vector whose direction is that in which the directional derivative of{{mvar| p}} is a maximum, and whose magnitude is that maximum.
Consistent with these alternative definitions of the gradient, we have defined the&amp;#8202;{{math| '''&amp;sdot;'''&amp;nabla;}} operator so that&amp;#8202; {{math|'''s&amp;#770;&amp;sdot;'''&amp;nabla;}} (for a ''unit'' vector {{math|'''s&amp;#770;'''}}) is the operator yielding the directional derivative in the direction of&amp;#8202; {{math|'''s&amp;#770;'''&amp;#8202;,}} and we have used that notation to bring theorem ({{EquationNote|5L}}) under theorem ({{EquationNote|19}}).

So far, we have said comparatively little about the curl. That imbalance will now be rectified.

{{cob}}

== Closed-circuit integrals per unit area ==

=== Instant integral theorems (on a condition) ===

{{cot}}

Theorems ({{EquationNote|5g}}) to ({{EquationNote|5L}}) are three-dimensional: each of them relates an integral over a volume {{mvar|V}}&amp;#8202; to an integral over its enclosing surface{{mvar| S}}. We now seek analogous ''two''-dimensional theorems, each of which relates an integral over a surface segment to an integral around its enclosing curve. For maximum generality, the surface segment should be allowed to be curved into a third dimension.{{efn|In mathematical jargon, it should be a two-dimensional ''manifold'' embedded in 3D Euclidean space.}} Theorems of the latter kind can be obtained as special cases of theorems ({{EquationNote|5g}}) to ({{EquationNote|5L}}) by suitably choosing {{mvar|V}} and {{mvar|S&amp;#8202;}}; this is another advantage of our &quot;volume first&quot; approach.

Let {{mvar|&amp;Sigma;}} be a surface segment enclosed by a curve {{mvar|C}} (a ''circuit'' or ''closed contour''), and let {{mvar|l}} be a parameter measuring arc length around {{mvar|C&amp;#8202;}}, so that a general element of{{mvar| C}}&amp;#8202; has length{{mvar| dl&amp;#8202;}}; and let a general element of the surface {{mvar|&amp;Sigma;}}&amp;#8202; have area {{mvar|d&amp;Sigma;}}. Let&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt; be the unit normal vector at a general point on {{mvar|&amp;Sigma;&amp;#8202;}}, and let &lt;math&gt;\mathbf{\hat{t}}&lt;/math&gt; be the unit ''tangent'' vector to{{mvar| C}} at a general point on {{mvar|C}}&amp;#8202; in the direction of increasing{{mvar| l}}. In the original case of a surface enclosing a volume, we had to decide whether the unit normal pointed into or out of the volume (we chose the latter). In the present case of a circuit enclosing a surface segment, we have to decide whether {{mvar|l}} is measured clockwise or counterclockwise as seen when looking in the direction of the unit normal, and we choose clockwise. So {{mvar|l }}''is measured clockwise about&lt;math&gt;~\boldsymbol{\hat{\nu}},&lt;/math&gt;'' and {{mvar|C }}is ''traversed'' clockwise about&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt;.

From {{mvar|&amp;Sigma;}}&amp;#8202; we can construct obvious candidates for {{mvar|V}} and{{mvar| S}}. From every point on {{mvar|&amp;Sigma;&amp;#8202;}}, erect a perpendicular with a uniform ''small''&amp;#8202; height {{mvar|h}} in the direction of&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt;. Then simply let {{mvar|V}} be the volume occupied by all the perpendiculars, and let {{mvar|S}} be its enclosing surface. Thus {{mvar|V}} is a (generally curved) thin slab of uniform thickness{{mvar| h}}, whose enclosing surface {{mvar|S}} consists of two close parallel (generally curved) broad faces connected by a perpendicular ''edge-face'' of uniform height{{mvar| h&amp;#8202;}}; and we can treat&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt; as a vector ''field''&amp;#8202; by extrapolating it perpendicularly from{{mvar| &amp;Sigma;}}. If we can arrange for {{mvar|h}} to cancel out, the volume{{mvar| V}}&amp;#8202; will serve as a 3D representation of the surface segment{{mvar| &amp;Sigma;}}&amp;#8202; while the ''edge-face'' will serve as a 2D representation of the curve{{mvar| C&amp;#8202;}}, so that our four theorems will relate an integral around {{mvar|C}}&amp;#8202; to an integral over {{mvar|&amp;Sigma;}}&amp;#8201; ''provided that there is no contribution from the broad faces to the integral over''{{mvar| S}}. For brevity, let us call this proviso the '''2D condition'''.

''If''&amp;#8202; the 2D condition is satisfied, an integral over the new {{mvar|S}}&amp;#8202; reduces to an integral over the edge-face, on which
:&lt;math&gt;dS = h\,dl \,,&lt;/math&gt;
so that the cancellation of{{mvar| h}} will leave an integral over {{mvar|C}}&amp;#8202; w.r.t. length. Meanwhile, in an integral over the new{{mvar| V}}, regardless of the 2D condition, we have
:&lt;math&gt;dV = h\,d\varSigma \,,&lt;/math&gt;
so that the cancellation of{{mvar| h}} will leave an integral over {{mvar|&amp;Sigma;}}&amp;#8202; w.r.t. area. So, substituting for {{mvar|dS}} and {{mvar|dV}}&amp;#8202; in ({{EquationNote|5g}}) to ({{EquationNote|5L}}), and canceling {{mvar|h}} as planned, we obtain respectively
{{NumBlk|:|&lt;math&gt;~~~~~\!\oint_C \mathbf{\hat{n}}~\!p \,dl
    \,= \iint_{\varSigma} \nabla p ~d\varSigma \qquad(?),
  &lt;/math&gt;|{{EquationRef|20g}}}}
{{NumBlk|:|&lt;math&gt;\oint_C \mathbf{\hat{n}}\times\mathbf{q} \,dl
    \,= \iint_{\varSigma} \operatorname{curl}\mathbf{q} ~d\varSigma \qquad(?),
  &lt;/math&gt;|{{EquationRef|20c}}}}
{{NumBlk|:|&lt;math&gt;~~\!\oint_C \mathbf{\hat{n}\cdot q} \,dl
    \,= \iint_{\varSigma} \operatorname{div}\mathbf{q} ~d\varSigma \qquad(?),
  &lt;/math&gt;|{{EquationRef|20d}}}}
{{NumBlk|:|&lt;math&gt;~~\oint_C \part_n \psi \;dl
    \,= \iint_{\varSigma} \triangle\psi ~d\varSigma \qquad(?),
  &lt;/math&gt;|{{EquationRef|20L}}}}
''all subject to the 2D condition'' (hence the question marks). In each equation, the circle on the left integral sign acknowledges that the integral is around a closed loop. The unit vector &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt;, which ''was'' normal to the edge-face, is now normal to both &lt;math&gt;\mathbf{\hat{t}}&lt;/math&gt; and&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt;; that is, &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is tangential to the surface segment {{mvar|&amp;Sigma;}}&amp;#8202; and projects perpendicularly outward from its bounding curve.

On the left side of ({{EquationNote|20g}}), the 2D condition is satisfied if (but not only if)&amp;#8202; &lt;math&gt;\mathbf{\hat{n}}p&lt;/math&gt; takes equal-and-opposite values at any two opposing points on opposing broad faces of{{mvar| S&amp;#8202;,}} i.e. if {{mvar|p}} takes the ''same'' value at such points, i.e. if {{mvar|p}} has a zero directional derivative normal to{{mvar| &amp;Sigma;}}.

Skipping forward to ({{EquationNote|20L}}), we see that the 2D condition is satisfied if&lt;math&gt;~\part_n \psi&lt;/math&gt; takes equal-and-opposite values at any two opposing points on opposing broad faces of{{mvar| S&amp;#8202;,}} i.e. if&lt;math&gt;~\part_{\nu}\psi&lt;/math&gt; (where &lt;math&gt;\nu&lt;/math&gt; measures distance in the direction of&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt;) takes the ''same'' value at such points, i.e. if&lt;math&gt;~\part^2_{\nu}\psi\!=\!0&lt;/math&gt;.

For ({{EquationNote|20c}}) and ({{EquationNote|20d}}), the 2D condition can be satisfied by construction, with more useful results&amp;mdash;as explained under the next two headings. To facilitate this process, we first make a minor adjustment to {{mvar|&amp;Sigma;}} and{{mvar| C}}. Noting that any curved surface segment can be approximated to any desired accuracy by a ''polyhedral'' surface enclosed by a ''polygon'', we shall indeed consider {{mvar|&amp;Sigma;}}&amp;#8202; to be a polyhedral surface made up of small planar elements, {{mvar|d&amp;Sigma;}}&amp;#8202; being the area of a general element, and we shall indeed consider {{mvar|C}} to be a polygon with short sides, {{mvar|dl}} being the length of a general side.{{efn|If any part of our argument requires {{mvar|&amp;Sigma;}} or {{mvar|C}} to be ''smooth'', this is not an impediment, because having approximated {{mvar|&amp;Sigma;}} or{{mvar| C}} to any desired accuracy by a polyhedron or polygon, we can then approximate the polyhedron or polygon to any desired ''higher'' accuracy by a smooth surface or curve!}} The benefit of this trick, as we shall see, is to make the unit normal &lt;math&gt;\boldsymbol{\hat{\nu}}&lt;/math&gt; uniform over each surface element, without forcing us to treat {{math|'''q'''}} (or any other field) as uniform over the same element. But, as the elements of{{mvar| C}}&amp;#8202; can ''independently'' be made as short as we like (dividing straight sides into shorter elements if necessary!), we can still consider &lt;math&gt;\boldsymbol{\hat{\nu}},&lt;/math&gt; {{math|'''q'''&amp;#8202;,}} and &lt;math&gt;\mathbf{\hat{t}}&lt;/math&gt; to be uniform over each element of{{mvar| C}}.

{{cob}}

=== Special case for the gradient ===

{{cot}}

In ({{EquationNote|20c}}), the 2D condition is satisfied by&lt;math&gt;~\mathbf{q}\!=\!p\boldsymbol{\hat{\nu}}&lt;/math&gt; (where {{mvar|p}} is a scalar field), because then the integrand on the left is zero on the broad faces of{{mvar| S&amp;#8202;}}, where {{math|'''n'''}} is parallel to&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt;. Equation ({{EquationNote|20c}}) then becomes
{{NumBlk|:|&lt;math&gt;
    \oint_C \mathbf{\hat{n}}{\times}\boldsymbol{\hat{\nu}}~\!p \;dl \,=
    \iint_{\varSigma}\operatorname{curl}p\boldsymbol{\hat{\nu}}\;d\varSigma \,.
  &lt;/math&gt;|{{EquationRef|21n}}}}
Now on the left,&amp;#8201; &lt;math&gt;\mathbf{\hat{n}}\!\times\!\boldsymbol{\hat{\nu}}\!=\!-\mathbf{\hat{t}}~\!;\,&lt;/math&gt; and on the right, over each surface element, the unit normal &lt;math&gt;\boldsymbol{\hat{\nu}}&lt;/math&gt; is uniform so that, by ({{EquationNote|8p}}),&amp;#8201; &lt;math&gt;\operatorname{curl}p\boldsymbol{\hat{\nu}}=~\!\!\nabla p \!\times\!\boldsymbol{\hat{\nu}}=-\boldsymbol{\hat{\nu}}\!\times\!\nabla p&lt;/math&gt;.  With these substitutions, the minus signs cancel and we get
{{NumBlk|:|&lt;math&gt;
    \oint_C p\mathbf{\hat{t}} \,dl \,=
    \iint_{\varSigma} \boldsymbol{\hat{\nu}}\times\nabla p \;d\varSigma
  &lt;/math&gt;|{{EquationRef|21g}}}}
or, if we write&amp;#8201; &lt;math&gt;d\mathbf{r}\!=\!\mathbf{\hat{t}}~\!dl&lt;/math&gt;&amp;#8201; and&amp;#8201; &lt;math&gt;\boldsymbol{d\varSigma}\!=\!\boldsymbol{\hat{\nu}}\,d\varSigma~\!,&lt;/math&gt;
{{NumBlk|:|&lt;math&gt;
    \oint_C p \,d\mathbf{r} \,=
    \iint_{\varSigma} \big(\boldsymbol{d\varSigma}\times\!\nabla p\big) \,.
  &lt;/math&gt;|{{EquationRef|21r}}}}
This result, although well attested in the literature,&lt;ref&gt;E.g., [[#gibbs-1881-4|Gibbs, 1884]], &amp;sect;&amp;#8239;165, eq.&amp;#8239;(1); [[#wilson-1901|Wilson, 1901]], p.&amp;#8239;255, Ex.&amp;#8239;1; [[#kemmer-77|Kemmer, 1977]], p.&amp;#8239;99, eq.&amp;#8239;(6); [[#hsu-84|Hsu, 1984]], p.&amp;#8239;146, eq.&amp;#8239;(7.31).&lt;/ref&gt; does not seem to have a name&amp;mdash;unlike the next result.

{{cob}}

=== Special case for the curl ===

{{cot}}

In ({{EquationNote|20d}}), the 2D condition is satisfied if {{math|'''q'''}} is replaced by&lt;math&gt;~\boldsymbol{\hat{\nu}}{\times}\mathbf{q}~\!,\,&lt;/math&gt; because then (again) the integrand on the left is zero on the broad faces of{{mvar| S&amp;#8202;}}, where {{math|'''n'''}} is parallel to&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt;. Equation ({{EquationNote|20d}}) then becomes
{{NumBlk|:|&lt;math&gt;
    \oint_C \mathbf{\hat{n}}\cdot\boldsymbol{\hat{\nu}}{\times}\mathbf{q} \;dl
    \,=
    \iint_{\varSigma}
      \operatorname{div}(\boldsymbol{\hat{\nu}}\!\times\!\mathbf{q})
    \,d\varSigma \,.
  &lt;/math&gt;|{{EquationRef|22n}}}}
Now on the left, the integrand can be written&amp;#8201; &lt;math&gt;\mathbf{\hat{n}}{\times}\boldsymbol{\hat{\nu}}\!\cdot\!\mathbf{q}\!=\!-\mathbf{\hat{t}}\!\cdot\!\mathbf{q}~\!;\,&lt;/math&gt; and on the right,&amp;#8201; &lt;math&gt;\operatorname{div}(\boldsymbol{\hat{\nu}}\!\times\!\mathbf{q})\!=\!-\operatorname{div}(\mathbf{q}\!\times\!\boldsymbol{\hat{\nu}})\!=\!-\operatorname{curl}\mathbf{q}\cdot\boldsymbol{\hat{\nu}}\,&lt;/math&gt; by identity ({{EquationNote|8c}}), since &lt;math&gt;\boldsymbol{\hat{\nu}}&lt;/math&gt; is uniform over each surface element.  With these substitutions, the minus signs cancel and we get
{{NumBlk|:|&lt;math&gt;
    \oint_C \mathbf{q} \cdot \mathbf{\hat{t}} \,dl \,=
    \iint_{\varSigma}
      \operatorname{curl}\mathbf{q}\cdot\boldsymbol{\hat{\nu}}
    \,d\varSigma
  &lt;/math&gt;|{{EquationRef|22c}}}}
or, if we again write&amp;#8201; &lt;math&gt;d\mathbf{r}\!=\!\mathbf{\hat{t}}~\!dl&lt;/math&gt;&amp;#8201; and&amp;#8201; &lt;math&gt;\boldsymbol{d\varSigma}\!=\!\boldsymbol{\hat{\nu}}\,d\varSigma~\!,&lt;/math&gt;
{{NumBlk|:|&lt;math&gt;
    \oint_C \mathbf{q} \cdot d\mathbf{r} \,=
    \iint_{\varSigma}
      \operatorname{curl}\mathbf{q}\cdot\boldsymbol{d\varSigma} \,.
  &lt;/math&gt;|{{EquationRef|22r}}}}
This result&amp;mdash;the best-known theorem relating an integral over a surface segment to an integral around its enclosing curve, and the best-known theorem involving the curl&amp;mdash;is called ''[[w:Sir George Stokes, 1st Baronet|Stokes]]' theorem'' or, more properly, the '''[[w:Lord Kelvin|Kelvin]]&amp;ndash;Stokes theorem''',&lt;ref&gt;''Cf''. [[#katz-79|Katz, 1979]], pp.&amp;#8239;149–50.&lt;/ref&gt; or simply the ''curl theorem''.&lt;ref&gt;Although Hsu ([[#hsu-84|1984]], p.&amp;#8239;141) applies that name to our theorem ({{EquationNote|5c}}).&lt;/ref&gt;

The integral on the left of ({{EquationNote|22c}}) or ({{EquationNote|22r}}) is called the '''circulation''' of the vector field {{math|'''q'''}} around the closed curve{{mvar| C}}. So, &lt;span id=&quot;kelvin-stokes-verbal&quot;&gt;in words&lt;/span&gt;, the Kelvin&amp;ndash;Stokes theorem says that ''the circulation of a vector field around a closed curve is equal to the flux of the curl of that vector field through any surface spanning that closed curve''.

Now let a general element of {{mvar|&amp;Sigma;}} (with area {{mvar|d&amp;Sigma;&amp;#8202;}}) be enclosed by the curve {{mvar|&amp;delta;C}}, traversed in the same direction as the outer curve {{mvar|C}}. Then, applying ({{EquationNote|22c}}) to the single element, we have
:&lt;math&gt;
    \oint_{\delta C} \!\mathbf{q} \cdot \mathbf{\hat{t}} \,dl \,=\,
    \operatorname{curl}\mathbf{q}\cdot\boldsymbol{\hat{\nu}} \,d\varSigma \,;
  &lt;/math&gt;
that is,
{{NumBlk|:|&lt;math&gt;
    \operatorname{curl}\mathbf{q}\cdot\boldsymbol{\hat{\nu}} \,=\,
    \frac{1}{d\varSigma}\oint_{\delta C}\!\mathbf{q}\cdot\mathbf{\hat{t}}\,dl\,,
  &lt;/math&gt;|{{EquationRef|23c}}}}
where the right-hand side is simply the ''circulation per unit area''.

Equation ({{EquationNote|23c}}) is an alternative definition of the curl: it says that ''the curl of''{{math| '''q'''}} ''is the vector whose scalar component in any direction is the circulation of''{{math| '''q'''}} ''per unit area of a surface whose normal points in that direction''. For ''real''{{math| '''q''',}} this component has its maximum, namely {{math|{{abs|curl&amp;#8201;'''q'''}}&amp;#8202;,}} in the direction of{{math| curl&amp;#8239;'''q'''&amp;#8202;}}; thus ''the curl of''{{math| '''q'''}} ''is the vector whose direction is that which a surface must face if the circulation of''{{math| '''q'''}} ''per unit area of that surface is to be a maximum, and whose magnitude is that maximum''. This is the usual conceptual definition of the curl.&lt;ref&gt;E.g., [[#gibbs-1881-4|Gibbs, 1881]], &amp;sect;&amp;#8239;61; [[#hsu-84|Hsu, 1984]], pp.&amp;#8239;117–18.&lt;/ref&gt;

[Notice, however, that our original volume-based definition ({{EquationNote|4c}}) is more succinct: the curl is the closed-surface circulation per unit volume, i.e. the skew surface integral per unit volume.]

It should now be clear where the curl gets its name (coined by [[w:James Clerk Maxwell|Maxwell]]), and why it is also called the ''rotation'' (indeed the {{math|curl}} operator is sometimes written &quot;{{math|rot}}&quot;, especially in Continental languages, in which &quot;rot&quot; does not have the same unfortunate everyday meaning as in English).

[[File:Vorticity_Figure_03_a-m.gif|thumb|Animation of a non-vortex-like velocity field whose curl (like its circulation around the red loop) is non-zero due to shear.]]
[[File:Vorticity_Figure_02_a-m.gif|thumb|Animation of a vortex-like velocity field whose curl is zero because the shear compensates for the rotation.]]

And it should now be unsurprising that ''a vector field with zero curl is described as '''irrotational''''' (which one must carefully pronounce differently from &quot;{{nowrap|irr''i&amp;#8202;''tational}}&quot;!), and that the curl of the velocity of a medium is called the '''vorticity'''.

However, a field does not need to be vortex-like in order to have a non-zero curl. For example, by identity ({{EquationNote|8p}}), in Cartesian coordinates, the velocity field {{math|''x'''''j'''}} has a curl equal to&amp;#8201; {{math|&amp;nabla;''x''&amp;#8201;&amp;times;&amp;#8239;'''j'''&amp;#8201;{{=}}&amp;#8201;'''i'''&amp;#8202;&amp;times;&amp;#8202;'''j'''&amp;#8201;{{=}}&amp;#8201;'''k'''&amp;#8202;,}}&amp;#8201; although it describes a ''shearing'' motion rather than a rotating motion. This is understandable because if you hold a pencil between the palms of your hands and slide one palm over the other (a shearing motion), the pencil rotates.

Conversely, we can have a vortex-like field whose curl is zero everywhere except on or near the axis of the vortex. For example, the '''Maxwell&amp;ndash;Amp&amp;egrave;re law''' in magnetostatics says that&amp;#8201; {{math|curl&amp;#8201;'''H'''&amp;#8201;{{=}}&amp;#8201;'''J'''&amp;#8202;,}} where {{math|'''H'''}} is the '''magnetizing field''' and {{math|'''J'''}} is the current density.{{efn|In the general case, there is an extra term {{math|{{sfrac|''&amp;part;''&amp;#8202;'''D'''|''&amp;part;t''}}}} on the right; but this term is zero in the magneto''static'' case.}} So if the current is confined to a wire, {{math|curl&amp;#8201;'''H'''&amp;#8202;}} is zero outside the wire&amp;mdash;although, as is well known, the field lines circle the wire. The resolution of the paradox is that {{math|'''H'''}} gets stronger as we approach the wire, making a shearing pattern, whose effect on the curl counteracts that of the rotation.

{{cob}}

=== The curl-grad and div-curl operators ===

{{cot}}

We have seen from ({{EquationNote|9L}}) that the Laplacian of a scalar field is the divergence of the gradient. Four more such second-order combinations make sense, namely the curl of the gradient (of a scalar field), and the divergence of the curl, the gradient of the divergence, and the curl of the curl (of a vector field). The first two&amp;#8239;&amp;mdash;&quot;curl grad&quot; and &quot;div curl&quot;&amp;mdash;&amp;#8239;can now be disposed of.

Let the surface segment {{mvar|&amp;Sigma;}} enclosed by the curve{{mvar| C}}&amp;#8202; be a segment of the closed surface {{mvar|S}} surrounding the volume{{mvar| V}}, and let {{mvar|&amp;Sigma;}} expand across {{mvar|S}} until it engulfs{{mvar| V}}, so that {{mvar|C}} shrinks to a point on the far side of{{mvar| S}}. Then, in the nameless theorem ({{EquationNote|21g}}) and the Kelvin&amp;ndash;Stokes theorem ({{EquationNote|22c}}), the integral on the left becomes zero while {{mvar|&amp;Sigma;}} and &lt;math&gt;\boldsymbol{\hat{\nu}}&lt;/math&gt; on the right become {{mvar|S}} and &lt;math&gt;\mathbf{\hat{n}},&lt;/math&gt; so that the theorems respectively reduce to
:&lt;math&gt;
    \iint_S \mathbf{\hat{n}}\times\nabla p \;dS
     \,=\, \mathbf{0}
  &lt;/math&gt;
and
:&lt;math&gt;
    \iint_S
      \mathbf{\hat{n}}\cdot\operatorname{curl}\mathbf{q}
    \;dS \,=\, 0 \,.
  &lt;/math&gt;
Applying theorem ({{EquationNote|5c}}) to the first of these two equations, and the divergence theorem ({{EquationNote|5d}}) to the second, we obtain respectively
:&lt;math&gt;\iiint_V \operatorname{curl}\nabla p \;dV ~\!=\, \mathbf{0} \,,&lt;/math&gt;
and
:&lt;math&gt;
    \iiint_{V}
      \operatorname{div}\operatorname{curl}\mathbf{q}
    \;dV ~\!=\, 0 \,.
  &lt;/math&gt;
As the integrals are zero for ''any'' volume {{mvar|V}}&amp;#8202; in which the integrands are defined, the integrands must be zero wherever they are defined; that is,
{{NumBlk|:|&lt;math&gt;
    \operatorname{curl}\nabla p \equiv \mathbf{0}
  &lt;/math&gt;|{{EquationRef|24c}}}}
and
{{NumBlk|:|&lt;math&gt;
    \operatorname{div}\operatorname{curl}\mathbf{q} \equiv 0 \,.
  &lt;/math&gt;|{{EquationRef|24d}}}}
In words, ''the curl of the gradient is zero'', and ''the divergence of the curl is zero''; or, more concisely, ''any gradient is irrotational'', and ''any curl is solenoidal''.

We might well ask whether the converses are true. Is every irrotational vector field the gradient of something? And is every solenoidal vector field the curl of something? The answers are affirmative, but the proofs require more preparation.

Meanwhile we may note, as a mnemonic aid, that when the left-hand sides of the last two equations are rewritten in the del-cross and del-dot notations, they become&amp;#8201; {{math|&amp;nabla;&amp;#8201;&amp;times;&amp;#8201;&amp;nabla;''p''}}&amp;#8201; and&amp;#8201; {{math|&amp;nabla;&amp;#8201;'''&amp;sdot;'''&amp;#8201;&amp;nabla;&amp;#8202;&amp;times;&amp;#8202;'''q'''&amp;#8202;,}} respectively. The former ''looks like'' (but isn't) a cross-product of two parallel vectors, and the latter ''looks like'' (but isn't) a scalar triple product with a repeated factor, so that each expression ''looks like'' it ought to be zero (and it is). But such appearances can lead one astray, because {{math|&amp;nabla;}} is an operator, not a self-contained vector quantity; for example,&amp;#8201; {{math|&amp;nabla;''p''&amp;#8201;&amp;times;&amp;#8201;&amp;nabla;''&amp;phi;''}}&amp;#8201; is ''not'' identically zero, because two gradients are not necessarily parallel.&lt;ref&gt;''Cf''. [[#feynman-63|Feynman, 1963]], vol.&amp;#8239;2, &amp;sect;2-8.&lt;/ref&gt;

We should also note, to tie a loose end, that identity ({{EquationNote|24d}}) was to be expected from our [[#kelvin-stokes-verbal|verbal statement]] of the Kelvin&amp;ndash;Stokes theorem ({{EquationNote|22c}}). That statement implies that the flux of the curl through any two surfaces spanning the same closed curve is the same. So if we make a ''closed'' surface from two spanning surfaces, the flux into one spanning surface is equal to the flux out of the other, i.e. the net flux out of the closed surface is zero, i.e. the integral of the divergence over the enclosed volume is zero; and since ''any'' simple volume in which the divergence is defined can be enclosed this way, the divergence itself (of the curl) must be zero wherever it is defined.

{{cob}}

== Change per unit length ==

{{cot}}

Continuing (and concluding) the trend of reducing the number of dimensions, we now seek ''one''-dimensional theorems, each of which relates an integral over a ''path'' to values at the endpoints of the path. For maximum generality, the path should be allowed to be curved into a second and a third dimension.

We ''could'' do this by further specializing theorems ({{EquationNote|5g}}) to ({{EquationNote|5L}}). We could take a curve {{math|&amp;Gamma;}} with a unit tangent vector{{math| '''s&amp;#770;'''}}. At every point on{{math| &amp;Gamma;}} we could mount a circular disk with a uniform ''small'' area{{mvar| &amp;alpha;&amp;#8202;,}} centered on{{math| &amp;Gamma;}} and orthogonal to it. We could let {{mvar|V}} be the volume occupied by all the disks and let {{mvar|S}} be its enclosing surface; thus {{mvar|V}} would be a thin right circular cylinder, except that its axis could be curved. If we could arrange for {{mvar|&amp;alpha;}} to cancel out, our four theorems would indeed be reduced to the desired form, ''provided'' that there were no contribution from the curved face of the &quot;cylinder&quot; to the integral over{{mvar| S}} (the &quot;1D proviso&quot;). But, as it turns out, this exercise yields only one case in which the &quot;1D proviso&quot; can be satisfied by a construction involving {{math|'''s&amp;#770;'''}} and a general field, and we have already ''almost'' discovered that case by a simpler and more conventional argument&amp;mdash;which we shall now continue.

{{cob}}

=== Fundamental theorem ===

{{cot}}

Equation ({{EquationNote|9g}}) is applicable where {{math|''p''('''r''')}} is a scalar field,&amp;#8201; {{mvar|s}} is a parameter measuring arc length along a curve{{math| &amp;Gamma;,}} and {{math|'''s&amp;#770;'''}} is the unit tangent vector to{{math| &amp;Gamma;}} in the direction of increasing{{mvar| s}}. Let {{mvar|s}} take the values {{math|''s''&lt;sub&gt;1&lt;/sub&gt;}} and {{math|''s''&lt;sub&gt;2&lt;/sub&gt;}} at the endpoints of{{math| &amp;Gamma;,}} where the position vector {{math|'''r'''}} takes the values {{math|'''r'''&lt;sub&gt;1&lt;/sub&gt;}} and {{math|'''r'''&lt;sub&gt;2&lt;/sub&gt;}} respectively. Then, integrating ({{EquationNote|9g}}) w.r.t.{{mvar| s}} from {{math|''s''&lt;sub&gt;1&lt;/sub&gt;}} to {{math|''s''&lt;sub&gt;2&lt;/sub&gt;}} and applying the fundamental theorem of calculus, we get
{{NumBlk|:|&lt;math&gt;
    \int_{s_1}^{s_2} \nabla p \cdot \mathbf{\hat{s}} \,ds
      \,=\, p(\mathbf{r}_2) - p(\mathbf{r}_1) \,.
  &lt;/math&gt;|{{EquationRef|25g}}}}
This is our third integral theorem involving the gradient, and the best-known of the three: it is commonly called simply the '''[[w:gradient theorem|gradient theorem]]''',&lt;ref&gt;Although Hsu ([[#hsu-84|1984]], p.&amp;#8239;141) applies that name to our theorem ({{EquationNote|5g}}).&lt;/ref&gt; or the ''fundamental theorem of the gradient'', or the ''fundamental theorem of line integrals''; it generalizes the fundamental theorem of calculus to a curved path.&lt;ref&gt;''Cf''. [[#gibbs-1881-4|Gibbs, 1881]], &amp;sect;&amp;sect;&amp;#8239;50,&amp;#8239;59; presumably this is one reason why Gibbs called the gradient simply the ''derivative''.&lt;/ref&gt; If we write {{math|''d'''''r'''}}&amp;#8202; for&amp;#8239; {{math|'''s&amp;#770;'''&amp;#8239;''ds''}} (the change in the position vector), we get the theorem in the alternative form
{{NumBlk|:|&lt;math&gt;
    \int_{\mathbf{r}_1}^{\mathbf{r}_2} \nabla p \cdot d\mathbf{r}
      \,=\, p(\mathbf{r}_2) - p(\mathbf{r}_1) \,.
  &lt;/math&gt;|{{EquationRef|25r}}}}
As the right-hand side of ({{EquationNote|25g}}) or ({{EquationNote|25r}}) obviously depends on the endpoints but ''not on the path in between'', so does the integral on the left. This integral is commonly called the '''work integral''' of{{math| &amp;nabla;''p''}} over the path&amp;mdash;because if {{math|&amp;nabla;''p''}} is a force, the integral is the work done by the force over the path. So, in words, the gradient theorem says that ''the change in value of a scalar field from one point to another is the work integral of the gradient of that field field over any path from the one to the other''.

Applying ({{EquationNote|25r}}) to a single element of the curve, we get
{{NumBlk|:|&lt;math&gt;\nabla p \cdot d\mathbf{r} = dp \,,
  &lt;/math&gt;|{{EquationRef|26g}}}}
which is reminiscent of&amp;#8201; &lt;math&gt;y'(x)~\!dx\,{=}\,dy\,&lt;/math&gt; in elementary calculus.&lt;ref&gt;''Cf''. [[#gibbs-1881-4|Gibbs, 1881]], &amp;sect;&amp;sect;&amp;#8239;50,&amp;#8239;51; presumably this is another reason why Gibbs called the gradient the ''derivative''.&lt;/ref&gt; Alternatively, we could have obtained ({{EquationNote|26g}}) by multiplying both sides of ({{EquationNote|9g}}) by{{mvar| ds}}, and then obtained ({{EquationNote|25r}}) by adding ({{EquationNote|26g}}) over all the elemental displacements{{math| ''d'''''r'''}} on any path from {{math|'''r'''&lt;sub&gt;1&lt;/sub&gt;}} to{{math| '''r'''&lt;sub&gt;2&lt;/sub&gt;}}.

If we ''close'' the path by setting&amp;#8201; {{math|'''r'''&lt;sub&gt;2 &lt;/sub&gt;{{=}} '''r'''&lt;sub&gt;1&lt;/sub&gt;&amp;#8202;,}} the gradient theorem reduces to
{{NumBlk|:|&lt;math&gt;\oint \nabla p \cdot d\mathbf{r} \,=\, 0 \,,
  &lt;/math&gt;|{{EquationRef|27g}}}}
where the integral is around ''any'' closed loop. Applying the Kelvin&amp;ndash;Stokes theorem then gives
{{NumBlk|:|&lt;math&gt;
    \iint_{\varSigma}
      \operatorname{curl}\nabla p \cdot \boldsymbol{\hat{\nu}}
    \,d\varSigma \,=\, 0 \,,
  &lt;/math&gt;|{{EquationRef|28g}}}}
where {{mvar|&amp;Sigma;}}&amp;#8202; is any surface spanning the loop and&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt; is the unit normal to{{mvar| &amp;Sigma;}}.&amp;#8201; As this applies to any loop spanned by any surface on which the integrand is defined,&amp;#8201; {{math|curl&amp;#8201;&amp;nabla;''p''}}&amp;#8202; must be zero wherever it is defined. This is a second proof (more conventional than the first) of theorem ({{EquationNote|24c}}).

{{cob}}

=== Scalar potential: field with given gradient ===

{{cot}}

'''Lemma''':  If&amp;#8201; {{math|curl&amp;#8239;'''q'''&amp;#8201;{{=}}&amp;#8201;'''0'''}}&amp;#8201; in a simply connected region{{mvar| V}},&amp;#8201; then&amp;#8239; &lt;math&gt;\textstyle\int\!\mathbf{q}\!\cdot\!d\mathbf{r}\,&lt;/math&gt; over any path in{{mvar| V}}&amp;#8239; depends only on the endpoints of the path.

''Proof:''  Suppose, on the contrary, that there are two paths {{math|&amp;Gamma;}} and {{math|&amp;Lambda;}} in{{mvar| V}},&amp;#8202; with a common starting point and a common finishing point, such that
:&lt;math&gt;\textstyle\int_{\Gamma}\mathbf{q}\cdot d\mathbf{r}
       \,\neq \textstyle\int_{\Lambda}\mathbf{q}\cdot d\mathbf{r} \,.&lt;/math&gt;
Let&amp;#8201; {{math|&amp;minus;&amp;Lambda;}} denote {{math|&amp;Lambda;}} traversed backwards. Then for every {{math|''d'''''r'''}} on {{math|&amp;Lambda;}}&amp;#8201; there is an equal and opposite{{math| ''d'''''r'''}} on&amp;#8202; {{math|&amp;minus;&amp;Lambda;&amp;#8202;,}} and vice versa, so that we have
:&lt;math&gt;\textstyle\int_{\Gamma}\mathbf{q}\cdot d\mathbf{r} \,\neq\,
       \textstyle-\!\int_{-\Lambda}\mathbf{q}\cdot d\mathbf{r} \,,&lt;/math&gt;
i.e.
:&lt;math&gt;\textstyle\int_{\Gamma}\mathbf{q}\cdot d\mathbf{r} \,+
  \textstyle\int_{-\Lambda}\mathbf{q}\cdot d\mathbf{r} \,\neq\, 0 \,,&lt;/math&gt;
where the left-hand side is now a work integral of{{math| '''q'''}} around a closed loop in{{mvar| V}}.&amp;#8201; By the simple connectedness of{{mvar| V}},&amp;#8202; this loop is spanned by some surface{{mvar| &amp;Sigma;}} in{{mvar| V}}.&amp;#8201; So we can apply the Kelvin&amp;ndash;Stokes theorem and conclude that the flux integral of&amp;#8202; {{math|curl&amp;#8239;'''q'''}}&amp;#8202; through{{mvar| &amp;Sigma;}}&amp;#8202; is non-zero, in which case&amp;#8202; {{math|curl&amp;#8239;'''q'''}}&amp;#8202; must be non-zero somewhere on{{mvar| &amp;Sigma;&amp;#8202;,}} hence somewhere in{{mvar| V}}&amp;#8201;&amp;mdash;&amp;#8239;contradicting the hypothesis of the lemma. &amp;#9724;

'''Corollary''':  If&amp;#8201; {{math|curl&amp;#8239;'''q'''&amp;#8201;{{=}}&amp;#8201;'''0'''}}&amp;#8201; in a simply connected region{{mvar| V}},&amp;#8202; there exists a scalar field {{mvar|p}} such that&amp;#8239; {{math|'''q'''&amp;#8201;{{=}}&amp;#8201;&amp;nabla;''p''}}&amp;#8201; in{{mvar| V}}.

''Proof:''  We shall show that a suitable candidate is
:&lt;math&gt;p(\mathbf{r}) \,=
    \int_{\mathbf{r}_0}^{\mathbf{r}} \!\mathbf{q}\cdot d\boldsymbol{\rho} \,,
  &lt;/math&gt;
where {{math|'''r'''&lt;sub&gt;0&lt;/sub&gt;}} is the position vector of any fixed point in{{mvar| V}},&amp;#8202; and {{mvar|'''&amp;rho;'''}} is the position vector of a general point on the path of integration, which may be any path in{{mvar| V}}. First note that {{math|''p''('''r''')}} is unambiguous because, by the preceding lemma, it is independent of the path for given {{math|'''r'''&lt;sub&gt;0&lt;/sub&gt;}} and{{math| '''r''',}} provided that the path is in{{mvar| V}}.&amp;#8201; Now to find&amp;#8202; {{math|&amp;nabla;''p''('''r'''),}}&amp;#8202; let {{mvar|&amp;sigma;}} be the arc length along the path from {{math|'''r'''&lt;sub&gt;0&lt;/sub&gt;}} to{{mvar| '''&amp;rho;'''&amp;#8202;}}, so that {{mvar|&amp;sigma;}} ranges from 0 to (say){{mvar| s}}&amp;#8201; as {{mvar|'''&amp;rho;'''}} ranges from {{math|'''r'''&lt;sub&gt;0&lt;/sub&gt;}} to{{math| '''r'''&amp;#8202;}}; and let {{math|'''s&amp;#770;'''}} be the unit vector tangential to the path at{{mvar| '''&amp;rho;'''&amp;#8202;}}, in the direction of increasing{{mvar| &amp;sigma;}}.&amp;#8201; Then&amp;#8239; {{math|''d'''&amp;rho;'''''&amp;#8201;{{=}}&amp;#8201;'''s&amp;#770;'''&amp;#8239;''d&amp;sigma;''&amp;#8202;,}} so that the above equation becomes
:&lt;math&gt;p\big(\mathbf{r}(s)\big) \,=
    \int_0^s \!\mathbf{q}\cdot\mathbf{\hat{s}} \,d\sigma \,.
  &lt;/math&gt;
Differentiating w.r.t.{{mvar| s}} gives
:&lt;math&gt;\part_s p = \mathbf{q}\cdot\mathbf{\hat{s}} \,,&lt;/math&gt;
where {{math|'''s&amp;#770;'''}} is evaluated at&amp;#8201; {{mvar|&amp;sigma;&amp;#8201;{{=}}&amp;#8201;s}}&amp;#8201; and is therefore in the direction in which the path reaches{{math| '''r'''}}.&amp;#8201; By the generality of the path, this can be ''any'' direction. So the last equation says that {{math|'''q'''}} is the vector whose (scalar) component in any direction is the derivative of{{mvar| p}} w.r.t. arc length in that direction; that is, {{math|'''q'''&amp;#8201;{{=}}&amp;#8201;&amp;nabla;''p''&amp;#8202;,}} as required. &amp;#9724;

This is the promised converse of theorem ({{EquationNote|24c}}). ''But'', given an irrotational vector field {{math|'''q'''&amp;#8202;,}} we usually prefer to find a scalar field whose ''negative'' gradient is{{math| '''q'''&amp;#8202;}};&amp;#8201; that is, we usually prefer a scalar field &lt;math&gt;\varphi&lt;/math&gt; such that&amp;#8201;
&lt;math&gt;\mathbf{q}~\!\!=\!-\nabla\varphi&lt;/math&gt;.&amp;#8201; Such a field &lt;math&gt;\varphi&lt;/math&gt; is called a '''scalar potential''' for{{math| '''q'''}}.&amp;#8201; From the above expression for {{math|''p''('''r'''),}} a suitable candidate is
{{NumBlk|:|&lt;math&gt;\varphi(\mathbf{r}) \,=\,
    -\!\int_{\mathbf{r}_0}^{\mathbf{r}} \!\mathbf{q}\cdot d\boldsymbol{\rho} \,.
  &lt;/math&gt;|{{EquationRef|29}}}}

A scalar field has zero gradient if and only if it is uniform, so that adding a uniform field, but ''only'' a uniform field, to a given scalar field leaves its gradient unchanged. Thus ''the scalar potential is determined up to an arbitrary additive uniform field''. This would be the case with or without the minus sign in front of the gradient. The reason for preferring the minus sign appears next.

{{cob}}

=== Conservative fields ===

{{cot}}

An irrotational vector field&amp;mdash;or, equivalently, a field that is (plus or minus) the gradient of something&amp;mdash;is described as '''conservative''', because if the field is a force, it does zero work around a closed loop, and consequently ''conserves energy'' around the loop (at least if the field does not change during traversal of the loop).

If the only force acting on a particle is&amp;#8201; {{math|'''F'''&amp;#8201;{{=}}&amp;#8201;&amp;minus;&amp;nabla;''U'',}}&amp;#8201; then, by the gradient theorem, the work done on the particle over a path is the increase in {{mvar|&amp;minus;U}},&amp;#8201; i.e. the ''decrease'' in{{mvar| U&amp;#8202;}}; and this work is the increase in the particle's kinetic energy{{mvar| T}}.&amp;#8201; Hence, if we identify {{mvar|U}} with the ''potential'' energy, the total energy&amp;#8202; {{mvar|U&amp;#8202;+&amp;#8202;T}}&amp;#8201; is conserved. This interpretation of the scalar potential is possible only if the force is ''minus'' the gradient of the potential.

The minus sign is also used if the conservative vector field is an '''electric field''' (force per unit charge) or a gravitational acceleration (force per unit mass); the scalar potential is potential energy per unit charge, or potential energy per unit mass, respectively.

{{cob}}

== Some special fields ==

=== The 1/''r'' scalar potential ===

{{cot}}

For the potential energy field {{NumBlk|:|&lt;math&gt;U = \frac{1}{\,r\,} \,&lt;/math&gt;|{{EquationRef|30}}}} where {{mvar|r}} is the distance from the origin (and {{math|''r''&amp;#8201;&amp;ne;&amp;#8201;0}}), let us find the corresponding force&amp;#8201; {{math|'''F'''&amp;#8201;{{=}}&amp;#8201;&amp;minus;&amp;nabla;''U''}}.&amp;#8201; The direction of&amp;#8202; {{math|&amp;nabla;''U''}}&amp;#8202; is that of the steepest increase of{{mvar| U}}, which, by the spherical symmetry, can only be parallel or antiparallel to &lt;math&gt;\mathbf{\hat{r}}&lt;/math&gt; (the unit vector pointing away from the origin). So {{math|&amp;nabla;''U''}} is equal to its vector component in the direction of&amp;nbsp;&lt;math&gt;\mathbf{\hat{r}}&lt;/math&gt;, or&amp;nbsp;&lt;math&gt;\mathbf{\hat{r}}&lt;/math&gt; times its scalar component in that direction; that is,
:&lt;math&gt;\nabla U = \big(\nabla U \cdot \mathbf{\hat{r}}\big)~\!\mathbf{\hat{r}}
                = \part_r U \,\mathbf{\hat{r}}
                = \frac{d}{dr}\Big(\!\frac{1}{\,r\,}\!\Big)~\!\mathbf{\hat{r}}
                = -\frac{1}{\,r^2}~\!\mathbf{\hat{r}} \,,&lt;/math&gt;
whence
{{NumBlk|:|&lt;math&gt;\mathbf{F} = \frac{\mathbf{\hat{r}}}{\,r^2} \,.
  &lt;/math&gt;|{{EquationRef|31}}}}
So the negative gradient of the {{math|1/''r''}}&amp;#8202; scalar potential ({{EquationNote|30}}) is the unit '''inverse-square radial vector field'''. Multiplying the numerator and denominator by {{mvar|r}} gives the alternative form
:&lt;math&gt;\mathbf{F} = \frac{\mathbf{r}}{\,r^3} \,,&lt;/math&gt;
which is convenient if the center of the force is shifted from the origin to position{{math| '''r&amp;prime;'''}}: in that case we simply replace {{math|'''r'''}} by {{math|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;''',}} and {{mvar|r}} by {{math|{{abs|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''}},}} so that the force becomes
:&lt;math&gt;\mathbf{F} = \frac{\mathbf{r}\!-\!\mathbf{r}'}
                         {|\mathbf{r}\!-\!\mathbf{r}'|^3}&lt;/math&gt;
and the corresponding scalar potential becomes
:&lt;math&gt;U = \frac{1}{|\mathbf{r}\!-\!\mathbf{r}'|} \,.&lt;/math&gt;

{{cob}}

=== Inverse-square radial vector field ===

{{cot}}

We derived the vector field ({{EquationNote|31}}) as the negative gradient of the scalar potential ({{EquationNote|30}}). Conversely, given the inverse-square radial vector field ({{EquationNote|31}}), we could derive its scalar potential from ({{EquationNote|29}}). At a general point on the path, let the position vector be&amp;#8201; &lt;math&gt;\boldsymbol{\rho}~\!\!=\!\rho\boldsymbol{\hat{\rho}}\,&lt;/math&gt; so that, by ({{EquationNote|31}}),&amp;#8201; &lt;math&gt;\mathbf{F}\!=\!\boldsymbol{\hat{\rho}}/\rho^2&lt;/math&gt;.&amp;#8201; Then ({{EquationNote|29}}) becomes
:&lt;math&gt;\begin{align}U(\mathbf{r})
    \,&amp;=\, -\!\int_{\mathbf{r}_0}^{\mathbf{r}} \!
               \mathbf{F} \cdot d\boldsymbol{\rho} \\[1ex]
      &amp;=\, -\!\int_{\mathbf{r}_0}^{\mathbf{r}} \!
               \tfrac{1}{\,\rho^2}~\!
              \boldsymbol{\hat{\rho}} \!\cdot\! d\boldsymbol{\rho} \\[1ex]
      &amp;=\, -\!\int_{r_0}^r \! \tfrac{1}{\,\rho^2} \,d\rho
     \ =\, \tfrac{1}{\,\rho\,}\bigg|^r_{r_0}
     \,=\, \frac{1}{\,r\,}-\frac{1}{\,r_0} \,,
  \end{align}&lt;/math&gt;
so that, if we choose&amp;#8201; {{math|''r''&lt;sub&gt;0&lt;/sub&gt;&amp;#8202;&amp;rarr;&amp;#8239;&amp;infin;&amp;#8202;,}} we recover ({{EquationNote|30}}).

Because {{math|'''F''',}} given by ({{EquationNote|31}}), has a scalar potential,&amp;#8201; {{math|curl&amp;#8201;'''F'''}}&amp;#8202; must be zero. This is independently obvious in that the spherical symmetry of{{math|&amp;#8202; '''F'''}} seems to rule out any resemblance of rotation or shear&amp;mdash;even at the origin, where {{math|'''F'''}} becomes infinite. On the last point, let us check whether&amp;#8201; {{math|curl&amp;#8201;'''F'''}}&amp;#8202; has a meaningful integral over a volume containing the origin. If the volume {{mvar|V}}&amp;#8202; is enclosed by the surface {{mvar|S}}&amp;#8202; whose outward unit normal is &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt;, then, by theorem ({{EquationNote|5c}}),
:&lt;math&gt;\iiint_V \operatorname{curl}\mathbf{F} ~dV
    \,= \iint_S \mathbf{\hat{n}}\times\mathbf{F} \,dS
    \,= \iint_S \mathbf{\hat{n}}\times\frac{\mathbf{\hat{r}}\,}{r^2} \,dS \,.
  &lt;/math&gt;
If {{mvar|V}} contains the origin, then, because&amp;#8201; {{math|curl&amp;#8201;'''F'''}}&amp;#8239; is zero everywhere ''except'' at the origin, the volume {{mvar|V}}&amp;#8202; can be replaced by any ''element'' of{{mvar| V}}&amp;#8202; containing the origin, whatever the shape of that element may be. If we choose that element to be a spherical ball centered on the origin, then &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is parallel to &lt;math&gt;\mathbf{\hat{r}}&lt;/math&gt;, so that the cross-product in the integrand on the right is zero. Thus the volume integral on the left is not only meaningful, but is ''zero'', even if the volume contains the point where the integrand is infinite. In this sense, the field {{math|'''F'''}} is ''so'' irrotational that its curl may be taken as zero even where the field itself is undefined!

The situation concerning the ''divergence'' of{{math|&amp;#8202; '''F'''}} is more complicated. Again, let the volume {{mvar|V}}&amp;#8202; be enclosed by the surface {{mvar|S}} whose outward unit normal is &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt;.&amp;#8201; By the divergence theorem ({{EquationNote|5d}}),
:&lt;math&gt;\begin{align}\iiint_V \operatorname{div}\mathbf{F} ~dV
    \,= \iint_S \mathbf{\hat{n}}\cdot\mathbf{F} \,dS
   \,&amp;= \iint_S \mathbf{\hat{n}}\cdot\frac{\mathbf{\hat{r}}\,}{r^2} \,dS\\[1ex]
     &amp;= \iint_S \frac{\mathbf{\hat{r}}\cdot\mathbf{\hat{n}}\,dS}{\,r^2} \\[1ex]
     &amp;= \iint_S d\Omega \,,
  \end{align}&lt;/math&gt;
where {{math|''d''&amp;Omega;}} is the ''solid angle'' subtended at the origin by the surface element of area{{mvar| dS&amp;#8202;}}, and is taken as positive if the outward unit normal &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; has a positive component ''away from'' the origin &lt;math&gt;(\mathbf{\hat{r}}\!\cdot\!\mathbf{\hat{n}}&gt;0)&lt;/math&gt;, and negative if&amp;#8202; &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; has a positive component ''toward'' the origin &lt;math&gt;(\mathbf{\hat{r}}\!\cdot\!\mathbf{\hat{n}}&lt;0)&lt;/math&gt;. If the volume enclosed by {{mvar|S}} does ''not'' include the origin, then for every positive contribution {{math|''d''&amp;Omega;}}&amp;#8201; there is a compensating negative contribution, so that the integral of&amp;#8201; {{math|div&amp;#8201;'''F'''}}&amp;#8202; over the volume is zero. As this applies to every such volume,&amp;#8201; {{math|div&amp;#8201;'''F'''}}&amp;#8202; must be zero everywhere except at the origin. If, on the contrary, the volume ''does'' include the origin, then the contributions {{math|''d''&amp;Omega;}} add up to the total solid angle subtended by the enclosing surface, which is{{math| 4''&amp;pi;''}}. In summary,
{{NumBlk|:|&lt;math&gt;\mathrm{div}\Big(\frac{\mathbf{\hat{r}}}{\,r^2}\Big)
  =~\! 4\pi~\!\delta(\mathbf{r}) \,,&lt;/math&gt;|{{EquationRef|32d}}}}
where {{math|''&amp;delta;''('''r'''),}} the 3D '''unit delta function''', is zero everywhere except at the origin, but has an integral of&amp;#8202; {{math|1}} over any volume that includes the origin. For example, a unit point-mass at the origin has the density {{math|''&amp;delta;''('''r''')}}, and a point-mass {{mvar|m}} at position {{math|'''r&amp;prime;'''}} has the density&amp;#8202; {{math|''m&amp;delta;''('''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;''')}}. As the argument of&amp;#8202; {{math|div}}&amp;#8202; in ({{EquationNote|32d}}) is&amp;#8202; {{math|&amp;minus;&amp;nabla;(1/''r''),}} we also have
{{NumBlk|:|&lt;math&gt;\triangle\Big(\frac{1}{\,r\,}\Big)
  = -4\pi~\!\delta(\mathbf{r}) \,.&lt;/math&gt;|{{EquationRef|32L}}}}
If we shift the centers from the origin to {{math|'''r&amp;prime;''',}} the last two results become
{{NumBlk|:|&lt;math&gt;
    \mathrm{div}\bigg(\frac{ \mathbf{r}\!-\!\mathbf{r}'}
                           {|\mathbf{r}\!-\!\mathbf{r}'|^3} \!\bigg)
    =~\! 4\pi~\!\delta(\mathbf{r}\!-\!\mathbf{r}')
  &lt;/math&gt;|{{EquationRef|33d}}}}
and
{{NumBlk|:|&lt;math&gt;
    \triangle\bigg(\frac{1}{|\mathbf{r}\!-\!\mathbf{r}'|}\bigg)
    = -4\pi~\!\delta(\mathbf{r}\!-\!\mathbf{r}') \,.
  &lt;/math&gt;|{{EquationRef|33L}}}}

{{cob}}

=== Field with given divergence (and zero curl) ===

{{cot}}

It follows from '''Coulomb's law''' that the electric field due to a point-charge {{mvar|Q}} at the origin, in a vacuum, is
:&lt;math&gt;\mathbf{E}
  = \frac{Q}{4\pi\epsilon_0 r^2} ~\!\mathbf{\hat{r}} \,,&lt;/math&gt;
where {{math|''&amp;epsiv;''&lt;sub&gt;0&lt;/sub&gt;}} is a physical constant (called the '''vacuum permittivity''' or simply the '''electric constant'''). In a ''vacuum'', the '''electric displacement field''', denoted by{{math| '''D'''&amp;#8202;,}} is {{math|''&amp;epsiv;''&lt;sub&gt;0&lt;/sub&gt;'''E'''}}.&amp;#8201; So it is convenient to multiply the above equation by {{math|''&amp;epsiv;''&lt;sub&gt;0&lt;/sub&gt;&amp;#8202;,}} obtaining
:&lt;math&gt;\mathbf{D}
  = \frac{Q}{4\pi} ~\!\frac{\mathbf{\hat{r}}\,}{r^2} \,.&lt;/math&gt;
This is a inverse-square radial vector field and therefore has zero curl.

Now suppose that, instead of a charge {{mvar|Q}} at the origin, we have a static ''charge density'' {{math|''&amp;rho;''('''r&amp;prime;''')}} in a general elemental volume {{mvar|dV&amp;prime;}}&amp;#8202; at position{{math| '''r&amp;prime;'''}} (the standard symbol for ''charge'' density being unfortunately the same as for ''mass'' density). Then the contribution from that element to the field{{math| '''D'''}} at position{{math| '''r'''}}&amp;#8202; is
:&lt;math&gt;d\mathbf{D}(\mathbf{r}) ~\!=~\!
    \frac{\,\rho(\mathbf{r}')\,dV'}{4\pi}\,
    \frac{\mathbf{r}\!-\!\mathbf{r}'}{|\mathbf{r}\!-\!\mathbf{r}'|^3}
  &lt;/math&gt;
provided that, for each {{math|'''r''',}} the dimensions of each volume element are small compared with {{math|{{abs|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''}}}}. This contribution likewise has zero curl. The total field due to static charges is then the sum of the contributions:
{{NumBlk|:|&lt;math&gt;\mathbf{D}(\mathbf{r}) \,= \iiint
    \frac{\,\rho(\mathbf{r}')}{4\pi}\,
    \frac{\mathbf{r}\!-\!\mathbf{r}'}{|\mathbf{r}\!-\!\mathbf{r}'|^3} \,dV'
  &lt;/math&gt;|{{EquationRef|34}}}}
where the integral is over all space. And {{math|'''D'''('''r''')}} has zero curl because all the contributions have zero curl.

Independently of the physical significance of&amp;#8202; {{math|'''D'''('''r'''),}} we can take its divergence &quot;term by term&quot; (or &quot;under the integral sign&quot;), obtaining
:&lt;math&gt;\begin{align}\operatorname{div}\mathbf{D}(\mathbf{r})
  \,&amp;= \iiint \frac{\,\rho(\mathbf{r}')}{4\pi}~\!\mathrm{div}\bigg(
         \frac{\mathbf{r}\!-\!\mathbf{r}'}{|\mathbf{r}\!-\!\mathbf{r}'|^3}
       \!\bigg) ~\!dV' \\[3pt]
    &amp;= \iiint \frac{\,\rho(\mathbf{r}')}{4\pi}\,
         4\pi~\!\delta(\mathbf{r}\!-\!\mathbf{r}') \,dV'
         \quad \big[\mathsf{by~eq.(33d)}\big] \\[3pt]
    &amp;= \iiint \rho(\mathbf{r}')\,\delta(\mathbf{r}\!-\!\mathbf{r}')\,dV'\\[3pt]
    &amp;= \iiint \rho(\mathbf{r})\,\delta(\mathbf{r}\!-\!\mathbf{r}') \,dV'
           ~~~ \begin{bmatrix}~\!\!
                 \mathsf{since}~\delta(\mathbf{r}\!-\!\mathbf{r}')\!=\!0\\
                 \mathsf{unless}~\,\mathbf{r}'{=}~\!\mathbf{r}
               ~\!\!\end{bmatrix} \\
    &amp;= \,\rho(\mathbf{r})\!\iiint\delta(\mathbf{r}\!-\!\mathbf{r}')\,dV'\\[3pt]
    &amp;= \,\rho(\mathbf{r})\!\iiint\delta(\mathbf{r}'{-}~\!\mathbf{r})\,dV',
  \end{align}&lt;/math&gt;
where the last step is permitted because the volume integral of the delta function of{{math| '''r&amp;prime;'''}} is not changed by a &quot;[[w:point reflection|point reflection]]&quot; (inversion) across {{math|'''r'''}}.&amp;#8201; As the volume of integration (all space) includes the shifted origin of the delta function, the integral is simply{{math| 1&amp;#8202;,}} so that
{{NumBlk|:|&lt;math&gt;
    \operatorname{div}\mathbf{D}=\rho \,,
  &lt;/math&gt;|{{EquationRef|35}}}}
where both sides are evaluated at{{math| '''r'''}}.

Mathematically, this result is an identity which applies if&amp;#8202; {{math|'''D'''}} is given by ({{EquationNote|34}}); substituting for{{math| '''D'''&amp;#8202;,}} we can write the identity in full as
{{NumBlk|:|&lt;math&gt;\rho(\mathbf{r}) \,\equiv\, \mathrm{div}\iiint
    \frac{\,\rho(\mathbf{r}')}{4\pi}\,
    \frac{\mathbf{r}\!-\!\mathbf{r}'}{|\mathbf{r}\!-\!\mathbf{r}'|^3}
  \,dV',&lt;/math&gt;|{{EquationRef|36}}}}
where the integral is over all space, or at least all of the space in which {{mvar|&amp;rho;}} may be non-zero. Subject to the convergence of the integral, this shows that ''we can construct an irrotational vector field whose divergence is a given scalar field''{{math| ''&amp;rho;''('''r''')}}. And of course, by theorem ({{EquationNote|24d}}), ''any curl'' can be added to that vector field without changing its divergence.

In ''electrostatics'', ({{EquationNote|34}}) is a generalization of Coulomb's law; and ({{EquationNote|35}}), which follows from ({{EquationNote|34}}), is '''Gauss's law''' expressed in ''differential form''. If we integrate ({{EquationNote|35}}) over a volume enclosed by a surface{{mvar| S}} (with outward unit normal &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt;) and apply the divergence theorem on the left, we get the ''integral form'' of Gauss's law:
{{NumBlk|:|&lt;math&gt;
    \iint_S \mathbf{D}\cdot\mathbf{\hat{n}}\,dS \,=\, Q_{\mathrm{e}} \,,
  &lt;/math&gt;|{{EquationRef|37}}}}
where {{math|''Q''&lt;sub&gt;e&lt;/sub&gt;}} is the total charge ''enclosed''&amp;#8202; by{{mvar| S}}.

{{cob}}

=== Field with given Laplacian ===

{{cot}}

In ({{EquationNote|36}}), we can recognize the {{math|'''r'''}}-dependent factor&amp;#8202; {{math|{{sfrac|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''|{{abs|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''}}&lt;sup&gt;3&lt;/sup&gt;}}}}&amp;#8201; as&amp;#8202; {{math|&amp;minus;&amp;nabla;{{sfrac|1|&amp;#8202;{{abs|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''}}&amp;#8202;}}}}&amp;#8201; and take the gradient operator outside the integral, obtaining
&lt;div style=&quot;margin-top: 1em&quot;&gt;
:&lt;math&gt;\rho(\mathbf{r}) \,\equiv\, \mathrm{div}\bigg(\!{-}\nabla\!\iiint
    \frac{\,\rho(\mathbf{r}')}{4\pi}\,
    \frac{1}{|\mathbf{r}\!-\!\mathbf{r}'|}
  \,dV' \!\bigg) \,,&lt;/math&gt;
&lt;/div&gt;
i.e.
{{NumBlk|:|&lt;math&gt;\rho(\mathbf{r}) \,\equiv\, \triangle\bigg(\!{-}\!\iiint
    \frac{\,\rho(\mathbf{r}')}{4\pi}\,
    \frac{1}{|\mathbf{r}\!-\!\mathbf{r}'|}
  \,dV' \!\bigg) \,,&lt;/math&gt;|{{EquationRef|38}}}}
where again the integral is over all space, or at least all of the space in which {{mvar|&amp;rho;}} may be non-zero. Subject to the convergence of the integral, this shows that ''we can construct a field whose Laplacian is a given field''. More precisely, it shows that we can construct a ''scalar'' field whose Laplacian is a given ''scalar''&amp;#8202; field{{math| ''&amp;rho;''('''r''')}}. But, due to the linearity of the Laplacian, the same applies to any given linear combination of scalar fields, including any combination whose coefficients are uniform vectors, uniform matrices, or uniform tensors of any order; that is, the same applies to any field that we can express with a uniform basis.

Mathematically, ({{EquationNote|38}}) is simply an identity. To find its significance in electrostatics, we can multiply it by&amp;#8202; {{math|&amp;minus;1&amp;#10744;''&amp;epsiv;''&lt;sub&gt;0&lt;/sub&gt;&amp;#8239;,}} obtaining
{{NumBlk|:|&lt;math&gt;-\frac{\rho(\mathbf{r})}{\epsilon_0} \,\equiv\,
  \triangle\iiint
    \frac{\,\rho(\mathbf{r}')}{4\pi\epsilon_0}\,
    \frac{1}{|\mathbf{r}\!-\!\mathbf{r}'|}
  \,dV',&lt;/math&gt;|{{EquationRef|39}}}}
which is also an identity. But the negative gradient of the expression after the integral sign is
:&lt;math&gt;\frac{\,\rho(\mathbf{r}')\,dV'}{4\pi\epsilon_0}\,
  \frac{\mathbf{r}\!-\!\mathbf{r}'}{|\mathbf{r}\!-\!\mathbf{r}'|^3}\,,&lt;/math&gt;
which is the contribution to the electric field at position{{math| '''r'''}} due to a charge&amp;#8202; {{math|''&amp;rho;''('''r&amp;prime;''')&amp;#8239;''dV&amp;prime;''}}&amp;#8239; at position{{math| '''r&amp;prime;'''}} in a vacuum. So the expression after the integral sign is the corresponding contribution to the electrostatic potential, and the whole integral is the whole electrostatic potential. Denoting this by &lt;math&gt;\varphi~\!,\,&lt;/math&gt; we can rewrite ({{EquationNote|39}}) as
{{NumBlk|:|&lt;math&gt;
    \triangle\varphi = -\frac{\rho}{\,\epsilon_0} \,.
  &lt;/math&gt;|{{EquationRef|40}}}}
This is '''Poisson's equation''' in electrostatics, treating the medium as a vacuum (so that {{mvar|&amp;rho; }}must be taken as the ''total'' charge density, including any contributions caused by the effect of the field on the medium). In a region in which&amp;#8201; {{math|''&amp;rho;''&amp;#8201;{{=}}&amp;#8201;0&amp;#8202;,}}&amp;#8201; Poisson's equation ({{EquationNote|40}}) reduces to
{{NumBlk|:|&lt;math&gt;
    \triangle\varphi = 0 \,,
  &lt;/math&gt;|{{EquationRef|41}}}}
which is '''Laplace's equation''' in electrostatics.

{{cob}}

=== The wave equation ===

{{cot}}

It is an empirical fact that a compressible fluid, such as air, carries waves of a mechanical nature: sound waves. In establishing the unambiguity of the gradient and the divergence, we have already derived equations dealing with the inertia and continuity (mass-conservation) of non-viscous fluids. So, by introducing a relation describing the compressibility, and eliminating variables, we should be able to get ''one'' equation (the &quot;wave equation&quot;) in ''one'' scalar or vector field (the &quot;wave function&quot;), with recognizably &quot;wavelike&quot; solutions. And we should expect this equation to be analogous to equations describing other kinds of waves.

If we suppose, for simplicity, that the only force acting on an element of fluid is the pressure force, then the applicable equation of motion is ({{EquationNote|6g}}). But, for reasons which will soon be apparent, let us call the pressure{{mvar| P}}, so that ({{EquationNote|6g}}) becomes
:&lt;math&gt;
    \rho\,\frac{d\mathbf{v}}{dt} = -\nabla P \,.
  &lt;/math&gt;
Then at ''equilibrium'' we have
:&lt;math&gt;
    0 ~\!= -\nabla P_0 \,,
  &lt;/math&gt;
where {{math|''P''&lt;sub&gt;0&lt;/sub&gt;}} is the equilibrium pressure. Subtracting this equation from the previous one and defining
:&lt;math&gt;p = P - P_0 \,,&lt;/math&gt;
we get
:&lt;math&gt;
    \rho\,\frac{d\mathbf{v}}{dt} = -\nabla p \,,
  &lt;/math&gt;
which looks like ({{EquationNote|6g}}), except that {{mvar|p}} is now the '''sound pressure''' (also called &quot;acoustic pressure&quot;, or sometimes &quot;excess pressure&quot;), i.e. the pressure rise above equilibrium.

For the equation of continuity we can use ({{EquationNote|7d'}}), which we repeat for convenience:
:&lt;math&gt;
    \rho\operatorname{div}\mathbf{v} = -\frac{d\rho}{dt} \,.
  &lt;/math&gt;

Eliminating {{math|'''v'''}} between the last two equations is fraught because {{math|'''v''' }}is evaluated at a moving point in the former and at a fixed point in the latter; and introducing any relation between {{mvar|p}} and {{mvar|&amp;rho;}} is similarly fraught because {{mvar|p}} is evaluated at a fixed point and {{mvar|&amp;rho;}} at a moving point. The obvious remedy is to apply the advection rule ({{EquationNote|16}}) to the last two equations, obtaining respectively
:&lt;math&gt;\begin{align}
    \rho\Big(\tfrac{\part\mathbf{v}}{\part t}
             + \mathbf{v}\cdot\nabla\mathbf{v}\Big) \,&amp;=\, -\nabla p ~; \\
    \rho\operatorname{div}\mathbf{v} \,&amp;=\,
      -\tfrac{\part\rho}{\part t} - \mathbf{v}\cdot\nabla\rho \,.
  \end{align}&lt;/math&gt;
That gets all the variables evaluated at fixed points, at the cost of making the equations more complicated and more obviously non-linear. But the equations and be simplified and linearized by '''small-amplitude approximations'''. In the parentheses in the first equation, the first term is proportional to the amplitude of the vibrations while the second term is a product of ''two'' factors proportional to the amplitude, so that, for sufficiently small amplitudes, the second term is negligible. Similarly, in the second equation, for sufficiently small amplitudes and a ''homogeneous medium'', we can neglect the second term on the right. Then, on the left side of each equation, we are left with a factor proportional to the amplitude, multiplied by{{mvar| &amp;rho;}}. But {{mvar|&amp;rho;}} is not proportional to the amplitude; only its deviation from the equilibrium density is so proportional. Hence, for small amplitudes,&amp;#8201; {{mvar|&amp;rho; }}can be replaced by the equilibrium density, which we shall call{{math| ''&amp;rho;''&lt;sub&gt;0&lt;/sub&gt;&amp;#8202;,}} which is independent of time and (in a homogeneous medium) independent of position. With these approximations, our equations of motion and continuity become
:&lt;math&gt;\begin{align}
    \rho_0 \mathbf{\dot{v}} &amp;= -\nabla p \,, \\[.5ex]
    \rho_0 \operatorname{div}\mathbf{v} &amp;= -\dot{\rho} \,,
  \end{align}&lt;/math&gt;
where, for brevity, we use an overdot to denote ''partial'' differentiation w.r.t. time (i.e., at a ''fixed'' point, not a point moving with the fluid).

Now we can eliminate {{math|'''v'''}}. Taking divergences in the first equation, and differentiating the second ''partially'' w.r.t. time (which can be done inside the {{math|div}} operator, which represents a linear combination), we get
:&lt;math&gt;\begin{align}
    \rho_0 \operatorname{div}\mathbf{\dot{v}} &amp;= -\triangle p \,, \\[.5ex]
    \rho_0 \operatorname{div}\mathbf{\dot{v}} &amp;= -\ddot{\rho} \,,
  \end{align}&lt;/math&gt;
so that we can equate the right-hand sides, obtaining
{{NumBlk|:|&lt;math&gt;\ddot{\rho} = \triangle p \,.&lt;/math&gt;|{{EquationRef|42}}}}

Maintaining the small-amplitude assumption, we can now consider compressibility. For ''small'' compressions in a ''homogeneous'' medium, we may suppose that the pressure change {{mvar|dp}} is some constant times the density change{{mvar| d&amp;rho;}}. It is readily verified that such a constant must have the dimension of velocity squared. So we can say&amp;#8201; {{math|''dp''&amp;#8201;{{=}}&amp;#8201;''c''&amp;sup2;&amp;#8202;''d&amp;rho;''&amp;#8202;,}} where {{mvar|c}} is a constant with the units of velocity.{{efn|When a gas is compressed, work is done on it, causing its temperature to rise, so that the ratio of {{mvar|dp}} to{{mvar| d&amp;rho;}} is higher than if the compression were isothermal. In sound waves, there is typically not enough time for a significant part of the heat of compression to be conducted away; that is, the compression is near enough to '''adiabatic'''. The words &quot;not enough time&quot; may suggest that the adiabatic approximation is a high-frequency approximation. But in fact, in free air, it is a ''low''-frequency approximation, because as the frequency is reduced, the equalization of temperature is hindered more by the longer wavelength than it is helped by the longer period. Only in a confined space, which limits the required distance of conduction, does the adiabatic assumption require the frequency to be ''above'' some lower limit. In a musical wind instrument, that lower limit tends to be far below the audible range. Meanwhile the upper limit, due to easier heat conduction within a shorter wavelength, tends to be very far above the audible range. Thus, under typical conditions, for the purpose of calculating{{mvar| c&amp;#8202;}}, the adiabatic assumption is reasonable. (See [[#fletcher-74|Fletcher, 1974]].)}} Dividing by {{mvar|dt}} gives&amp;#8201; &lt;math&gt;\dot{p}\!=\!c^2\dot{\rho}~\!,\,&lt;/math&gt; whence
{{NumBlk|:|&lt;math&gt;\ddot{p} = c^2~\!\ddot{\rho} \,.&lt;/math&gt;|{{EquationRef|43}}}}
Substituting from  ({{EquationNote|42}}) then gives the desired '''wave equation''':
{{NumBlk|:|&lt;math&gt;\ddot{p} = c^2 \triangle p \,.&lt;/math&gt;|{{EquationRef|44}}}}

This is the 3D classical wave equation with the sound pressure {{mvar|p}} as the wave function. For a generic wave function {{mvar|&amp;psi;&amp;#8202;,}} in a homogeneous isotropic medium, we would expect the equation to be
{{NumBlk|:|&lt;math&gt;\ddot{\psi} = c^2 \triangle\psi \,,&lt;/math&gt;|{{EquationRef|45}}}}
which may be written more compactly as
{{NumBlk|:|&lt;math&gt;\Box\psi =~\! 0 \,,&lt;/math&gt;|{{EquationRef|46}}}}
where {{math|&amp;#9744;,}} pronounced &quot;wave&quot; or &quot;box&quot;,{{efn|Or sometimes &quot;quabla&quot;, by analogy with &quot;nabla&quot;.}} is called the '''D'Alembertian''' operator and is defined by
{{NumBlk|:|&lt;math&gt;
    \Box\psi := \triangle\psi - \frac{1}{\,c^2}\frac{\part^2 \psi}{\part t^2}
  &lt;/math&gt;|{{EquationRef|47}}}}
in this paper, although other conventions exist.{{efn|In particular, some authorities change the sign, defining {{math|&amp;#9744;}} as&amp;#8201; &lt;math&gt;\tfrac{1}{\,c^2}\tfrac{\part^2}{\part t^2}\!-\!\triangle&lt;/math&gt;&amp;#8239;,&amp;#8201; and some write the operator (however defined) as{{math| &amp;#9744;&lt;sup&gt;2&lt;/sup&gt;}}.}}

In a ''static'' situation, the second term on the right of ({{EquationNote|47}}) is zero. So one advantage of definition ({{EquationNote|47}}), over any alternative definition that changes the sign or the scale factor, is that ''in the static case, the D'Alembertian is reduced to the Laplacian'', making it especially obvious that ''in the static case, the wave equation is reduced to Laplace's equation'' [compare ({{EquationNote|46}}) and ({{EquationNote|41}})]. Also notice that the D'Alembertian, being a linear combination of two linear operators, is itself ''linear''.

{{cob}}

=== Spherical waves ===

{{cot}}

Having established that there are wavelike time-dependent fields described by equation ({{EquationNote|45}}), in which the constant {{mvar|c}} has the units of velocity, we can now make an informed guess at an elementary solution of the equation. Consider the candidate
{{NumBlk|:|&lt;math&gt;
    \psi(\mathbf{r},t) = \tfrac{1}{\,r\,}~\!f\big(t-r/c\big) \,,
  &lt;/math&gt;|{{EquationRef|48}}}}
where&amp;#8201; &lt;math&gt;\mathbf{r}=r\mathbf{\hat{r}}&lt;/math&gt;&amp;#8201; is the position vector (so that {{mvar|r}} is distance from the origin),&amp;#8201; {{mvar|f}}&amp;#8239; is an arbitrary function (arbitrary except that it will need to be twice differentiable),&amp;#8201; {{mvar|t }}is time, and {{mvar|c }}is a constant (and obviously {{mvar|&amp;psi; }}is not defined at the origin even if {{mvar|f&amp;#8202; }}is.)

If, at the origin, the function {{mvar|f}}&amp;#8239; has a certain argument at time&amp;#8201; {{math|''t&amp;#8201;{{=}}&amp;#8201;&amp;tau;''&amp;#8202;,}}&amp;#8201; then at any distance{{mvar| r}}&amp;#8202; from the origin, it has the same argument at time&amp;#8201; {{math|''t&amp;#8201;{{=}}&amp;#8201;&amp;tau;&amp;#8239;+&amp;#8202;r''&amp;#10744;''c''&amp;#8202;,}}&amp;#8201; which is&amp;#8201; {{math|''r''&amp;#10744;''c'' }}&amp;#8239;''later''&amp;#8202; than at the origin. Hence, if {{mvar|f}}&amp;#8239; has a certain feature (e.g., a zero-crossing) at the origin, the time taken for that feature to reach any distance{{mvar| r}}&amp;#8239; is{{math| ''r''&amp;#10744;''c''&amp;#8202;,}}&amp;#8239; implying that the feature travels outward from the origin at speed{{mvar| c}}.&amp;#8201; Another way to perceive this is to set the argument of{{mvar| f}}&amp;#8201; equal to a constant (corresponding to some feature of the function) and differentiate w.r.t.{{mvar| t&amp;#8202;,}} obtaining&amp;#8201; &lt;math&gt;\dot{r}\!=\!c&lt;/math&gt;&amp;#8202; (the speed at which the feature recedes from the origin). Thus equation ({{EquationNote|48}}) describes ''waves''&amp;#8202; radiating outward from the origin with speed{{mvar| c}}. (The symbol {{mvar|c}} comes from a general-purpose Latin word for speed, but has become the usual symbol for ''wave'' speed.)

Equation ({{EquationNote|48}}) further implies that there are surfaces over which the wave function {{mvar|&amp;psi;}}&amp;#8239; is uniform&amp;mdash;namely surfaces of constant{{mvar| r}},&amp;#8201; i.e. spheres centered on the origin. These are the '''wavefronts'''. So ({{EquationNote|48}}) describes '''spherical waves'''.

Because the surface area of a sphere is proportional to the square of its radius, we should expect the radiated '''intensity''' (power per unit area) to satisfy an ''inverse-square law'' (if the medium is ''lossless''&amp;mdash;neither absorbing nor scattering the radiated power). That does ''not'' mean that the wave function itself should satisfy an inverse-square law. In a traveling wave in 3D space, there will be an &quot;effort&quot; variable (e.g., sound pressure) and a &quot;flow&quot; variable (e.g., fluid velocity), and the instantaneous intensity will be proportional to the product of the two. If the two are proportional to each other, the instantaneous intensity will be proportional to the square of one or the other. Hence if the instantaneous intensity falls off like{{math| 1/''r''&amp;#8202;&amp;sup2;,}} the effort and flow variables&amp;mdash;and the wave function, if it is proportional to one or the other&amp;mdash;will fall off like{{math| 1/''r''}}. That suggests the attenuation factor {{math|1/''r''}}&amp;#8202; in ({{EquationNote|48}}).

But there are big ''if''&amp;#8202;s in that argument. For all we know so far, the relation between effort and flow could involve a lag, so that the ''instantaneous'' product of the two could swing negative although it averages to something positive. And for all we know so far, the lag could vary with{{mvar| r}}, allowing at least one of the two (effort or flow) to depart from the {{math|1/''r''}}&amp;#8202; law, even if their average product still falls off like{{math| 1/''r''&amp;#8202;&amp;sup2;}}. The {{math|1/''r''}}&amp;#8202; factor in ({{EquationNote|48}}) is therefore only an &quot;informed guess&quot;. Notwithstanding these complications, we have also guessed that the form of the function {{mvar|f}}&amp;#8202; (the '''waveform''') does not change as {{mvar|r}} increases; we have not considered whether this behavior might depend on the medium, or the waveform, or the geometry of the wavefronts.

So let us carefully check whether ({{EquationNote|48}}) satisfies ({{EquationNote|45}}) or, equivalently, ({{EquationNote|46}}).

As a first step, and as a useful inquiry in its own right, we find {{math|&amp;#9651;''&amp;psi;''}} from definition ({{EquationNote|4L}}), given that {{mvar|&amp;psi;}} is a function of {{math|(''r'',&amp;#8202;''t'') }}only. For the surface {{mvar|&amp;delta;S}}&amp;#8202; let us start with
* a cone (''not'' a double cone) with its apex at the origin, subtending a ''small'' solid angle {{mvar|&amp;omega;}} at the origin,
* a sphere centered on the origin, with radius {{math|''r''}}, and
* a sphere centered on the origin, with radius {{mvar|r&amp;#8202;+&amp;#8202;dr&amp;#8202;}};
and let the volume element be the region inside the cone and between the spheres, so that its enclosing surface {{mvar|&amp;delta;S}}&amp;#8202; has three faces: a segment of the cone, a segment of the inner sphere with area{{math| ''r''&amp;#8202;&amp;sup2;''&amp;#8202;&amp;omega;''&amp;#8202;,}} and a segment of the outer sphere with area{{math| (''r&amp;#8202;+&amp;#8202;dr'')&lt;sup&gt;2&lt;/sup&gt;''&amp;omega;''&amp;#8202;}}. By the symmetry of{{mvar| &amp;psi;&amp;#8202;}}, the outward normal derivative {{mvar|&amp;part;&lt;sub&gt;n&lt;/sub&gt;&amp;#8202;&amp;psi;}}&amp;#8202; is equal to zero on the conical face,&amp;#8201; {{math|+''&amp;part;&lt;sub&gt;r&lt;/sub&gt;&amp;#8202;&amp;psi;''(''r&amp;#8202;+&amp;#8202;dr'',&amp;#8202;''t'')}} on the outer spherical face, and&amp;#8202; {{math|&amp;minus;''&amp;part;&lt;sub&gt;r&lt;/sub&gt;&amp;#8202;&amp;psi;''(''r'',&amp;#8202;''t'')}} on the inner spherical face. The volume of the element is&amp;#8201; {{math|''dV''&amp;#8201;{{=}}&amp;#8201;''r''&amp;#8202;&amp;sup2;''&amp;#8202;&amp;omega;&amp;#8239;dr''}}. So, assembling the pieces of definition ({{EquationNote|4L}}), we get
:&lt;!-- SUBSCRIPTS ENLARGED FOR LEGIBILITY: --&gt;&lt;math&gt;\begin{align}\triangle\psi
    &amp;= \frac{1}{r^2 \omega \,dr}\Big(\!
         (r\!+\!dr)^2 \omega ~\!\part_{\textstyle r} \psi(r\!+\!dr,t)
         - r^2 \omega ~\!\part_{\textstyle r} \psi(r,t)
       \!\Big) \\[1ex]
    &amp;= \frac{1}{\,r^2}~\!
       \frac{(r\!+\!dr)^2 \part_{\textstyle r}\psi(r\!+\!dr,t)
             - r^2 \part_{\textstyle r}\psi(r,t)}{dr} \\[.5ex]
    &amp;= \frac{1}{\,r^2}~\!
       \frac{\part}{\part r}\Big(r^2 \part_{\textstyle r}\psi(r,t)\Big) \,,
  \end{align}&lt;/math&gt;
i.e.
{{NumBlk|:|&lt;math&gt;
    \triangle\psi(r,t)
    \equiv \frac{1}{\,r^2}~\!\frac{\part}{\part r}
             \Big(r^2 \frac{\part\psi}{\part r}\Big)
             \qquad \big[\mathsf{if}\,\,r\!\neq~\!\!0\big]\,.
  &lt;/math&gt;|{{EquationRef|49}}}}

Now we can verify our &quot;informed guess&quot;. Differentiating ({{EquationNote|48}}) twice w.r.t.{{mvar| t}}&amp;#8202; by the chain rule gives
{{NumBlk|:|&lt;math&gt;
    \frac{\part^2\psi}{\part t^2} = \frac{1}{\,r\,}~\!f''\!\big(t-r/c\big) \,,
  &lt;/math&gt;|{{EquationRef|50}}}}
where each prime {{math|(&amp;prime;)}} denotes differentiation of the function w.r.t. its own argument. Differentiating ({{EquationNote|48}}) once w.r.t.{{mvar| r}}&amp;#8202; by the product rule and chain rule, we get
{{NumBlk|:|&lt;math&gt;
    \frac{\part\psi}{\part r}
    \,=\, -\frac{1}{cr}~\!f'\!\big(t-r/c\big)
          -\frac{1}{\,r^2}~\!f\big(t-r/c\big) \,.
  &lt;/math&gt;|{{EquationRef|51}}}}
Proceeding as specified in ({{EquationNote|49}}), we multiply this by {{math|''r''&amp;#8202;&amp;sup2;}}, differentiate again w.r.t.{{mvar| r}} (giving three terms, of which two cancel), and divide by {{math|''r''&amp;#8202;&amp;sup2;}}, obtaining
{{NumBlk|:|&lt;math&gt;
    \triangle\psi = \frac{1}{c^2 r}~\!f''\!\big(t-r/c\big) \,.
  &lt;/math&gt;|{{EquationRef|52}}}}
Then if we substitute ({{EquationNote|52}}) and ({{EquationNote|50}}) into ({{EquationNote|47}}), we obviously get&amp;#8201; {{math|&amp;#9744;''&amp;psi;''&amp;#8201;{{=}}&amp;#8201;0&amp;#8202;,}} satisfying ({{EquationNote|46}}). So we have guessed correctly.

Having shown that the D'Alembertian of{{mvar| &amp;psi;&amp;#8202;}}, as given by ({{EquationNote|48}}), is zero everywhere except at the origin (where it is not defined), let us now find its integral over a volume{{mvar| V}} (enclosed by a surface{{mvar| S}}) that includes the origin. From ({{EquationNote|47}}),
:&lt;math&gt;\begin{align}
    \iiint_V \Box\psi \,dV
      \,&amp;= \iiint_V \triangle\psi \,dV
           - \frac{1}{\,c^2}\iiint_V \frac{\part^2 \psi}{\part t^2}\,dV\\[.5em]
        &amp;= \iint_S \part_n \psi \,dS
           - \frac{1}{\,c^2}\iiint_V \frac{\part^2 \psi}{\part t^2}\,dV\,,
  \end{align}&lt;/math&gt;
where the second equality follows from theorem ({{EquationNote|5L}}). Now because the integrand on the left is zero except at the origin, ''any''{{mvar| V}} containing the origin will give the same integral. So for convenience, let {{mvar|V}} be a spherical ball of radius{{mvar| R}} centered on the origin. Then, by the spherical symmetry of{{mvar| &amp;psi;&amp;#8202;,}} integration over{{mvar| S}} reduces to multiplication by{{math| 4''&amp;pi;R''&amp;#8202;&lt;sup&gt;2&lt;/sup&gt;,}} and {{mvar|&amp;part;&lt;sub&gt;n&lt;/sub&gt;}} is equivalent to{{mvar| &amp;part;&lt;sub&gt;r&lt;/sub&gt;&amp;#8202;,}} and {{mvar|dV}} can be taken as{{math| 4''&amp;pi;r''&lt;sup&gt;&amp;#8202;2&lt;/sup&gt;''dr''}}. With these substitutions we have
:&lt;math&gt;
    \iiint_V \Box\psi \,dV \,=\,
      4\pi R^2 \frac{\part\psi}{\part r}\bigg|_{r=R} \!
      - \frac{1}{\,c^2}\!\int_0^R \!\frac{\part^2 \psi}{\part t^2}\,4\pi r^2\,dr
  &lt;/math&gt;
or, substituting from ({{EquationNote|51}}) and ({{EquationNote|50}}),
:&lt;math&gt;\begin{align}
    \iiint_V \!\Box\psi \,dV ~\!\!
     =\,&amp; 4\pi R^2 \!\Big(\!{-}\tfrac{1}{cR} f'\!\big(t\!-\!R/c\big)
                   - \tfrac{1\,}{R^2} f\big(t\!-\!R/c\big)\!\Big) \\
        &amp;- \tfrac{1}{\,c^2}\!\int_0^R \!\tfrac{1}{\,r\,}~\!
                                      f''\!\big(t-r/c\big)\,4\pi r^2\,dr \\[1ex]
     =\,&amp;-\tfrac{4\pi R}{\,c\,}~\!f'\!\big(t\!-\!R/c\big)
           - 4\pi f\big(t\!-\!R/c\big) \\
        &amp;- \tfrac{4\pi}{\,c^2}\!\int_0^R \!rf''\!\big(t-r/c\big) \,dr \,.
  \end{align}&lt;/math&gt;
Again noting that any {{mvar|V}} containing the origin will give the same volume integral, we can let {{mvar|R}} approach zero, with the result that the right-hand side approaches&amp;#8239;{{math| &amp;minus;4''&amp;pi;f''&amp;#8202;(''t'')}}. This is the integral of{{math| &amp;#9744;''&amp;psi;''}} over any volume containing the origin, for {{mvar|&amp;psi;}} given by ({{EquationNote|48}}). Meanwhile {{math|&amp;#9744;''&amp;psi;''}} is zero everywhere except that the origin. In summary,
{{NumBlk|:|&lt;math&gt;
    \Box~\!\Big\{\!\tfrac{1}{\,r\,}~\!f\big(t-r/c\big)\!\Big\}
    \equiv -4\pi f(t)\,\delta(\mathbf{r}) \,.
  &lt;/math&gt;|{{EquationRef|53}}}}
Shifting the center of the spherical waves from the origin to position{{math| '''r&amp;prime;''',}} we get
{{NumBlk|:|&lt;math&gt;
    \Box~\!\Big\{\tfrac{1}{|\mathbf{r}-\mathbf{r}'|}
                 ~\!f\big(t-\tfrac{|\mathbf{r}-\mathbf{r}'|\,}{c}\big)\Big\}
    \equiv -4\pi f(t)\,\delta(\mathbf{r}\!-\!\mathbf{r}') \,.
  &lt;/math&gt;|{{EquationRef|54}}}}

We shall refer to the field given by ({{EquationNote|48}}) as the wave function due to a '''monopole''' source with '''strength''' {{math|''f''&amp;#8202;(''t'')}} at the origin. The D'Alembertian of this wave function is given by ({{EquationNote|53}}).&lt;ref&gt;Our definition of ''strength'' follows the old convention used by Baker &amp;amp; Copson ([[#baker-copson-39|1939, p.&amp;#8239;42]]), Born &amp;amp; Wolf ([[#born-wolf-02|2002, p.&amp;#8239;421]]), and Larmor ([[#larmor-1904|1904, p.&amp;#8239;5]]). The newer convention followed by Miller ([[#miller-91|1991, p.&amp;#8239;1371]]) would use the denominator {{math|4''&amp;pi;r''}} instead of our {{mvar|r}} in ({{EquationNote|48}}); this would have the advantage of eliminating the factor{{math| 4''&amp;pi;''}} from the D'Alembertian of the wave function, and the disadvantage of introducing that factor into the (denominator of the) wave function itself.&lt;/ref&gt; Hence the field whose D'Alembertian is given by ({{EquationNote|54}}) is the wave function due to a monopole source with strength {{math|''f''&amp;#8202;(''t'')}} at position{{math| '''r&amp;prime;'''}}. In each case, the D'Alembertian is zero everywhere except at the source; that is, the field satisfies the wave equation except at the source.

''A note in passing:&amp;#8202;'' The above verification that the field ({{EquationNote|48}}) satisfies the wave equation (except at the origin) did not depend on whether {{mvar|c}} was positive or negative. Nor did the demonstration that its D'Alembertian is given by ({{EquationNote|53}}). Hence we may replace{{mvar| c}} by{{mvar| &amp;minus;c}} in ({{EquationNote|54}}) and conclude that, even for positive{{mvar| c&amp;#8202;}}, the field
{{NumBlk|:|&lt;math&gt;
    \tfrac{1}{|\mathbf{r}-\mathbf{r}'|}
    ~\!f\big(t+\tfrac{|\mathbf{r}-\mathbf{r}'|\,}{c}\big)
  &lt;/math&gt;|{{EquationRef|54a}}}}
also satisfies the wave equation (except at{{math| '''r&amp;prime;'''}}), and has the same D'Alembertian as in ({{EquationNote|54}}) and the same limiting behavior as{{math| '''r'''&amp;rightarrow;'''r&amp;prime;'''}} and the argument of{{mvar| f}}&amp;#8201; approaches{{mvar| t}}. In this case, however, for positive{{mvar| c&amp;#8202;}}, we can hardly speak of a &quot;source&quot; at{{math| '''r&amp;prime;'''}}, because expression ({{EquationNote|54a}}) describes ''inward''-bound spherical waves converging on{{math| '''r&amp;prime;'''}}, which would violate causality if {{math| '''r&amp;prime;'''}} were the location of the source. But even if ({{EquationNote|54a}}) is dismissed as an &quot;acausal&quot; or &quot;unphysical&quot; solution of the wave equation, it nevertheless ''is'' a solution, and we are free to exploit this fact (such as it is) in derivations and proofs.

{{cob}}

=== Field with given D'Alembertian ===

{{cot}}

Now suppose that, instead of a monopole wave source with strength {{math|''f''&amp;#8202;(''t'')}} at the general position{{math| '''r&amp;prime;''',}} we have at that position a source strength ''density&lt;math&gt;~w(\mathbf{r}'\!,t)&lt;/math&gt;'' in an elemental volume {{mvar|dV&amp;prime;}}, whose (causal!) contribution to the wave function {{mvar|&amp;psi;}} at position{{math| '''r'''}}&amp;#8202; is therefore
:&lt;math&gt;d\psi(\mathbf{r},t)
    = \tfrac{1}{|\mathbf{r}-\mathbf{r}'|}~\!
      w\big(\mathbf{r}',\,t-\tfrac{|\mathbf{r}-\mathbf{r}'|\,}{c}\big)\,dV' ,
  &lt;/math&gt;
where for each {{math|'''r''',}} the dimensions of each volume element are small compared with {{math|{{abs|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''}}}}. Then the total wave function is the sum of the contributions:
{{NumBlk|:|&lt;math&gt;\psi(\mathbf{r},t)
    = \iiint \tfrac{1}{|\mathbf{r}-\mathbf{r}'|}~\!
      w\big(\mathbf{r}',\,t-\tfrac{|\mathbf{r}-\mathbf{r}'|\,}{c}\big)\,dV' ,
  &lt;/math&gt;|{{EquationRef|55}}}}
where the integral is over all space.

Independently of the physical significance of{{math| ''&amp;psi;''('''r''',&amp;#8202;''t''),}} we can take its D'Alembertian &quot;under the integral sign&quot; by rule ({{EquationNote|54}}), obtaining
:&lt;math&gt;\begin{align}\Box\psi(\mathbf{r},t)
   &amp;= \iiint \Big({-}4\pi~\!w(\mathbf{r}'\!,t)\,
                  \delta(\mathbf{r}\!-\!\mathbf{r}')\Big)\,dV' \\[.5ex]
   &amp;= \iiint \Big({-}4\pi~\!w(\mathbf{r},t)\,
                  \delta(\mathbf{r}\!-\!\mathbf{r}')\Big)\,dV' \\[.5ex]
   &amp;= -4\pi~\!w(\mathbf{r},t)\!
       \iiint\delta(\mathbf{r}\!-\!\mathbf{r}')\,dV' \\[.5ex]
   &amp;= -4\pi~\!w(\mathbf{r},t)\!
       \iiint\delta(\mathbf{r}'{-}~\!\mathbf{r})\,dV' ;
  \end{align}&lt;/math&gt;
that is,
{{NumBlk|:|&lt;math&gt;
    \Box\psi(\mathbf{r},t) = -4\pi~\!w(\mathbf{r},t) \,.
  &lt;/math&gt;|{{EquationRef|56}}}}

Mathematically, equation ({{EquationNote|56}}) is an identity which applies if {{math|''&amp;psi;''('''r''',&amp;#8202;''t'')}} is given by ({{EquationNote|55}}). Substituting from ({{EquationNote|55}}) and solving for&lt;math&gt;~w,&lt;/math&gt; we can write the identity in full as
{{NumBlk|:|&lt;math&gt;w(\mathbf{r},t) \equiv \Box\bigg({-}\tfrac{1}{4\pi}\!\iiint
      \tfrac{1}{|\mathbf{r}-\mathbf{r}'|}~\!
      w\big(\mathbf{r}',\,t-\tfrac{|\mathbf{r}-\mathbf{r}'|\,}{c}\big)
    \,dV'\bigg) \,,
  &lt;/math&gt;|{{EquationRef|57}}}}
where the integral is over all space, or at least all of the space in which&lt;math&gt;~w&lt;/math&gt; may be non-zero. Subject to the convergence of the integral, this shows that ''we can construct a wave function with a given D'Alembertian''.

Physically, equation ({{EquationNote|56}}) gives the D'Alembertian of the wave function for a source density&amp;nbsp;&lt;math&gt;w&lt;/math&gt;. It is the ''inhomogeneous wave equation'', which applies in the presence of an arbitrary source density&amp;mdash;in contrast to the ''homogeneous wave equation'' ({{EquationNote|46}}), which applies in a region where the source density is zero. In this context the word ''homogeneous'' or ''inhomogeneous'' describes the equation, not the medium (which has been assumed homogeneous and isotropic).

In a ''static'' situation, in which the D'Alembertian is reduced to the Laplacian, the inhomogeneous wave equation ({{EquationNote|56}}) is reduced to the form of Poisson's equation ({{EquationNote|40}}). As written, equation ({{EquationNote|40}}) is Poisson's equation in electro''statics''; it applies to the charge density{{math| ''&amp;rho;''('''r''')}}, for which the scalar potential [in ({{EquationNote|39}})] is
:&lt;math&gt;\varphi(\mathbf{r}) = \tfrac{1}{4\pi\epsilon_0}
    \iiint \tfrac{1}{|\mathbf{r}-\mathbf{r}'|}
    \rho(\mathbf{r}') \,dV'.
  &lt;/math&gt;
In electro''dynamics'', which takes time-dependence into account, the scalar potential due to the charge density{{math| ''&amp;rho;''('''r''',&amp;#8202;''t'')}} is
:&lt;math&gt;\varphi(\mathbf{r},t) = \tfrac{1}{4\pi\epsilon_0}
    \iiint \tfrac{1}{|\mathbf{r}-\mathbf{r}'|}
    \rho\big(\mathbf{r}',\,t-\tfrac{|\mathbf{r}-\mathbf{r}'|\,}{c}\big) \,dV',
  &lt;/math&gt;
where the wave speed {{mvar|c}} is the speed of light; this is the same as in the static case except for the delay {{math|{{sfrac|&amp;#8202;{{abs|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''}}&amp;#8202;|''c''}}&amp;#8202;,}} indicating that the influence of the change density at{{math| '''r&amp;prime;'''}} travels outward from that point at the speed of light. In the dynamic case, by rule ({{EquationNote|57}}), the D'Alembertian of the scalar potential is
:&lt;math&gt;
    \Box\varphi = -\frac{\rho(\mathbf{r},t)}{\,\epsilon_0} \,.
  &lt;/math&gt;
This result is the inhomogeneous wave equation in the scalar potential&amp;mdash;the equation which, in the electro''static'' case, reduces to Poisson's equation ({{EquationNote|40}}).

In electro''dynamics'', however, the electric field&amp;#8202; {{math|'''E'''}} is ''not'' simply&lt;math&gt;\,\,{-}\nabla\varphi~\!,\,&lt;/math&gt; but&lt;math&gt;~\,{-}\nabla\varphi~\!\!-\!\tfrac{\part\mathbf{A}}{\part t}~\!,\,&lt;/math&gt; where {{math|'''A'''}} is the '''magnetic vector potential''', whose defining property is that its curl is the '''magnetic flux density''':
:&lt;math&gt;\mathbf{B} = \operatorname{curl}\mathbf{A} \,.&lt;/math&gt;
By identity ({{EquationNote|24d}}), this property implies
:&lt;math&gt;\operatorname{div}\mathbf{B} = 0 \,,&lt;/math&gt;
which is '''Gauss's law for magnetism'''. We have noted in passing&amp;mdash;but not yet proven&amp;mdash;that ({{EquationNote|24d}}) has a converse, whereby the solenoidality of{{math|&amp;#8202; '''B'''}} implies the ''existence'' of the vector potential{{math| '''A'''}}.  Precedents suggest we might be able to prove this by finding a vector field whose curl is a delta function&amp;mdash;perhaps through new identities relating it to a field whose divergence is a delta function&amp;mdash;and using it to construct a vector field with a given curl. In fact we shall prove our &quot;converse&quot; differently, but we shall still need some new identities for the purpose. And to obtain those identities (among others), we must take the detour that we have made a virtue of ''not'' taking until now&amp;hellip;

{{cob}}

== Cartesian coordinates ==

=== Indicial notation; implicit summation ===

{{cot}}

Considering that a scalar field is a function of three coordinates, while a vector field has three components each of which is a function of three coordinates, we can readily imagine that coordinate-based derivations of vector-analytic identities are likely to be excruciatingly repetitive&amp;mdash;unless perhaps we choose a notation that concisely specifies the repetition. So, instead of writing the Cartesian coordinates as {{math|''x'',&amp;#8239;''y'',&amp;#8239;''z''&amp;#8202;,}}&amp;#8239; we shall usually write them as {{mvar|x&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; where&amp;#8202; {{math|''i''&amp;#8201;{{=}}&amp;#8239;1,&amp;#8202;2,&amp;#8202;3&amp;#8202;,}}&amp;#8201; respectively;&amp;#8202; and instead of writing the unit vectors in the directions of the respective axes as {{math| '''i''',&amp;#8202;'''j''','''k'''&amp;#8202;,}}&amp;#8239; we shall usually write them as {{math|'''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8202;}}.&amp;#8201; And for partial differentiation w.r.t.{{math| ''x&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;,}} instead of writing {{mvar|{{sfrac|&amp;part;|&amp;part;x&lt;sub&gt;i&lt;/sub&gt;}}}} or even {{math|''&amp;part;&lt;sub&gt;x&lt;sub&gt;i&lt;/sub&gt;&lt;/sub&gt;''&amp;#8202;,}} we shall write {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;}}.

Now comes a stroke of genius for which we are indebted to Einstein (although he used it in a more sophisticated context!). Instead of writing the position vector as
:&lt;math&gt;\mathbf{r} = x_1\mathbf{e}_1 + x_2\mathbf{e}_2 + x_3\mathbf{e}_3&lt;/math&gt;
or even as
:{{big|&lt;math&gt;\mathbf{r} = \textstyle\sum_i x_i \mathbf{e}_i \,,&lt;/math&gt;}}
we shall write it simply as
:{{big|&lt;math&gt;\mathbf{r} = x_i \mathbf{e}_i \,,&lt;/math&gt;}}
where it is ''understood''&amp;#8202; that we ''sum over the repeated index''. More generally, we shall write the vector field {{math|'''q'''}} as
:{{big|&lt;math&gt;\mathbf{q} = q_i \mathbf{e}_i&lt;/math&gt;}}
with implicit summation, and the vector field {{math|'''v'''}} as
:{{big|&lt;math&gt;\mathbf{v} = v_i \mathbf{e}_i&lt;/math&gt;}}
with implicit summation, and so on. (By that nomenclature, the position vector in Cartesian coordinates should be, and often is, called {{math|'''x'''&amp;#8202;}}; but we called it {{math|'''r'''}} because we wanted to call its magnitude {{mvar|r}}, for ''radius''.)

Implicit summation not only avoids writing the {{big|{{math|&amp;Sigma;}}}} symbol and specifying the index of summation, but also allows a summation over ''two'' repeated indices, say {{mvar|i}} and {{mvar|j&amp;#8202;}}, to be considered as summed first over {{mvar|i}} and then over {{mvar|j}} or vice versa, removing the need for an explicit regrouping of terms. Of course, if we hide messy details behind a notation, we need to make sure that it handles those details correctly. In particular, when we perform an operation on an implicit sum, we implicitly perform it ''term-by-term'', and must therefore make sure that the operation is valid when interpreted that way.

{{cob}}

=== Formulation of operators ===

{{cot}}

'''Gradient''':  Putting&amp;#8201; {{mvar|s&amp;#8201;{{=}}&amp;#8201;x&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8201; in ({{EquationNote|9g}}), we find that the scalar component of{{math|&amp;#8202; &amp;nabla;''p''}} in the direction of each {{math|'''e'''&lt;sub&gt;''i''&lt;/sub&gt;}}&amp;#8202; is{{mvar|&amp;#8202; &amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8201;p}}.&amp;#8201; To obtain the vector component in that direction, we multiply by {{math|'''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8202;}}.&amp;#8201; Assembling the components, we have (with implicit summation)
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla p = \mathbf{e}_i ~\!\part_i p
  &lt;/math&gt;}}|{{EquationRef|58g}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla =~\! \mathbf{e}_i \part_i
  &lt;/math&gt;}}|{{EquationRef|58o}}}}
or, in traditional longhand notation,
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla =~\! \mathbf{i}~\!\tfrac{\part}{\part x}
           +~\! \mathbf{j}~\!\tfrac{\part}{\part y}
           +~\! \mathbf{k}~\!\tfrac{\part}{\part z} \,.
  &lt;/math&gt;}}|{{EquationRef|58t}}}}
It is also worth noting, from ({{EquationNote|58g}}), that the squared magnitude of{{math|&amp;#8202; &amp;nabla;''p'' &amp;#8239;}}is
{{NumBlk|:|{{big|&lt;math&gt;
    |\nabla p|^2 =~\! \part_i p \;\part_i p \,,
  &lt;/math&gt;}}|{{EquationRef|58s}}}}
where we write&amp;#8239; {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;p&amp;#8201;&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;p}}&amp;#8201; rather than {{math|(''&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;p'')&lt;sup&gt;2&lt;/sup&gt;}}&amp;#8239; to ensure that implicit summation applies.

As reported by Tai ([[#tai-94|1994]]), there are unfortunately some textbooks in which the del operator is defined as
:{{big|&lt;math&gt;\nabla =~\!
    \tfrac{\part}{\part x}~\!\mathbf{i} +
    \tfrac{\part}{\part y}~\!\mathbf{j} +
    \tfrac{\part}{\part z}~\!\mathbf{k} \quad\qquad
  &lt;/math&gt;}}{{big|1=[''sic!''&amp;#8239;]}}
&amp;mdash;which, on its face, is not an operator at all, but a self-contained expression whose value is the zero vector (because it is a sum of derivatives of constant vectors). Among the offenders is Erwin Kreyszig, who, in the 6th edition of his bestselling ''Advanced Engineering Mathematics'' ([[#kreyszig-62-|1988]], p.&amp;#8239;486), misdefines the del operator thus and then rewrites the gradient of{{mvar|&amp;#8202; f}}&amp;#8239; as {{math|&amp;nabla;&amp;#8202;''f'',}} apparently imagining that the differentiation operators look ''through'' the constant vectors rather than ''at''&amp;#8202; them. Six pages later, he defines the divergence in Cartesian coordinates (which we shall do shortly) and then immediately informs us that &quot;Another common notation for the divergence of{{math| '''v'''}} is {{math|&amp;nabla;'''&amp;sdot;&amp;#8202;v'''}},&quot; where {{math|&amp;nabla;}} is defined as before, but the resulting {{math|&amp;nabla;'''&amp;sdot;&amp;#8202;v'''}} is apparently not identically zero!&lt;ref&gt;The latter passage, as it appears in the 5th edition (p.&amp;#8239;397), is the one cited by Tai ([[#tai-94|1994]], p.&amp;#8239;6).&lt;/ref&gt; These errors persist in the 10th edition ([[#kreyszig-62-|2011]], pp.&amp;#8239;396,&amp;#8239;402–3). Tai finds similar howlers in mathematics texts by Wilfred Kaplan, Ladis D. Kovach, and Merle C. Potter, and in electromagnetics texts by William H. Hayt and Martin A. Plonus.&lt;ref&gt;Quoted by Tai ([[#tai-94|1994]]), in alphabetical order within each category. For Kovach he could have added p.&amp;#8239;308.&amp;#8201; Potter he misnames as Porter.&lt;/ref&gt;&amp;#8201; Knudsen &amp;amp; Katz, in ''Fluid Dynamics and Heat Transfer'' (1958), avoid the misdefinition of{{math| &amp;nabla;,}} but implicitly define the divergence of{{math| '''V'''}} as {{math|1='''V&amp;sdot;'''&amp;nabla;}}&amp;#8202; (which, as we have seen, is actually an operator), and then somehow reduce it to the correct expression for{{math|&amp;#8202; div&amp;#8239;'''V'''}}.&amp;#8239;&lt;ref&gt;Quoted by Tai ([[#tai-94|1994]], p.&amp;#8239;23).&lt;/ref&gt;  But I digress.

'''Curl and divergence''':  Expressing the operand of the curl in components, and noting that the unit vectors are ''uniform'', we can apply ({{EquationNote|8p}}):
:{{big|&lt;math&gt;\begin{align}
    \operatorname{curl}\mathbf{q}
    &amp;= ~\!\mathrm{curl}(q_j ~\!\mathbf{e}_j) &amp;&amp; \\
    &amp;= \nabla q_j \times \mathbf{e}_j &amp;&amp; [\mathsf{\scriptstyle by~eq.(8p)}] \\
    &amp;= ~\!\mathbf{e}_i \part_i q_j \times \mathbf{e}_j
       &amp;&amp; [\mathsf{\scriptstyle by~eq.(58g)}] \\
    &amp;= ~\!\mathbf{e}_i ~\!\!\times \part_i ~\!q_j \mathbf{e}_j \,. &amp;&amp;
  \end{align}&lt;/math&gt;}}
If we sum over {{mvar|j}} first, this is
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}\mathbf{q} =~\! \mathbf{e}_i ~\!\!\times\part_i\mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|59c}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl} =~\! \mathbf{e}_i ~\!\!\times\part_i
  &lt;/math&gt;}}|{{EquationRef|59o}}}}
or, in traditional longhand,
:{{big|&lt;math&gt;
    \operatorname{curl}
    \,=\, \mathbf{i} \times ~\!\!\tfrac{\part}{\part x}
     +~\! \mathbf{j} \times ~\!\!\tfrac{\part}{\part y}
     +~\! \mathbf{k} \times ~\!\!\tfrac{\part}{\part z} \,.
  &lt;/math&gt;}}
For the ''divergence'' we proceed as for the curl except that, instead of ({{EquationNote|8p}}), we use ({{EquationNote|8g}}):
:{{big|&lt;math&gt;\begin{align}
    \operatorname{div}\mathbf{q}
    &amp;= ~\!\mathrm{div}(q_j ~\!\mathbf{e}_j) &amp;&amp; \\
    &amp;= \nabla q_j \cdot \mathbf{e}_j &amp;&amp; [\mathsf{\scriptstyle by~eq.(8g)}] \\
    &amp;= ~\!\mathbf{e}_i \part_i q_j \cdot \mathbf{e}_j
       &amp;&amp; [\mathsf{\scriptstyle by~eq.(58g)}] \\
    &amp;= ~\!\mathbf{e}_i ~\!\!\cdot \part_i ~\!q_j \mathbf{e}_j \,; &amp;&amp;
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{div}\mathbf{q} =~\! \mathbf{e}_i ~\!\!\cdot \part_i\mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|60d}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{div} =~\! \mathbf{e}_i ~\!\!\cdot \part_i
  &lt;/math&gt;}}|{{EquationRef|60o}}}}
or, in traditional longhand,
:{{big|&lt;math&gt;
    \operatorname{div}
    \,=\, \mathbf{i} \cdot \tfrac{\part}{\part x}
     +~\! \mathbf{j} \cdot \tfrac{\part}{\part y}
     +~\! \mathbf{k} \cdot \tfrac{\part}{\part z} \,.
  &lt;/math&gt;}}
It follows from ({{EquationNote|59c}}) and ({{EquationNote|60d}}), if it was not already obvious, that ''a uniform vector field has zero curl and zero divergence''.

Although the above expressions for the divergence and curl will surprise many modern readers, they match the ''initial definitions'' of the divergence and curl given by the founder of vector analysis as we know it, [[w:Josiah Willard Gibbs|J.&amp;#8239;Willard Gibbs]] ([[#gibbs-1881-4|1881]], &amp;sect;&amp;#8239;54). Gibbs even uses the {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; notations on the left sides of the defining equations, and only ''after''&amp;#8202; the equations (albeit immediately after) does he announce that&amp;#8202; &quot;{{math|&amp;#8202;&amp;nabla;'''&amp;sdot;'''&amp;#8202;''&amp;omega;''}} is called the ''divergence'' of{{mvar| &amp;omega;}}&amp;#8239; and {{math|&amp;nabla;&amp;#8202;&amp;times;''&amp;omega;''}}&amp;#8239; its ''curl''.&quot; (He uses Greek letters for vectors.) Our notation and Cartesian expression for the gradient ({{EquationNote|58g}}) also match Gibbs ([[#gibbs-1881-4|1881]], &amp;sect;&amp;#8239;52). Hence, using the Gibbs notations, we can merge definitions ({{EquationNote|58g}}), ({{EquationNote|59c}}), and ({{EquationNote|60d}}) into the general Cartesian formula
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla~\!\! * \psi =~\! \mathbf{e}_i ~\!\! * \part_i \psi
  &lt;/math&gt;}}|{{EquationRef|60s}}}}
(with implicit summation), where the {{math|&amp;lowast;}} operator may be a null (for the gradient), a cross (for the curl), or a dot (for the divergence).

Gibbs does not offer any justification for the {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; notations, but nor is it difficult to find such a justification based on his definitions. As{{math| '''e'''&lt;sub&gt;''i''&lt;/sub&gt;}} is a ''uniform'' vector, we can rewrite ({{EquationNote|59c}}) ''rigorously'' as
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}\mathbf{q} = \part_i(\mathbf{e}_i ~\!\!\times\mathbf{q})
  &lt;/math&gt;}}|{{EquationRef|61c}}}}
and thence ''operationally'' as
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}\mathbf{q} =~\! \mathbf{e}_i\part_i \times \mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|61o}}}}
or, recalling ({{EquationNote|58o}}),
:&lt;math&gt;
    \operatorname{curl}\mathbf{q} ~\!= \nabla \times \mathbf{q} \,,
  &lt;/math&gt;
which can be evaluated in the usual manner as
:&lt;math&gt;
    \operatorname{curl}\mathbf{q} \,=\,
      \begin{vmatrix}
        \mathbf{i} &amp; \part_x &amp; q_x \\
        \mathbf{j} &amp; \part_y &amp; q_y \\
        \mathbf{k} &amp; \part_z &amp; q_z
      \end{vmatrix} \,,
  &lt;/math&gt;
where {{mvar|q&lt;sub&gt;x&lt;/sub&gt;}} is the {{mvar|x}} component of{{math| '''q'''&amp;#8202;}}, etc. This indeed is how one evaluates the curl of a given field in Cartesian coordinates, although we shall find ({{EquationNote|59c}}) more convenient for deriving identities. Similarly, we can rewrite ({{EquationNote|60d}}) ''rigorously'' as
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{div}\mathbf{q} = \part_i(\mathbf{e}_i ~\!\!\cdot \mathbf{q})
  &lt;/math&gt;}}|{{EquationRef|62d}}}}
and thence ''operationally'' as
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{div}\mathbf{q} =~\! \mathbf{e}_i\part_i \cdot \mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|62o}}}}
or, recalling ({{EquationNote|58o}}),
:&lt;math&gt;
    \operatorname{div}\mathbf{q} ~\!= \nabla \!\cdot \mathbf{q} ~.
  &lt;/math&gt;
For evaluating the divergence of a given field, however, we simplify ({{EquationNote|62d}}) to
:{{big|&lt;math&gt;
    \operatorname{div}\mathbf{q} = \part_i q_i
  &lt;/math&gt;}}
or, in traditional longhand,
:&lt;math&gt;
    \operatorname{div}\mathbf{q}
    ~\!= \frac{\part q_x}{\part x}
       + \frac{\part q_y}{\part y}
       + \frac{\part q_z}{\part z} \,,
  &lt;/math&gt;
although we shall find ({{EquationNote|60d}}) more convenient for deriving identities. But the longhand form makes it especially obvious that if{{math|&amp;#8202; '''r'''}} is the position vector,
{{NumBlk|:|&lt;math&gt;
    \operatorname{div}\mathbf{r} = 3 \,.
  &lt;/math&gt;|{{EquationRef|62r}}}}

Notice that we can get from ({{EquationNote|62o}}) back to ({{EquationNote|60d}}) by permuting the {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}} with the dot, and from ({{EquationNote|61o}}) back to ({{EquationNote|59c}}) by permuting the {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}} with the cross, as if the differentiation operator could, as it were, look through the dot or the cross&amp;mdash;or, as Gibbs's student [[w:Edwin Bidwell Wilson|Edwin B.&amp;#8201;Wilson]] puts it, &quot;pass by&quot; the dot and the cross, yielding Gibbs's original definitions.&lt;ref&gt;[[#wilson-1901|Wilson, 1901]], p.&amp;#8239;150.&lt;/ref&gt; Hence Wilson considers it helpful to regard Gibbs's {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; and {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; notations as &quot;the (formal) scalar product and the (formal) vector product of{{math|&amp;#8202; &amp;nabla;}} into&quot; the operand, or &quot;the symbolic scalar and vector products of{{math|&amp;#8202; &amp;nabla;}} into&quot; the operand, and to regard {{math|&amp;nabla;}} as a &quot;symbolic vector&quot;&lt;ref&gt;[[#wilson-1901|Wilson, 1901]], pp.&amp;#8239;150,&amp;#8239;152. Wilson does not announce this idea in his preface (p.&amp;#8239;xii), although Tai ([[#tai-95|1995, p.&amp;#8239;26]]) gets the contrary impression by omitting a comma from the relevant quote.&lt;/ref&gt; (not to be confused with Tai's symbolic vector&lt;math&gt;~\nabla\!\!\!\!^{\textstyle_-}&lt;/math&gt;).

Tai ([[#tai-94|1994]], [[#tai-95|1995]]) rejects Wilson's argument together with the entire tradition of treating {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; as compound operators. Of formal products, Tai says that the concept &quot;has had a tremendously detrimental effect upon the learning of vector analysis&quot;; he calls such a product a &quot;meaningless assembly&quot;.&lt;ref&gt;[[#tai-95|Tai, 1995]], pp.&amp;#8239;26,&amp;#8239;38.&lt;/ref&gt; Of the &quot;pass by&quot; step, he complains that &quot;standard books on mathematical analysis do not have such a theorem.&quot;&lt;ref&gt;[[#tai-95|Tai, 1995]], p.&amp;#8239;28.&lt;/ref&gt;

I submit, however, that the intermediate steps ({{EquationNote|61c}}) and ({{EquationNote|62d}}), after which we take the constant multiplier outside the operator (eqs. {{EquationNote|61o}} &amp;amp; {{EquationNote|62o}}), support Wilson's &quot;pass by&quot; argument. In any event the reader may write out the sums on the right-hand sides of ({{EquationNote|59c}}) and ({{EquationNote|60d}}) and verify that they agree with the formal products {{math|&amp;nabla;&amp;#8202;&amp;times;&amp;#8202;'''q'''}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;&amp;#8239;q'''}}&amp;#8202; respectively&amp;mdash;and may notice that in the evaluation of each formal product, the cross or dot is eventually eliminated, leaving nothing to &quot;pass by&quot;.&lt;ref&gt;The latter observation is made, or at least suggested, by Kemin et al. ([[#kemin-et-al-00|2000]], p.&amp;#8239;605).&lt;/ref&gt; I further submit that the great generality of our derivation of equations ({{EquationNote|14}}), above, compels us to treat the {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; notations as more than mere notations. But the kicker is that Tai himself, having found the form of the del operator in ''general'' coordinates ([[#tai-95|1995]], p.&amp;#8239;64, eq.&amp;#8239;9.33), derives original corresponding forms of the {{math|div}} and {{math|curl}} operators (his eqs.&amp;#8239;9.35 &amp;amp; 9.40) which, upon reversal of the forbidden &quot;pass by&quot;, become del-dot and del-cross! Indeed his three equations, just cited, are reminiscent of our ({{EquationNote|58o}}), ({{EquationNote|60o}}), and ({{EquationNote|59o}}) respectively. That being said, I shall find some points of agreement with Tai, and some reasons to criticize Wilson.

'''Laplacian''':  If {{mvar|&amp;psi;}} is a ''scalar'' field, then
:{{big|&lt;math&gt;\begin{align}
    \triangle\psi
    &amp;= \operatorname{div}\nabla\psi \\
    &amp;= \mathbf{e}_i \cdot \part_i(\mathbf{e}_j ~\!\part_j \psi) \\
    &amp;= \;\!\mathbf{e}_i{\cdot}\;\!\mathbf{e}_j \;\part_i \part_j \psi \,.
  \end{align}&lt;/math&gt;}}
In this double summation, the only non-zero terms are those for which&amp;#8202; {{mvar|j&amp;#8239;{{=}}&amp;#8201;i&amp;#8202;}},&amp;#8239; in which case&amp;#8239; {{math|'''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8239;'''&amp;sdot;&amp;#8239;e'''&lt;sub&gt;''j''&lt;/sub&gt;&amp;#8201;{{=}}&amp;#8239;1}}.&amp;#8201; So we have
{{NumBlk|:|{{big|&lt;math&gt;
    \triangle\psi = \part_i \part_i \psi \,,
  &lt;/math&gt;}}|{{EquationRef|63L}}}}
where we write&amp;#8202; {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;&amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; rather than {{math|''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&lt;sup&gt;2&lt;/sup&gt;}}&amp;#8202; in order to maintain implicit summation. In traditional longhand, ({{EquationNote|63L}}) becomes
:&lt;math&gt;\triangle\psi ~\!= \frac{\part^2 \psi}{\part x ^2}
                        + \frac{\part^2 \psi}{\part y ^2}
                        + \frac{\part^2 \psi}{\part z ^2}
  &lt;/math&gt;
or, in operational terms,
:&lt;math&gt;\triangle ~\!= \frac{\part^2}{\part x ^2}
                    + \frac{\part^2}{\part y ^2}
                    + \frac{\part^2}{\part z ^2}
  &lt;/math&gt;
or, by comparison with ({{EquationNote|58t}}),
:&lt;math&gt;\triangle = \nabla{\cdot}\nabla &lt;/math&gt;
&amp;mdash;as expected.

By the linearity of the Laplacian, the same applies if {{mvar|&amp;psi;}} is any field expressible in terms of a uniform basis. For example, if {{mvar|&amp;psi;}} is a ''vector'' field given by&amp;#8239; {{math|''&amp;psi;&lt;sub&gt;j&lt;/sub&gt;''&amp;#8239;'''e'''&lt;sub&gt;''j''&lt;/sub&gt;}}&amp;#8239; (with implicit summation), then
:{{big|&lt;math&gt;\begin{align}
    \triangle\psi
    &amp;= \triangle(\psi_j ~\!\mathbf{e}_j) \\
    &amp;= \mathbf{e}_j ~\!\triangle\psi_j \\
    &amp;= \mathbf{e}_j \part_i \part_i \psi_j \\
    &amp;= \part_i \part_i (\psi_j ~\!\mathbf{e}_j)
     = \part_i \part_i \psi \,,
  \end{align}&lt;/math&gt;}}
where the third line follows from ({{EquationNote|63L}}) as applied to a scalar field. Thus ({{EquationNote|63L}}) is quite general.

After listing theorems ({{EquationNote|5g}}) to ({{EquationNote|5L}}) above, we gave reasons for describing {{math|&amp;nabla;,}} {{math|curl,}} and {{math|div}} as ''differential operators'', and {{math|&amp;#9651;}} as a ''2nd-order'' differential operator&amp;mdash;the implication being that the others are only 1st-order. We now have the promised &quot;additional reason&quot; for these descriptions: when expressed in Cartesian coordinates, the {{math|&amp;#9651;}} operator involves second derivatives, while the others involve (only) first derivatives. In the meantime we have acquired the {{math|'''q&amp;sdot;'''&amp;nabla;}} operator, which is also 1st-order, as we shall now confirm.

'''Advection, directional derivative, etc.''':   If {{mvar|&amp;psi;}} is a ''scalar'' field, then
:{{big|&lt;math&gt;\begin{align}
    \mathbf{q}\;\!{\cdot}\nabla\,\psi
    &amp;= (\mathbf{q}) \cdot (\nabla\psi) \\
    &amp;= (q_i~\!\mathbf{e}_i) \cdot (\mathbf{e}_j ~\!\part_j \psi) \\
    &amp;= \;\!\mathbf{e}_i{\cdot}\;\!\mathbf{e}_j \;q_i \part_j \psi \,.
  \end{align}&lt;/math&gt;}}
In this double summation, the only non-zero terms are those for which&amp;#8202; {{mvar|j&amp;#8239;{{=}}&amp;#8201;i&amp;#8202;}},&amp;#8239; in which case&amp;#8239; {{math|'''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8239;'''&amp;sdot;&amp;#8239;e'''&lt;sub&gt;''j''&lt;/sub&gt;&amp;#8201;{{=}}&amp;#8239;1}}.&amp;#8201; So we have
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,\psi = q_i ~\!\part_i \psi
  &lt;/math&gt;}}|{{EquationRef|64}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla = q_i ~\!\part_i
  &lt;/math&gt;}}|{{EquationRef|64o}}}}
or, in traditional longhand,
:{{big|&lt;math&gt;\mathbf{q}\;\!{\cdot}\nabla
    =~\! q_x\tfrac{\part}{\part x}
    +~\! q_y\tfrac{\part}{\part y}
    +~\! q_z\tfrac{\part}{\part z} \,,
  &lt;/math&gt;}}
which indeed is the &quot;formal&quot; or &quot;symbolic&quot; dot-product of&amp;#8202; {{math|'''q'''}} and{{math| &amp;nabla;}}.&amp;#8202; By the linearity of the directional derivative in ({{EquationNote|11}}), the same result applies if {{mvar|&amp;psi;}} is a vector field or any field expressible in terms of a uniform basis. In particular, if{{math| '''r'''}} is the position vector, we have
:{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,\mathbf{r}
    = q_i ~\!\part_i \mathbf{r}
    = q_i ~\!\mathbf{e}_i \,,
  &lt;/math&gt;}}
i.e.,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,\mathbf{r} = \mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|64r}}}}
&amp;mdash;which is also deducible from ({{EquationNote|11}}).

For convenience in the following discussion, we shall refer to the scaled-directional-derivative operator {{math|'''q&amp;sdot;'''&amp;nabla;}} as an &quot;advection&quot; operator although, physically, it represents advection only if {{math|'''q'''}} is the material velocity.

{{cob}}

=== Identities without pain ===

{{cot}}

In deriving the Cartesian expressions for the gradient, curl, divergence, Laplacian, and advection operators, we used the preceding identities ({{EquationNote|9g}}), ({{EquationNote|8p}}), ({{EquationNote|8g}}), ({{EquationNote|9L'}}), and ({{EquationNote|11}}) respectively, the last being a definition generalizing ({{EquationNote|9g}}). Thus we could have derived the Cartesian expressions quite early in the exposition, although we did not find that option convenient. The other vector-analytic identities that we have previously mentioned are:
* ({{EquationNote|8c}}), which showed the unambiguity of the curl;
* ({{EquationNote|8q}}), which has a question mark after it;
* ({{EquationNote|17}}), a product rule for the divergence, which is yet to be proven as a general identity;
* ({{EquationNote|24c}}) and ({{EquationNote|24c}}), concerning &quot;curl grad&quot; and &quot;div curl&quot;; and
* the identities showing that we can construct a field with a given divergence ({{EquationNote|36}}), Laplacian ({{EquationNote|38}}), or D'Alembertian ({{EquationNote|57}}).

The above list exposes the following shortcomings:
* we have not yet investigated &quot;grad div&quot; and &quot;curl curl&quot;;
* we have only one ''product rule''&amp;#8202;&amp;mdash;the unverified identity ({{EquationNote|17}})&amp;mdash;in which ''both'' factors are spatially variable fields; this needs to be verified and identities ({{EquationNote|8c}}) and ({{EquationNote|8p}}) need to be generalized;
* our collection of product rules does not yet include the curl of a cross-product, or the gradient of a dot-product or of a product of scalars, or the advection of a product; and
* we do not yet have any ''chain rules'' involving {{math|&amp;nabla;,}} {{math|curl,}} or {{math|div}}.
With the aid of the Cartesian forms of the various operators, we may now fill these gaps.

&lt;br /&gt;

The &quot;'''grad div'''&quot; and &quot;'''curl curl'''&quot; operators turn out to be related:
:{{big|&lt;math&gt;\begin{align}
    \operatorname{curl}\operatorname{curl}\mathbf{q} \;\!
    &amp;= \mathbf{e}_i \times\part_i(\operatorname{curl}\mathbf{q}) \\
    &amp;= \mathbf{e}_i \times\part_i(\mathbf{e}_j \times\part_j\mathbf{q}) \\
    &amp;= \mathbf{e}_i \times(\mathbf{e}_j \times\part_i\part_j\mathbf{q}) \,,
  \end{align}&lt;/math&gt;}}
whence expanding the vector triple product gives
:{{big|&lt;math&gt;\operatorname{curl}\operatorname{curl}\mathbf{q} \;\!
    = \mathbf{e}_i \!\cdot~\!\!\part_i\part_j\mathbf{q} ~\mathbf{e}_j
      - \mathbf{e}_i {\cdot}~\!\mathbf{e}_j \,\part_i\part_j\mathbf{q} \,.
  &lt;/math&gt;}}
In the first term on the right, we can switch the order of partial differentiation; and in the second term&amp;mdash;which, like the first, is a double summation&amp;mdash;the only non-zero contributions are those for which&amp;#8202; {{mvar|j&amp;#8239;{{=}}&amp;#8201;i}}&amp;#8239; and&amp;#8239; {{math|'''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8239;'''&amp;sdot;&amp;#8239;e'''&lt;sub&gt;''j''&lt;/sub&gt;&amp;#8201;{{=}}&amp;#8239;1}}.&amp;#8201; So we have
:{{big|&lt;math&gt;\begin{align}
    \operatorname{curl}\operatorname{curl}\mathbf{q} \;\!
    &amp;= \mathbf{e}_i \!\cdot~\!\!\part_j\part_i\mathbf{q} ~\mathbf{e}_j
       - \part_i\part_i\mathbf{q} \\
    &amp;= \mathbf{e}_j \,\part_j(\mathbf{e}_i \!\cdot~\!\!\part_i\mathbf{q})
      - \part_i\part_i\mathbf{q} \,;
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}\operatorname{curl}\mathbf{q}
    ~\!\equiv \nabla\operatorname{div}\mathbf{q} - \triangle\mathbf{q} \,.
  &lt;/math&gt;}}|{{EquationRef|65}}}}
This result may be memorized as &quot;''curl curl is grad div minus del squared''&amp;#8239;&quot; and written as
{{NumBlk|:|{{big|{{math|&amp;nabla;&amp;#8239;&amp;times;&amp;#8201;(&amp;nabla;&amp;#8239;&amp;times;&amp;#8201;'''q''') &amp;equiv; &amp;nabla; &amp;nabla;'''&amp;sdot;&amp;#8239;q''' &amp;minus; &amp;nabla;&lt;sup&gt;2&lt;/sup&gt;&amp;#8202;'''q'''}}&amp;#8201;,}}|{{EquationRef|66}}}}
which ''looks like'' the expansion of a vector triple product; and the key step in the above derivation, based on the Gibbs definitions of the operators,  ''really is''&amp;#8202; the expansion of a vector triple product.

&lt;br /&gt;

We now turn to ''product rules'' in which neither factor is assumed uniform.

The '''curl of a cross-product''' is
:{{big|&lt;math&gt;\begin{align}
    &amp;\operatorname{curl}(\mathbf{a}\!\times\!\mathbf{b}) \\
    &amp;~= \mathbf{e}_i ~\!\!\times\part_i(\mathbf{a}\times\mathbf{b}) \\
    &amp;~= \mathbf{e}_i ~\!\!\times(\part_i\mathbf{a}\times\mathbf{b} +
                                 \mathbf{a}\times\part_i\mathbf{b}) \\
    &amp;~= \mathbf{e}_i ~\!\!\times(\part_i\mathbf{a}\times\mathbf{b}) +
        \mathbf{e}_i ~\!\!\times(\mathbf{a}\times\part_i\mathbf{b}) \\
    &amp;~= \mathbf{e}_i{\cdot}~\!\mathbf{b} \,\part_i\mathbf{a}
        - \mathbf{e}_i{\cdot}~\!\part_i\mathbf{a} \;\mathbf{b}
        + \mathbf{e}_i{\cdot}~\!\part_i\mathbf{b} \;\mathbf{a}
        - \mathbf{e}_i{\cdot}~\!\mathbf{a} \,\part_i\mathbf{b} \\
    &amp;~= b_i\part_i\mathbf{a} - (\operatorname{div}\mathbf{a})~\!\mathbf{b}
        + (\operatorname{div}\mathbf{b})~\!\mathbf{a} - a_i\part_i\mathbf{b} \\
    &amp;~= \mathbf{b}~\!{\cdot}\nabla\,\mathbf{a}
        - \mathbf{b}\operatorname{div}\mathbf{a}
        + \mathbf{a}\operatorname{div}\mathbf{b}
        - \mathbf{a}~\!{\cdot}\nabla\,\mathbf{b} \,,
  \end{align}&lt;/math&gt;}}
i.e.,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}(\mathbf{a}\!\times\!\mathbf{b})
    \equiv \mathbf{a}\operatorname{div}\mathbf{b}
         - \mathbf{b}\operatorname{div}\mathbf{a}
         + \mathbf{b}~\!{\cdot}\nabla\,\mathbf{a}
         - \mathbf{a}~\!{\cdot}\nabla\,\mathbf{b} \,.
  &lt;/math&gt;}}|{{EquationRef|67c}}}}

The '''divergence of a cross-product''', as we might expect, is simpler:
:{{big|&lt;math&gt;\begin{align}\operatorname{div}(\mathbf{a}\!\times\!\mathbf{b})
    &amp;= \mathbf{e}_i ~\!\!\cdot\part_i(\mathbf{a}\times\mathbf{b}) \\
    &amp;= \mathbf{e}_i ~\!\!\cdot(\part_i\mathbf{a}\times\mathbf{b} +
                               \mathbf{a}\times\part_i\mathbf{b}) \\
    &amp;= \mathbf{e}_i ~\!\!\cdot\part_i\mathbf{a}\times\mathbf{b} +
       \mathbf{e}_i ~\!\!\cdot\mathbf{a}\times\part_i\mathbf{b} \\
    &amp;= \mathbf{e}_i ~\!\!\cdot\part_i\mathbf{a}\times\mathbf{b} -
       \mathbf{e}_i ~\!\!\cdot\part_i\mathbf{b}\times\mathbf{a} \\
    &amp;= \mathbf{b}\cdot\mathbf{e}_i ~\!\!\times\part_i\mathbf{a} -
       \mathbf{a}\cdot\mathbf{e}_i ~\!\!\times\part_i\mathbf{b} \,;
  \end{align}&lt;/math&gt;}}
i.e.,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{div}(\mathbf{a}\!\times\!\mathbf{b})
    \equiv \mathbf{b}\cdot\operatorname{curl}\mathbf{a}
         - \mathbf{a}\cdot\operatorname{curl}\mathbf{b} \,.
  &lt;/math&gt;}}|{{EquationRef|67d}}}}
In particular, in electromagnetics,&amp;#8201; {{math|div('''E'''&amp;#8239;&amp;times;&amp;#8239;'''H''') &amp;equiv; '''H&amp;#8239;&amp;sdot;'''&amp;#8201;curl&amp;#8201;'''E''' &amp;minus; '''E&amp;#8239;&amp;sdot;'''&amp;#8201;curl&amp;#8201;'''H'''}}&amp;#8239;;&amp;#8239; this is the identity on which [[w:Poynting's theorem|Poynting's theorem]] is based. But if&amp;#8202; {{math|'''b'''}} in ({{EquationNote|67d}}) is uniform, then ({{EquationNote|67d}}) reduces to ({{EquationNote|8c}}).

The '''gradient of a dot-product''', by comparison, is surprisingly messy:
:{{big|&lt;math&gt;\begin{align}\nabla\,\mathbf{a}{\cdot}\mathbf{b}
    &amp;= \mathbf{e}_i
       \part_i(\mathbf{a}\!\cdot\!\mathbf{b}) \\
    &amp;= \mathbf{e}_i
       (\mathbf{a}\!\cdot\!\part_i\mathbf{b} +
        \mathbf{b}\!\cdot\!\part_i\mathbf{a}) \\
    &amp;= \mathbf{a}\!\cdot\!\part_i\mathbf{b}
       \;\mathbf{e}_i +
       \mathbf{b}\!\cdot\!\part_i\mathbf{a}
       \;\mathbf{e}_i \,.
  \end{align}&lt;/math&gt;}}
Now the first term on the right can be recognized as&amp;#8201; {{math|'''a'''&amp;#8202;&amp;times;&amp;#8239;('''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8202;&amp;times;&amp;#8239;''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;'''b''')&amp;#8201;+&amp;#8201;'''a&amp;sdot;&amp;#8202;e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8201;''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;'''b'''&amp;#8202;}};&amp;#8201; that is,&amp;#8201; {{math|'''a'''&amp;#8202;&amp;times;&amp;#8239;('''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8202;&amp;times;&amp;#8239;''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;'''b''')&amp;#8201;+&amp;#8201;''a&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;'''b'''&amp;#8202;}};&amp;#8201; that is,&amp;#8201; &lt;math&gt;\mathbf{a}\!\times\!\operatorname{curl}\mathbf{b}+\mathbf{a}~\!{\cdot}\nabla\,\mathbf{b}&lt;/math&gt;.&amp;#8201; Similarly, the second term is&amp;#8201; &lt;math&gt;\mathbf{b}\!\times\!\operatorname{curl}\mathbf{a}+\mathbf{b}~\!{\cdot}\nabla\,\mathbf{a}&lt;/math&gt;.&amp;#8201; Thus we have
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla\,\mathbf{a}{\cdot}\mathbf{b}
    \equiv \mathbf{a}\!\times\!\operatorname{curl}\mathbf{b}
         + \mathbf{b}\!\times\!\operatorname{curl}\mathbf{a}
	 + \mathbf{a}~\!{\cdot}\nabla\,\mathbf{b}
	 + \mathbf{b}~\!{\cdot}\nabla\,\mathbf{a} \,.
  &lt;/math&gt;}}|{{EquationRef|68}}}}
For ''uniform''&amp;#8202; {{math|'''b'''&amp;#8202;,}} the first and third terms on the right vanish, and we can solve for the first term on the right, obtaining
:&lt;math&gt;\mathbf{b}\times\operatorname{curl}\mathbf{a} ~\!=
    \nabla\,\mathbf{b{\cdot}a} - \mathbf{b}~\!{\cdot}\nabla\,\mathbf{a}
  \qquad&lt;/math&gt;[&amp;#8202;for uniform {{math|'''b'''}}] ,
so that we can now drop the question mark after ({{EquationNote|8q}}). If we write the curl operator as&amp;#8202; {{math|&amp;nabla;&amp;#8202;&amp;times;&amp;#8239;,}}&amp;#8239; the last equation [or ({{EquationNote|8q}})]&amp;#8202; ''looks like'' the expansion of a vector triple product; but the identity is valid only for uniform{{math| '''b'''}}.

The '''gradient of a product of scalars''', unlike that of a dot-product, is as simple as the product rule for ordinary differentiation:
:{{big|&lt;math&gt;\begin{align}\nabla(p\varphi)
    &amp;= \mathbf{e}_i \part_i(p\varphi) \\
    &amp;= \mathbf{e}_i(p~\!\part_i\varphi + \varphi~\!\part_i p) \\
    &amp;= p~\!\mathbf{e}_i\part_i\varphi + \varphi~\!\mathbf{e}_i\part_i p \,;
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla(p\varphi) \equiv p\;\!\nabla\varphi + \varphi\;\!\nabla p \,.
  &lt;/math&gt;}}|{{EquationRef|69}}}}

The '''advection of a product''' is equally simple, ''regardless of the type of product'', except that the order of a cross-product matters. Let {{mvar|&amp;psi;}} and {{mvar|&amp;chi;}} be scalar or vector fields, and let {{math|''&amp;psi;''&amp;#8202;&amp;lowast;''&amp;chi;''}} denote any meaningful product of the two. Then, by ({{EquationNote|64}}),
:{{big|&lt;math&gt;\begin{align}
    \mathbf{q}\;\!{\cdot}\nabla\,(\psi~\!\! * \!\chi)
    &amp;= q_i \part_i (\psi~\!\! * \!\chi) \\
    &amp;= q_i (\psi * \part_i \chi + \part_i \psi * \chi) \\
    &amp;= \psi * q_i \part_i \chi + q_i \part_i \psi * \chi \,;
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,(\psi~\!\! * \!\chi)
    \equiv \psi * (\mathbf{q}\;\!{\cdot}\nabla\chi) +
           (\mathbf{q}\;\!{\cdot}\nabla\psi) * \chi \,.
  &lt;/math&gt;}}|{{EquationRef|70}}}}
The {{math|'''q&amp;sdot;'''&amp;nabla;}} operator is a ''scalar'' operator in the sense that it maps the operand field to a field of the same order&amp;mdash;a scalar field to a scalar field, a vector field to a vector field, a matrix field to a matrix field, etc.&amp;mdash;&amp;#8202;''as if''&amp;#8202; it were multiplication by a scalar or differentiation w.r.t. a scalar; and indeed a differentiation w.r.t. path length appears in the coordinate-free definition ({{EquationNote|11}}) of the operator. Moreover, we did not need coordinates to obtain rule ({{EquationNote|70}}); as the reader may verify, the same rule can be obtained directly from the definition ({{EquationNote|11}}) in a similar manner. From these points of view, the simplicity of the rule is unsurprising.

The '''curl of the product of a scalar and a vector''' is
:{{big|&lt;math&gt;\begin{align}\operatorname{curl}p\mathbf{b}
    &amp;= \mathbf{e}_i ~\!\!\times \part_i(p\mathbf{b}) \\
    &amp;= \mathbf{e}_i ~\!\!\times(p~\!\part_i\mathbf{b}+\part_i p\;\mathbf{b}) \\
    &amp;= p~\!\mathbf{e}_i{\times}~\!\part_i\mathbf{b} +
       \mathbf{e}_i\part_i p \times\mathbf{b} \\
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}p\mathbf{b}
    \,\equiv\, p\operatorname{curl}\mathbf{b} ~\!+ \nabla p \times\mathbf{b} \,.
  &lt;/math&gt;}}|{{EquationRef|71c}}}}
For uniform {{math|'''b'''&amp;#8202;,}} this reduces to ({{EquationNote|8p}}), which was used to derive the Cartesian form of the curl ({{EquationNote|59c}}).

For the '''divergence of the product of a scalar and a vector''', we proceed likewise except that we use a dot instead of a cross. The result is
{{NumBlk|:|{{big|&lt;math&gt;
     \operatorname{div}p\mathbf{b}
     \,\equiv\, p\operatorname{div}\mathbf{b} ~\!+ \nabla p \cdot \mathbf{b} \,,
  &lt;/math&gt;}}|{{EquationRef|71d}}}}
which has the same form as ({{EquationNote|17}}), delivering the promised confirmation that ({{EquationNote|17}}) is an identity. For uniform {{math|'''b'''&amp;#8202;,}}&amp;#8201; ({{EquationNote|71d}}) reduces to ({{EquationNote|8g}}), which was used to derive the Cartesian form of the divergence ({{EquationNote|60d}}).

That exhausts the first-order product rules. For curiosity's sake, we shall also derive one second-order rule.

The '''Laplacian of the product of a scalar field and a generic field''', by ({{EquationNote|63L}}), is
:{{big|&lt;math&gt;\begin{align}\triangle(p\psi)
    &amp;= \part_i \part_i (p\psi) \\
    &amp;= \part_i (p~\!\part_i \psi + \psi~\!\part_i p) \\
    &amp;= p\,\part_i \part_i \psi + \part_i \psi\,\part_i p +
       \psi~\!\part_i \part_i p + \part_i p\,\part_i \psi \\
    &amp;= p\,\part_i \part_i \psi + 2\part_i p\,\part_i \psi +
       \psi~\!\part_i \part_i p \\
    &amp;= p~\!\triangle\psi + 2\part_i p\,\part_i \psi +
       \psi~\!\triangle p \,.
  \end{align}&lt;/math&gt;}}
In the middle term, by ({{EquationNote|58g}}), {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;p}}&amp;#8202; is the {{mvar|i&amp;#8202;}}th component of{{math|&amp;#8202; &amp;nabla;''p''}}&amp;#8239; so that, by ({{EquationNote|64o}}),&amp;#8201; {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;p&amp;#8239;&amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8239; is the {{math|'''q&amp;sdot;'''&amp;nabla;}} operator for&amp;#8239; {{math|'''q'''&amp;#8239;{{=}}&amp;#8239;&amp;nabla;''p''}}.&amp;#8201; So we have
{{NumBlk|:|{{big|&lt;math&gt;
     \triangle(p\psi)
     \equiv p~\!\triangle\psi + 2(\nabla p \cdot~\!\! \nabla)\psi +
              \psi~\!\triangle p \,.
  &lt;/math&gt;}}|{{EquationRef|72}}}}
The argument assumes a scalar{{mvar| p}} but is indifferent to whether {{mvar|&amp;psi;}} is a scalar or a vector or a higher-order tensor.

&lt;br /&gt;

Finally we turn to ''chain rules''&amp;#8202;&amp;mdash;&amp;#8202;especially the simple cases of the gradient, curl, divergence, advection, and Laplacian of a function of a scalar field{{mvar| u}}. As usual, let {{mvar|p}} denote a scalar field, {{math|'''q'''}} a vector field, and {{mvar|&amp;psi;}} a generic field.

'''Gradient&amp;#8202;&amp;#10744;&amp;#8202;curl&amp;#8202;&amp;#10744;&amp;#8202;divergence of a function of a scalar''':  By the general Cartesian formula ({{EquationNote|60s}}) and the chain rule for{{math| ''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;,}}
:{{big|&lt;math&gt;\begin{align}\nabla~\!\! * \big(\psi(u)\big)
    &amp;= \mathbf{e}_i ~\!\! * \part_i \big(\psi(u)\big) \\
    &amp;= \mathbf{e}_i ~\!\! * \psi'~\!\!(u) ~\!\part_i u \\
    &amp;= \mathbf{e}_i \part_i u * \psi'~\!\!(u) \,;
  \end{align}&lt;/math&gt;}}
i.e., by ({{EquationNote|58g}}),
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla~\!\! * \big(\psi(u)\big)
    \equiv \nabla u * \psi'~\!\!(u) \,.
  &lt;/math&gt;}}|{{EquationRef|73}}}}
In particular, if&amp;#8202; {{math|&amp;lowast;}} is a null,
{{NumBlk|:|&lt;math&gt;
    \nabla\big(p(u)\big) \equiv \nabla u \;p'~\!\!(u) \,;
  &lt;/math&gt;|{{EquationRef|73g}}}}
and if&amp;#8202; {{math|&amp;lowast;}} is a cross,
{{NumBlk|:|&lt;math&gt;
    \mathrm{curl}\big(\mathbf{q}(u)\big)
    \equiv \nabla u \times \mathbf{q}'~\!\!(u) \,;
  &lt;/math&gt;|{{EquationRef|73c}}}}
and if&amp;#8202; {{math|&amp;lowast;}} is a dot,
{{NumBlk|:|&lt;math&gt;
    \mathrm{div}\big(\mathbf{q}(u)\big)
    \equiv \nabla u \cdot \mathbf{q}'~\!\!(u) \,.
  &lt;/math&gt;|{{EquationRef|73d}}}}

'''Advection of a function of a scalar''':
:&lt;!-- SUBSCRIPTS ENLARGED FOR LEGIBILITY: --&gt;&lt;math&gt;\begin{align}
    \mathbf{q}\;\!{\cdot}\nabla\,\big(\psi(u)\big)
    &amp;= q_{\textstyle i} \part_{\textstyle i} \big(\psi(u)\big) \\
    &amp;= q_{\textstyle i} ~\!\psi'~\!\!(u) ~\!\part_{\textstyle i} u \\
    &amp;= q_{\textstyle i} \part_{\textstyle i} u \;\psi'~\!\!(u) \,;
  \end{align}&lt;/math&gt;
i.e.,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,\big(\psi(u)\big)
    \equiv~\! \mathbf{q}\;\!{\cdot}\nabla u ~\psi'~\!\!(u) \,.
  &lt;/math&gt;}}|{{EquationRef|73q}}}}
This fits into the pattern set by ({{EquationNote|73}}) in that the gradient operator in ({{EquationNote|73g}}) is replaced by an advection operator.

Of the last four results, only ({{EquationNote|73c}}) is dependent on the order of the {{math|&amp;lowast;}} product; the others could equally well be written
{{NumBlk|:|&lt;math&gt;\begin{align}
    \nabla\big(p(u)\big) &amp;\equiv~\! p'~\!\!(u) ~\!\nabla u \\
    \mathrm{div}\big(\mathbf{q}(u)\big)
    &amp;\equiv~\! \mathbf{q}'~\!\!(u) \cdot~\!\! \nabla u \\
    \mathbf{q}\;\!{\cdot}\nabla\,\big(\psi(u)\big)
    &amp;\equiv~\! \psi'~\!\!(u)\;\mathbf{q}\;\!{\cdot}\nabla u ~.
  \end{align}&lt;/math&gt;|{{EquationRef|73z}}}}

The '''Laplacian of a function of a scalar''' departs from the above pattern.
:{{big|&lt;math&gt;\begin{align}\triangle\big(\psi(u)\big)
    &amp;= \part_i \part_i \big(\psi(u)\big) \\
    &amp;= \part_i\big(\psi'~\!\!(u) ~\!\part_i u\big) \\
    &amp;= \psi'~\!\!(u) ~\!\part_i\part_i u
       + \psi''~\!\!(u) ~\!\part_i u \,\part_i u \,,
  \end{align}&lt;/math&gt;}}
where the last line follows from the product rule for {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; and, in the second term, the chain rule for{{mvar| &amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;}}.&amp;#8201; In that second term, the implicit sum&amp;#8202; {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;u&amp;#8239;&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;u}}&amp;#8239; can be recognized as&amp;#8202; {{math|{{abs|&amp;nabla;''u''}}&lt;sup&gt;2&lt;/sup&gt;}}&amp;#8202; by ({{EquationNote|58s}}). So we have
{{NumBlk|:|{{big|&lt;math&gt;
    \triangle\big(\psi(u)\big)
    \equiv \psi'~\!\!(u)~\!\triangle u + \psi''~\!\!(u)~\!\big|\nabla u\big|^2.
  &lt;/math&gt;}}|{{EquationRef|74}}}}

'''Multivariate chain rule''':  The foregoing chain rules involve ''one'' intermediate function of ''one'' scalar variable. It will be useful to have an elementary chain rule that can handle more than one of each. Let {{math|''p''('''r''')}} be a smooth scalar field, and let {{math|'''r'''}} in turn be a smooth function of several variables, one of which, say{{mvar| t&amp;#8202;}}, is allowed to vary while the others are held constant, so that {{math|'''r'''}} changes by {{math|''d'''''r'''}} when {{mvar|t}} changes by {{mvar|dt}}. Then dividing ({{EquationNote|26g}}) by {{mvar|dt}}&amp;#8202; gives
:&lt;math&gt;\part_t p = \nabla p \cdot \part_t \mathbf{r}&lt;/math&gt;
or, in indicial Cartesian coordinates with implicit summation,
:{{big|&lt;math&gt;\part_t p = \part_i p \,\part_t x_i&lt;/math&gt;}}
or, in traditional longhand,
:{{big|&lt;math&gt;\tfrac{\part}{\part t}~\!p(x,y,z)
    = \tfrac{\part p}{\part x}~\!\tfrac{\part x}{\part t}
    + \tfrac{\part p}{\part y}~\!\tfrac{\part y}{\part t}
    + \tfrac{\part p}{\part z}~\!\tfrac{\part z}{\part t} \,.
  &lt;/math&gt;}}
This is the desired multivariate chain rule for a scalar function of three intermediate real variables. The assumption that these variables are Cartesian coordinates is not a loss of generality, because any three real quantities can be suitably scaled and represented by perpendicular axes, so that any scalar function of them becomes a function of position, to which ({{EquationNote|26g}}) applies; and then the scaling can be reversed without changing the products in the last equation. Moreover, by the linearity of{{mvar| &amp;part;&lt;sub&gt;t&lt;/sub&gt;&amp;#8239;}}, the scalar field {{mvar|p}} may be replaced by any field expressible in terms of a uniform basis. For example, for a vector field{{math| '''q'''&amp;#8202;}},
:{{big|&lt;math&gt;\begin{align}\part_t \mathbf{q}
    &amp;= \part_t (q_j \mathbf{e}_j) \\
    &amp;= \mathbf{e}_j \part_t q_j \\
    &amp;= \mathbf{e}_j \part_i q_j \,\part_t x_i \\
    &amp;= \part_i (q_j \mathbf{e}_j) ~\!\part_t x_i
     = \part_i \mathbf{q} \,\part_t x_i \,,
  \end{align}&lt;/math&gt;}}
where the third line is obtained by applying the multivariate chain rule for a scalar field. Thus, for a generic field {{mvar|&amp;psi;&amp;#8202;}},
{{NumBlk|:|{{big|&lt;math&gt;
    \part_t \psi = \part_i \psi \,\part_t x_i
  \qquad&lt;/math&gt;}}[&amp;#8202;for generic {{mvar|&amp;psi;}} and {{mvar|x&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;}}].|{{EquationRef|75}}}}

'''Gradient&amp;#8202;&amp;#10744;&amp;#8202;curl&amp;#8202;&amp;#10744;&amp;#8202;divergence of a function of a scaled position vector''':  We end this subsection by deriving a lemma for use in the next subsection. If{{mvar| k}} is a uniform scalar multiplier and {{math|'''r'''}} is the position vector,
:{{big|&lt;math&gt;
    \nabla * \psi(k\mathbf{r})
    = \mathbf{e}_i * \part_i \psi(k\mathbf{r})
    = k\mathbf{e}_i * \part_{(kx_{\scriptstyle i})} \psi(k\mathbf{r}) \,,
  &lt;/math&gt;}}
where the third expression is obtained by from the second by multiplying each denominator (change in{{mvar| x&lt;sub&gt;i&lt;/sub&gt;}}) by{{mvar| k}}&amp;#8239; and compensating. But now we have
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla * \psi(k\mathbf{r}) = k\,\big(\nabla {*}~\! \psi\big)_{k\mathbf{r}} \,,
  &lt;/math&gt;}}|{{EquationRef|76}}}}
where the parentheses and subscript indicate that the expression for &lt;math&gt;\nabla{*}~\!\psi&lt;/math&gt; is to be evaluated with {{math|'''r'''}} replaced by {{math| ''k''&amp;#8202;'''r'''}}. We shall be interested in the curl (for which {{math|&amp;lowast;}} is a cross).

{{cob}}

=== Field with given curl ===

{{cot}}

Consider the vector field
{{NumBlk|:|&lt;math&gt;
    \mathbf{v}(\mathbf{r}) = \mathbf{q}(\mathbf{r})\times\mathbf{r} \,,
  &lt;/math&gt;|{{EquationRef|77}}}}
where {{math|'''q'''}} is a ''solenoidal''&amp;#8202; vector field and {{math|'''r''' }}is the position vector. By identity ({{EquationNote|67c}}),
:&lt;math&gt;\operatorname{curl}\mathbf{v}
  =   \mathbf{q}\operatorname{div}\mathbf{r}
    - \mathbf{r}\operatorname{div}\mathbf{q}
    + \mathbf{r}~\!{\cdot}\nabla\,\mathbf{q}
    - \mathbf{q}~\!{\cdot}\nabla\,\mathbf{r}
  &lt;/math&gt;
where, by hypothesis, {{math|div&amp;#8239;'''q'''}}&amp;#8202; is zero. Applying identities ({{EquationNote|62r}}) and ({{EquationNote|64r}}) then yields
:&lt;math&gt;\begin{align}
  \operatorname{curl}\mathbf{v}
  &amp;= 3\mathbf{q} + \mathbf{r}~\!{\cdot}\nabla\,\mathbf{q} - \mathbf{q} \\
  &amp;= 2\mathbf{q} + \mathbf{r}~\!{\cdot}\nabla\,\mathbf{q} \,.
  \end{align}&lt;/math&gt;

[[File:Vorticity_Figure_01_a-m.gif|thumb|Animation of a rigid-body-like velocity field, whose curl is twice the angular velocity.]]

In the special case in which {{math|'''q'''}} is the ''angular velocity'' {{math|'''&amp;omega;'''}} of a '''rigid body''' about an axis through the origin,&amp;#8201; {{math|'''v''' }}is the velocity field ({{math|'''&amp;omega;'''&amp;#8239;&amp;times;&amp;#8239;'''r'''}}) and {{math|'''&amp;omega;'''}} is uniform, so that the last result reduces to&amp;#8201; {{math|curl&amp;#8239;'''v'''&amp;#8201;{{=}}&amp;#8201;2'''&amp;omega;'''&amp;#8202;}}; that is, ''the vorticity is twice the angular velocity''. As the vorticity in this case is uniform and therefore independent of position relative to the axis, it does not change if the axis is shifted, provided that the angular velocity has the same magnitude and direction. And because a uniform velocity field has zero curl, the vorticity is also unchanged if a translational motion is superposed on the rotation. This is the most direct connection that we have seen between curl and rotation. But again I digress.

Returning to the more general case in which {{math|'''q''' }}is not necessarily uniform, but merely solenoidal,&lt;ref&gt;The following explanation takes some hints from Christopher Ford's note on &quot;Vector Potentials&quot; at [https://www.maths.tcd.ie/~houghton/231/Notes/ChrisFord/vp.pdf maths.tcd.ie/~houghton/231/Notes/ChrisFord/vp.pdf] (2006).&lt;/ref&gt; we have
:&lt;math&gt;\operatorname{curl}\mathbf{v}(\mathbf{r})
    = 2\mathbf{q} + \mathbf{r}~\!{\cdot}\nabla\,\mathbf{q} \,,
  &lt;/math&gt;
to which we can apply our lemma ({{EquationNote|76}}) with a uniform real factor {{mvar|t&amp;#8202;}}, obtaining
:&lt;math&gt;
    \operatorname{curl}\mathbf{v}(t\mathbf{r})
    = t\Big(2\mathbf{q}
            + \mathbf{r}~\!{\cdot}\nabla\,\mathbf{q}\Big)_{t\mathbf{r}} \,.
  &lt;/math&gt;
On the left we can recall ({{EquationNote|77}}); and on the right we can apply ({{EquationNote|11}}), noting that the magnitude of{{math| {{abs|'''r'''}}}} is{{mvar| r&amp;#8202;}}, which measures distance in the direction of{{math| '''r'''}}. Thus we obtain
:&lt;math&gt;
  \mathrm{curl}\big(\mathbf{q}(t\mathbf{r})\times t\mathbf{r}\big)
    = t\Big(2\mathbf{q}
            + r~\! \part_{\textstyle r} \mathbf{q}\Big)_{t\mathbf{r}} \,,
  &lt;/math&gt;
where the effect of the operator {{mvar|r&amp;#8202;d&lt;sub&gt;r&lt;/sub&gt;}} is independent of the scaling of{{math| '''r'''}}, so that the overall dependence on {{math|''t''&amp;#8202;'''r'''}} is easily made explicit:
:&lt;math&gt;
  \mathrm{curl}\big(\mathbf{q}(t\mathbf{r})\times t\mathbf{r}\big)
    = t\Big(2\mathbf{q}(t\mathbf{r})
            + r~\! \part_{\textstyle r} \mathbf{q}(t\mathbf{r})\Big) \,.
  &lt;/math&gt;
Now if the direction of{{math| '''r'''}} is held constant,&amp;#8201; {{math|'''q'''(''t''&amp;#8202;'''r''')}} is a function of{{mvar| tr&amp;#8239;}}; and in general, by the chain rule,&amp;#8201; {{math|''r&amp;#8202;&amp;part;&lt;sub&gt;r&lt;/sub&gt; f''&amp;#8202;(''tr'') {{=}} ''t&amp;#8202;&amp;part;&lt;sub&gt;t&lt;/sub&gt; f''&amp;#8202;(''tr'')}}.&amp;#8201; So we have
:&lt;math&gt;\begin{align}
  \mathrm{curl}\big(\mathbf{q}(t\mathbf{r})\times t\mathbf{r}\big)
    &amp;= t\Big(2\mathbf{q}(t\mathbf{r})+t~\!\part_t\mathbf{q}(t\mathbf{r})\Big)\\
    &amp;= 2t\mathbf{q}(t\mathbf{r}) + t^2 \part_t \mathbf{q}(t\mathbf{r}) \\
    &amp;= \part_t \big(t^2 \mathbf{q}(t\mathbf{r})\big) \,.
  \end{align}&lt;/math&gt;
Integrating w.r.t. {{mvar|t}}&amp;#8239; from 0 to 1 gives
:&lt;math&gt;
  \int_0^1\!\mathrm{curl}\big(\mathbf{q}(t\mathbf{r})\times t\mathbf{r}\big)\,dt
    = \big(t^2 \mathbf{q}(t\mathbf{r})\big)\Big|_0^1
    =~\! \mathbf{q}(\mathbf{r}) \,;
  &lt;/math&gt;
that is,
{{NumBlk|:|&lt;math&gt;\mathbf{q}(\mathbf{r}) \equiv
  \mathrm{curl}\int_0^1\!\mathbf{q}(t\mathbf{r})\!\times\!t\mathbf{r}\;dt\qquad
  &lt;/math&gt;[&amp;#8202;for solenoidal {{math|'''q'''&amp;#8202;}}].|{{EquationRef|78}}}}
Thus for any solenoidal vector field{{math| '''q'''}}&amp;#8239; we can construct a '''vector potential'''&amp;mdash;that is, a field whose curl is{{math| '''q'''&amp;#8202;}}; such a field is given by the integral on the right. This is the long-promised proof of the &quot;converse&quot; of identity ({{EquationNote|24d}}). Of course the vector potential is not unique, because any conservative field&amp;mdash;but ''only'' a conservative field&amp;mdash;can be added to it without changing its curl. Hence the existence of ''one'' vector potential implies the existence of infinitely many. The above integral gives us ''one''.

The proof of ({{EquationNote|78}}) assumes that {{math|'''q''' }}is solenoidal not only at position{{math| '''r'''&amp;#8202;,}} but also at{{math| ''t''&amp;#8202;'''r'''}}&amp;#8202; where&amp;#8239; {{math|0&amp;#8239;&amp;leq;&amp;#8239;''t''&amp;#8239;&amp;leq;&amp;#8202;1}}, i.e. at every point on the line-segment from the origin to{{math| '''r'''}}.&amp;#8201; A '''star-shaped''' region is one that contains a point{{mvar| O}}&amp;#8202; such that for every point{{mvar| P}} in the region, the line-segment {{mvar|OP}} is entirely contained in the region. We may choose any such {{mvar|O}}&amp;#8202; as the origin in the proof of ({{EquationNote|78}}). So the proof tells us that if a vector field is solenoidal within a star-shaped region, it has a vector potential in that region. As a special case, a vector field that is solenoidal everywhere has a vector potential everywhere.

{{cob}}

=== Notes on the curl of the curl ===

{{cot}}

Identity ({{EquationNote|65}}), namely
:&lt;math&gt;
    \operatorname{curl}\operatorname{curl}\mathbf{q}
    ~\!\equiv \nabla\operatorname{div}\mathbf{q} - \triangle\mathbf{q}
  &lt;/math&gt;
(&quot;curl curl is grad div minus del squared&quot;), has at least three implications worth noting here.

First, it can be rearranged as
{{NumBlk|:|&lt;math&gt;\triangle\mathbf{q}
    ~\!\equiv \nabla\operatorname{div}\mathbf{q}
              - \operatorname{curl}\operatorname{curl}\mathbf{q}
  &lt;/math&gt;|{{EquationRef|79}}}}
(&quot;del squared is grad div minus curl curl&quot;). This would serve as a coordinate-free definition of the Laplacian of a vector, if we did not already have one.&lt;ref&gt;''Cf''. [[#gibbs-1881-4|Gibbs, 1881]], &amp;sect;&amp;#8202;71, and [[#moon-spencer-65|Moon &amp;amp; Spencer, 1965]], p.&amp;#8239;235; quoted in [[#tai-95|Tai, 1995]], pp.&amp;#8239;18,&amp;#8239;43.&lt;/ref&gt; But we do: we started with a coordinate-free definition ({{EquationNote|4L}}) for a generic field, established its unambiguity via ({{EquationNote|9L}}), and found its Cartesian form ({{EquationNote|63L}}), which we used in the derivation of ({{EquationNote|79}}). Wherever we start, we may properly assert by way of contrast that the Laplacian of a ''vector''&amp;#8202; is given by ({{EquationNote|79}}), whereas the Laplacian of a ''scalar''&amp;#8202; is given by the divergence of the gradient. But we should ''not'' conclude, as Moon &amp;amp; Spencer do, that representing the scalar and vector Laplacians by the same symbol is &quot;poor practice&amp;hellip; since the two are basically quite different&quot;,&lt;ref&gt;[[#moon-spencer-65|Moon &amp;amp; Spencer, 1965]], p.&amp;#8239;236.&lt;/ref&gt; because in fact the two have a common definition which is succinct, unambiguous, and coordinate-free: the Laplacian (of anything) is the closed-surface integral of the outward normal derivative, per unit volume.{{efn|Tai ([[#tai-95|1995]], pp.&amp;#8239;43–4) also disagrees with Moon &amp;amp; Spencer, but for a different reason: he regards the Laplacian as the divergence of the gradient even if the operand is a ''vector'' field. For better or worse, we do not consider the gradient of a vector in the present paper&amp;mdash;although the reader can probably work out how to modify ({{EquationNote|26g}}) if&amp;#8202; {{math|''d'''''r'''}} is written as a column vector and&amp;#8202; {{mvar|dp}}&amp;#8202; is ''replaced''&amp;#8202; by a column vector (compare the later footnote on ''dyadics'').}}

Second, by reason of identity ({{EquationNote|38}}) and the remarks thereunder, a given vector field{{math| '''v'''}} can be written
:&lt;math&gt;\mathbf{v}(\mathbf{r}) \,\equiv\, \triangle\bigg(\!{-}\!\iiint
    \frac{\,\mathbf{v}(\mathbf{r}')}{4\pi}\,
    \frac{1}{|\mathbf{r}\!-\!\mathbf{r}'|}
  \,dV' \!\bigg) \,,&lt;/math&gt;
where the integral is over all space, or at least all of the space in which {{math|'''v'''}} may be non-zero. So, subject to the convergence of the integral, there exists a vector field{{math| '''q'''}} such that
:&lt;math&gt;\mathbf{v} = \triangle\mathbf{q} \,;&lt;/math&gt;
that is, by ({{EquationNote|79}}), there exists{{math| '''q'''}} such that
:&lt;math&gt;\mathbf{v}
    = \nabla\operatorname{div}\mathbf{q}
      - \operatorname{curl}\operatorname{curl}\mathbf{q} \,,
  &lt;/math&gt;
which implies the existence of a scalar field, say&lt;math&gt;\,\varphi~\!,\,&lt;/math&gt; and a vector field, say{{math| '''&amp;Psi;'''}}, such that
:&lt;math&gt;\mathbf{v}
    = -\nabla\varphi+\operatorname{curl}\boldsymbol{\Psi}
  &lt;/math&gt;
(namely&amp;#8201; &lt;math&gt;\varphi\!=\!-\!\operatorname{div}\mathbf{q}\,&lt;/math&gt; and&amp;#8201; {{math|'''&amp;Psi;'''&amp;#8201;{{=}}&amp;#8201;&amp;minus;&amp;#8202;curl&amp;#8239;'''q'''}}). In short, subject to the convergence of the said integral,
* ''a given vector field can be resolved into [minus] a gradient plus a curl''.
Such a resolution is called a '''Helmholtz decomposition''', and the proposition that it exists is the ''Helmholtz decomposition theorem''. Of course the gradient is irrotational and the curl is solenoidal so that, subject to the same convergence,
* ''a given vector field can be resolved into an irrotational field plus a solenoidal field''.
This is a second statement of the theorem, and follows from the first. And the first follows from the second because an irrotational field has a scalar potential by ({{EquationNote|29}}) and a solenoidal field has a vector potential by ({{EquationNote|78}}).

Third, if {{math|'''q'''}} is ''solenoidal'', the term&amp;#8201; {{math|&amp;nabla;&amp;#8201;div&amp;#8239;'''q'''}}&amp;#8239; in ({{EquationNote|65}}) or ({{EquationNote|79}}) vanishes. Hence ''for a solenoidal field, the curl of the curl is minus the Laplacian''. For example, in the ''dynamic'' case, in a ''vacuum'', the Maxwell&amp;ndash;Amp&amp;egrave;re law says that&amp;#8201; {{math|curl&amp;#8201;'''H''' {{=}} ''&amp;epsiv;''&lt;sub&gt;0&lt;/sub&gt;&amp;#8202;'''E&amp;#775;'''}}.&amp;#8201; Multiplying this by the physical constant {{math|''&amp;mu;''&lt;sub&gt;0&lt;/sub&gt;}} (called the '''vacuum permeability''' or simply the '''magnetic constant''') gives&amp;#8201; {{math|curl&amp;#8201;'''B''' {{=}} ''&amp;mu;''&lt;sub&gt;0&lt;/sub&gt;&amp;#8202;''&amp;epsiv;''&lt;sub&gt;0&lt;/sub&gt;&amp;#8239;'''E&amp;#775;'''&amp;#8202;,}}&amp;#8239; whence
:&lt;math&gt;\operatorname{curl}\operatorname{curl}\mathbf{B}
    = \mu_0\epsilon_0\operatorname{curl}\mathbf{\dot{E}} \,.
  &lt;/math&gt;
But, by Gauss's law for magnetism, {{math|'''B'''}} is solenoidal so that, by ({{EquationNote|65}}), the left-hand side of the above is&amp;#8201; {{math|&amp;minus;&amp;#9651;'''B'''}}.&amp;#8201; And by '''Faraday's law''',&amp;#8201; &lt;math&gt;\operatorname{curl}\mathbf{E}=-\mathbf{\dot{B}}&lt;/math&gt;,&amp;#8201; so that&amp;#8201; &lt;math&gt;\operatorname{curl}\mathbf{\dot{E}}=-\mathbf{\ddot{B}}&lt;/math&gt;.&amp;#8201; Making these substitutions, we get&amp;#8201; &lt;math&gt;-\triangle\mathbf{B}=-\mu_0\epsilon_0\mathbf{\ddot{B}}~\!,\,&lt;/math&gt; i.e.
:&lt;math&gt;\mathbf{\ddot{B}}=\frac{1}{\mu_0\epsilon_0}~\!\triangle\mathbf{B} \,.&lt;/math&gt;
By comparison with ({{EquationNote|45}}), this is the wave equation with
:&lt;math&gt;c=\frac{1}{\sqrt{\mu_0\epsilon_0}} \,.&lt;/math&gt;
Thus the Maxwell&amp;ndash;Amp&amp;egrave;re law, Gauss's law for magnetism, and Faraday's law, with the aid of ({{EquationNote|65}}), predict the existence of '''electromagnetic waves''' together with their speed.

For these reasons, especially the last, one could hardly overstate the importance of identity ({{EquationNote|65}}).

{{cob}}

=== Digression: Proofs from formal products ===

{{cot}}

We have seen that Wilson ([[#wilson-1901|1901]], pp.&amp;#8239;150,&amp;#8239;152) interprets the divergence and curl as &quot;formal&quot; or &quot;symbolic&quot; scalar and vector products with the {{math|&amp;nabla; }}operator.&amp;#8201; {{nowrap|C.-T. Tai}}, in his [[#tai-95|1995 report]] (pp.&amp;#8239;26–9), alleges that this interpretation began with Wilson and not with Gibbs. Here I shall submit, on the contrary, that while the terminology may not be attributable to Gibbs, the concept certainly is.

Later in the same report, Tai confuses the picture by citing the first volume of [[w:Oliver Heaviside|Heaviside]]'s ''Electromagnetic Theory'' (1893), where Heaviside, although his notations for the scalar and vector products differ from those of Gibbs, nevertheless considers the {{math|&amp;nabla;}} operator as a factor in such products. Tai continues:
&lt;blockquote&gt;At the time of his writing he [Heaviside] was already aware of Gibbs' pamphlets on vector analysis but Wilson's book was not yet published. It seems, therefore, that Heaviside and Wilson independently introduced the misleading concept for the scalar and vector products between {{math|&amp;nabla;}} and a vector function. Both were, perhaps, induced by Gibbs' notations for the divergence and the curl. Heaviside did not even include the word 'formal' in his description of the products.&lt;ref&gt;[[#tai-95|Tai, 1995]], p.&amp;#8239;35.&lt;/ref&gt;
&lt;/blockquote&gt;
Whereas it was quite in character for Heaviside to treat an operator that way, the word &quot;independently&quot; would have surprised Wilson and is contradicted by Tai himself, who observes that Wilson's preface acknowledges Heaviside.&lt;ref&gt;[[#tai-95|Tai, 1995]], pp.&amp;#8239;25,&amp;#8239;29.&lt;/ref&gt; In Wilson's own words:
&lt;blockquote&gt;By far the greater part of the material used in the following pages has been taken from the course of lectures on Vector Analysis delivered annually at the University [Yale] by Professor Gibbs. Some use, however, has been made of the chapters on Vector Analysis in Mr. Oliver Heaviside's ''Electromagnetic Theory'' (Electrician Series, 1893) and in Professor F&amp;ouml;ppl's lectures on ''Die Maxwell'sche Theorie der Electricit&amp;auml;t'' (Teubner, 1894).&amp;#8239;....

Notwithstanding the efforts which have been made during more than half a century to introduce Quaternions into physics the fact remains that they have not found wide favor.{{efn|A ''quaternion''&amp;#8202; is a mathematical object invented by [[w:William Rowan Hamilton|William Rowan Hamilton]] in 1843, consisting of two parts which Hamilton later called the scalar part and the vector part. For most purposes the two parts were found to be more useful separately than together. By putting them together, however, Hamilton constructed a set which satisfied all the algebraic field axioms except commutativity of multiplication. This was, and is, considered a triumph.}} On the other hand there has been a growing tendency especially in the last decade toward the adoption of some form of Vector Analysis. The works of Heaviside and F&amp;ouml;ppl referred to before may be cited in evidence. As yet however no system of Vector Analysis which makes any claim to completeness has been published. In fact Heaviside says: &quot;I am in hopes that the chapter which I now finish may serve as a stopgap till regular vectorial treatises come to be written suitable for physicists, based upon the vectorial treatment of vectors&quot; (''Electromagnetic Theory'', Vol.&amp;#8239;{{serif|I}}., p.&amp;#8239;305). Elsewhere in the same chapter Heaviside has set forth the claims of vector analysis as against Quaternions, and others have expressed similar views.&lt;ref&gt;[[#wilson-1901|Wilson, 1901]], pp.&amp;#8239;ix,&amp;#8239;xi–xii.&lt;/ref&gt;
&lt;/blockquote&gt;

Most damaging to Tai's thesis, however, is Gibbs's original pamphlet, a copy of which Heaviside received from Gibbs himself in June 1888.&lt;ref&gt;[[#gibbs-1881-4|Gibbs, 1881–84]], privately printed version&amp;mdash;of which the scan in our bibliography is of the very copy that Gibbs sent to Heaviside, with annotations in Heaviside's hand. On the annotations see [[#rocci-20|Rocci, 2020]].&lt;/ref&gt; Sections 62 to 65 of the pamphlet appear under the heading
&lt;blockquote style=&quot;text-align: center&quot;&gt;{{math|&amp;nabla;,}} {{math|&amp;nabla;'''&amp;sdot;'''&amp;#8202;,}} ''and''&amp;#8202; {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8239; ''applied to Functions of Functions of Position''.
&lt;/blockquote&gt;In &amp;sect;&amp;#8239;62, Gibbs says that a constant scalar factor after such an operator may be placed before it (that is, taken outside the operator). {{nowrap|In &amp;sect;&amp;#8239;63}} he states our rule ({{EquationNote|73g}}) for the gradient of a function of a scalar field. His next section (in which I have bolded the vector field{{math| '''&amp;omega;'''}}) is worth quoting in full:
&lt;blockquote&gt;64.  If {{mvar|u}} or {{math|'''&amp;omega;'''}} is a function of several scalar or vector variables, which are themselves functions of the position of a single point, the value of&amp;#8202; {{math|&amp;nabla;''u''}} or {{math|&amp;nabla;'''&amp;sdot;&amp;#8239;&amp;omega;'''}}&amp;#8202; or {{math|&amp;nabla;&amp;#8202;&amp;times;&amp;#8202;'''&amp;omega;'''}}&amp;#8202; will be equal to the sum of the values obtained by making successively all but each one of these variables constant.
&lt;/blockquote&gt;
This proposition is a ''generalized product rule'' in the sense that the &quot;function of several scalar or vector variables&quot; may be, but is not restricted to, any sort of product of those variables. Gibbs continues:
&lt;blockquote&gt;65.  By the use of this principle, we easily derive the following identical equations:
&lt;/blockquote&gt;
Six &quot;equations&quot; follow. The first says that the gradient operation is distributive over addition, and the second says the same of the divergence and curl (on one line). The last four are our identities ({{EquationNote|69}}), ({{EquationNote|71d}}), ({{EquationNote|71c}}), and ({{EquationNote|67d}}), in that order (albeit with different symbols). Gibbs then remarks (with my italics):
&lt;blockquote&gt;The student will observe an analogy between these equations and the formul&amp;aelig; of ''multiplication''. (In the last four equations the analogy appears most distinctly when we regard all the factors but one as constant.) Some of the more curious features of this analogy are due to the fact that the {{math|&amp;nabla;}} contains implicitly the vectors {{math|'''i'''&amp;#8202;,}} {{math|'''j'''&amp;#8202;,}} and {{math|'''k'''&amp;#8202;,}} which are to be ''multiplied''&amp;#8202; into the following quantities.
&lt;/blockquote&gt;
Indeed, if the ''first''&amp;#8202; factor is constant, identities ({{EquationNote|69}}), ({{EquationNote|71d}}), ({{EquationNote|71c}}), and ({{EquationNote|67d}}) become
:&lt;math&gt;\begin{align}
    \nabla(p\varphi) &amp;= p\;\!\nabla\varphi \\
    \nabla\cdot p\mathbf{b} &amp;= p\,\nabla{\cdot}~\!\mathbf{b} \\
    \nabla\times p\mathbf{b} &amp;= p\,\nabla{\times}~\!\mathbf{b} \\
    \nabla\cdot(\mathbf{a}\!\times\!\mathbf{b})
    &amp;= -\mathbf{a}\cdot\nabla{\times}~\!\mathbf{b} \,,
  \end{align}&lt;/math&gt;
whereas if the ''second''&amp;#8202; factor is constant, they become respectively
:&lt;math&gt;\begin{align}
    \nabla(p\varphi) &amp;= \varphi\;\!\nabla p \\
    \nabla\cdot p\mathbf{b} &amp;= \nabla p \cdot \mathbf{b} \\
    \nabla\times p\mathbf{b} &amp;= \nabla p \times \mathbf{b} \\
    \nabla\cdot(\mathbf{a}\!\times\!\mathbf{b})
    &amp;= \nabla{\times}~\!\mathbf{a}\cdot\mathbf{b} \,.
  \end{align}&lt;/math&gt;
All eight equations look like rearrangements of ''products'' involving a vector{{math| &amp;nabla;}}.&amp;#8201; [Concerning the last ''three'' equations, we have made that observation before; see ({{EquationNote|15}}) above.]&amp;#8201; But only seven of the eight are explained by taking the constant outside the operator ({{nowrap|as in &amp;sect;&amp;#8239;62}}); the exception is the fourth, in which the minus sign is not explained by that step alone, but ''is'' explained by the change in the cyclic order of the formal triple product. And if we add the two right-hand sides corresponding to each of the four left-hand sides, we get the identities in which both factors are variable&amp;mdash;as claimed {{nowrap|in &amp;sect;&amp;#8239;64}}.

If &amp;sect;&amp;#8239;65 leaves any doubt that Gibbs approved of formal products with the symbolic vector{{math| &amp;nabla;}} (albeit without using those terms), this is dispelled {{nowrap|by &amp;sect;&amp;#8202;166}}, where he writes:
&lt;blockquote&gt;166.&amp;#8201; To the equations in No.&amp;#8239;65 may be added many others&amp;hellip;
&lt;/blockquote&gt;
followed by a list of seven identities terminated by &quot;etc.&quot; Six of the seven are beyond the scope of the present paper,{{efn|They involve ''dyadics'', i.e. 2nd-order tensors written in a vector-friendly notation. The fourth of the seven is
:{{math|&amp;nabla;('''&amp;tau;&amp;sdot;&amp;#8202;&amp;omega;''') {{=}} &amp;nabla;'''&amp;tau;&amp;#8239;&amp;sdot;&amp;#8201;&amp;omega;''' + &amp;nabla;'''&amp;omega;&amp;#8201;&amp;sdot;&amp;#8201;&amp;tau;'''&amp;#8239;,}}
which is our ({{EquationNote|68}}) expressed in terms of the dyadics {{math|&amp;nabla;'''&amp;tau;'''}} and{{math| &amp;nabla;'''&amp;omega;'''&amp;#8202;}}; the right-hand side is not to be confused with
:{{math|('''&amp;omega;&amp;#8202;&amp;sdot;'''&amp;nabla;)'''&amp;tau;''' + '''(&amp;tau;&amp;sdot;'''&amp;nabla;)'''&amp;omega;'''&amp;#8239;,}}
which would contradict our ({{EquationNote|68}}).}} while the third of the seven is our ({{EquationNote|67c}}). After that list comes the smoking gun ({{nowrap|&amp;sect;&amp;#8202;166, continued}}):
&lt;blockquote&gt;The principle in all these cases is that if we have one of the operators&amp;#8202; {{math|&amp;nabla;,}} {{math|&amp;nabla;'''&amp;sdot;'''&amp;#8202;,}} {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8239; prefixed to a ''product'' of any kind, and we make any transformation of the expression which would be allowable if the {{math|&amp;nabla;}} were a ''vector'', (viz: by changes in the order of the ''factors'', in the signs of ''multiplication'', in the parentheses written or implied, etc.,) by which changes the {{math|&amp;nabla;}} is brought into connection with one particular factor, the expression thus transformed will represent the part of the value of the original expression which results from the variation of that factor.
&lt;/blockquote&gt;
The italics are mine, but I have refrained from italicizing those instances of the word &quot;factor&quot; which are not applicable to{{math| &amp;nabla;}}. In particular, at the stage when &quot;the {{math|&amp;nabla;}} is brought into connection with one particular factor,&quot; the &quot;part of the value&amp;hellip; which results from the variation of that factor&quot; evidently means the term of the sum {{nowrap|in &amp;sect;&amp;#8239;64}}&amp;#8202;&amp;mdash;which, as we have noted, amounts to a generalized product rule. But, according to the stated &quot;principle', we reach that stage by treating{{math| &amp;nabla;}} as one of the &quot;''factors''&quot;. I rest my case.

&lt;br /&gt;

Wilson ([[#wilson-1901|1901]], p.&amp;#8239;157) gives a comprehensive list of sum and product rules for the gradient, divergence, and curl, and properly states (p.&amp;#8239;158) that the rules may be proven &quot;most naturally&quot; from Gibbs's definitions of the operators&amp;mdash;our equations ({{EquationNote|58g}}), ({{EquationNote|60d}}), and ({{EquationNote|59c}}). Understandably, Wilson uses a {{math|&amp;sum;}} sign rather than implicit summation. Less understandably, and less fortunately, he does not sum over a numerical index; e.g., he defines the curl operator as
:{{big|&lt;math&gt;\nabla\times
    \,=\, \textstyle\sum\,\mathbf{i}~\!\!\times\!\frac{\part}{\part x}
          \qquad\quad
  &lt;/math&gt;[sic]}}
and explains that &quot;The summation extends over {{math|''x'',&amp;#8239;''y'',&amp;#8239;''z''}}.&quot;&amp;#8201; With these definitions he proves our identities ({{EquationNote|71c}}) and ({{EquationNote|68}}) essentially as we have done, but inevitably with greater difficulty, which may explain why he then says &quot;The other formul&amp;aelig; are demonstrated in a similar manner&quot; before reverting to Gibbs's strategy of varying one factor at a time. He announces (p.&amp;#8239;159) that the variable held constant will be written as a subscript after the product, and he combines this notation with his {{math|&amp;sum;}} notation in a rigorous proof that varying one factor at a time is valid for our ({{EquationNote|68}}), i.e. the gradient of a dot-product. Noting that this result is analogous to
:&lt;math&gt;d(\mathbf{u}\cdot\mathbf{v})
    = \mathbf{u}\cdot d\mathbf{v} + d\mathbf{u}\cdot\mathbf{v} \,,
  &lt;/math&gt;
he then jumps to the conclusion that varying one factor at a time is valid for ''all''&amp;#8202; of his product rules&amp;mdash;notwithstanding that the simple relation between the gradient and a small change due to spatial displacement ({{EquationNote|26g}}) has no counterpart for the divergence or curl.

That ''per saltum''&amp;#8202; conclusion is his cue to go formal and symbolic. To obtain the curl of a cross-product as in our ({{EquationNote|67c}}), he &quot;formally&quot; expands a vector triple product to obtain the curl when the first factor is constant, states the curl when the second factor is held constant, and adds the two partial curls ([[#wilson-1901|Wilson, 1901]], p.&amp;#8239;161). Next he gives various arrangements of our ({{EquationNote|8q}}), except that he presents the first vector not as strictly uniform, but as merely ''held'' constant for the gradient operation. He states in passing that a proof may be effected by &quot;expanding in terms of&amp;#8202; {{math|'''i'''&amp;#8202;,&amp;#8239;'''j''',&amp;#8239;'''k'''}}&quot;; but instead of such a proof, he offers a &quot;method of remembering the result&quot; by expanding the &quot;product&quot;&amp;#8239; {{math|'''u'''&amp;#8239;&amp;times;&amp;#8239;(&amp;nabla;&amp;#8239;&amp;times;&amp;#8239;'''v''')}}&amp;#8201; &quot;formally as if&amp;#8202; {{math|&amp;nabla;,&amp;#8239;'''u'''&amp;#8202;,&amp;#8202;'''v'''}}&amp;#8202; were all real vectors&quot; (pp.&amp;#8239;161–2). Concerning the curl of the gradient, and the divergence of the curl (pp.&amp;#8239;167,&amp;#8239;168), he recommends expanding in terms of&amp;#8202; {{math|'''i'''&amp;#8202;,&amp;#8239;'''j''',&amp;#8239;'''k'''&amp;#8202;,}}&amp;#8239; but does not elaborate. Concerning the curl of the curl, however, he shows what would happen if it were &quot;expanded formally according to the law of the triple vector product&quot; (p.&amp;#8239;169).

In defense of the &quot;formal product&quot; method, we should note that the operators {{math|''&amp;part;&lt;sub&gt;x&lt;/sub&gt;''&amp;#8202;,}} {{math|''&amp;part;&lt;sub&gt;y&lt;/sub&gt;''&amp;#8202;,}} and{{math| ''&amp;part;&lt;sub&gt;z&lt;/sub&gt;''&amp;#8202;}} are ''linear'', so that they are distributive over addition and may be permuted with multiplication by a constant, as if the operators themselves were multipliers (like components of vectors). They may be similarly permuted with other like operators&amp;mdash;explaining why the formal-product method correctly deals with the curl of the gradient, the divergence of the curl, and the curl of the curl. But such an operator ''cannot'' be permuted with multiplication by a ''variable'', because then the product rule of differentiation applies, yielding an extra term. The formal-product system responds to this difficulty by generalizing the product rule as in &amp;sect;&amp;sect;&amp;#8239;64 &amp;amp;&amp;#8239;166 of Gibbs ([[#gibbs-1881-4|1881–84]]). As Borisenko &amp;amp; Tarapov put it ([[#borisenko-tarapov-68|1968]], p.&amp;#8239;169),
&lt;blockquote&gt;the operator {{math|&amp;nabla;}} acts on each factor separately with the other held fixed. Thus {{math|&amp;nabla;}} should be written after any factor regarded as a constant in a given term and before any factor regarded as variable.
&lt;/blockquote&gt;
In this they differ inconsequentially from Gibbs, who requires that the operator be &quot;brought into connection&quot; with the factor considered variable.

To illustrate, let us find the gradient of a dot-product, essentially in the manner of Borisenko &amp;amp; Tarapov ([[#borisenko-tarapov-68|1968]], p.&amp;#8239;180), quoted by Tai ([[#tai-95|1995]], p.&amp;#8239;46; the next five equation numbers are Tai's). In this case the generalized product rule gives
{{NumBlk|:|&lt;math&gt;\nabla(\mathbf{A ~\!\!\cdot B})
    = \nabla(\mathbf{A}_c {\cdot}~\!\mathbf{B}) +
      \nabla(\mathbf{A} ~\!\!\cdot \mathbf{B}_c) \,,
  &lt;/math&gt;|{{EquationRef|7.26}}}}
where the subscript {{mvar|c}} marks the factor held ''constant'' during the differentiation. In Wilson's notation, this equation would be written
:{{midsize|&lt;math&gt;\nabla(\mathbf{A ~\!\!\cdot B})
    = \nabla(\mathbf{A ~\!\!\cdot B})_{\mathbf{A}} +
      \nabla(\mathbf{A ~\!\!\cdot B})_{\mathbf{B}} \,,
  &lt;/math&gt;}}
where a trailing subscript indicates which factor is held constant.&amp;#8201; In the ''Feynman'' subscript notation, the subscript is attached to the {{math|&amp;nabla;}} operator and indicates which factor is allowed to ''vary'', so that the same equation would be written
:{{midsize|&lt;math&gt;\nabla(\mathbf{A ~\!\!\cdot B})
    = \nabla_{\mathbf{B}}(\mathbf{A ~\!\!\cdot B}) +
      \nabla_{\!\mathbf{A}}(\mathbf{A ~\!\!\cdot B}) \,.
  &lt;/math&gt;}}
But, as we are discussing Borisenko &amp;amp; Tarapov, we press on with ({{EquationNote|7.26}}).&amp;#8201; By the algebraic identity
{{NumBlk|:|&lt;math&gt;\mathbf{c}(\mathbf{a\cdot b})
    \,=\, (\mathbf{a\cdot c})\mathbf{b}
    \,-\, \mathbf{a}\times(\mathbf{b}\times\mathbf{c}) \,,
  &lt;/math&gt;|{{EquationRef|7.27}}}}
i.e.
:&lt;math&gt;\mathbf{c}(\mathbf{a\cdot b})
    \,=\, (\mathbf{a\cdot c})\mathbf{b}
    \,+\, \mathbf{a}\times(\mathbf{c}\times\mathbf{b}) \,,
  &lt;/math&gt;
we can say
{{NumBlk|:|&lt;math&gt;\nabla(\mathbf{A}_c {\cdot}~\!\mathbf{B})
    \,=\, (\mathbf{A}_c {\cdot}\nabla)\mathbf{B}
    \,+\, \mathbf{A}_c\times(\nabla\times\mathbf{B}) \,.
  &lt;/math&gt;|{{EquationRef|7.28}}}}
Similarly,&lt;ref&gt;In the next equation as printed in Borisenko &amp;amp; Tarapov ([[#borisenko-tarapov-68|1968]], p.&amp;#8239;180), the first cross should be &quot;&amp;equals;&quot;; Tai ([[#tai-95|1995]], p.&amp;#8239;46) quotes it with the correction.&lt;/ref&gt;
{{NumBlk|:|&lt;math&gt;\nabla(\mathbf{B}_c {\cdot}~\!\mathbf{A})
    \,=\, (\mathbf{B}_c {\cdot}\nabla)\mathbf{A}
    \,+\, \mathbf{B}_c\times(\nabla\times\mathbf{A}) \,.
  &lt;/math&gt;|{{EquationRef|7.29}}}}
Substituting ({{EquationNote|7.28}}) and ({{EquationNote|7.29}}) into ({{EquationNote|7.26}}), in which the order of the dot-products is immaterial, and dropping the {{mvar|c }}subscripts (because they are now outside the differentiations), we get the correct result
{{NumBlk|:|{{midsize|&lt;math&gt;\nabla(\mathbf{A{\cdot}B})
    =     (\mathbf{A}{\cdot}\nabla)\mathbf{B}
      +   (\mathbf{B}\;\!{\cdot}\nabla)\mathbf{A}
      +   \mathbf{A}{\times}(\nabla{\times}\mathbf{B})
      +   \mathbf{B}{\times}(\nabla{\times}\mathbf{A}) \,,
  &lt;/math&gt;}}|{{EquationRef|7.30}}}}
corresponding to our ({{EquationNote|68}}).

Tai ([[#tai-95|1995]], p.&amp;#8239;47) is unimpressed, asking why we cannot apply ({{EquationNote|7.27}}) directly to the left side of ({{EquationNote|7.26}}). The answer to that is obvious: on the left side, the {{math|&amp;nabla;}} operator is applied to a product of ''two variables'', and the variations of ''both'' must be taken into account. But there is a harder question which Tai does not ask: in ({{EquationNote|7.28}}), why can't we have {{math|&amp;nabla;'''&amp;sdot;A'''&lt;sub&gt;c&lt;/sub&gt;}} instead of{{math| '''A'''&lt;sub&gt;c&lt;/sub&gt;'''&amp;sdot;'''&amp;nabla;}}&amp;#8239;? (Or, in terms of Feynman subscripts, why can't we have {{math|&amp;nabla;'''&lt;sub&gt;B&lt;/sub&gt;&amp;#8202;&amp;sdot;&amp;#8202;A'''}} instead of{{math| '''A&amp;sdot;'''&amp;nabla;&lt;sub&gt;'''B'''&lt;/sub&gt;}}?) Because that would make the term vanish? Yes, it would; but, as there is only one variable factor on the left side, why do we need two terms on the right? Because the rule says {{math|&amp;nabla;}} should be written after the constant but before the variable? Yes, but that rule serves the purpose of varying ''each'' variable, whereas there is only one variable to vary on the left of ({{EquationNote|7.28}}). The same issue arises in ({{EquationNote|7.29}}). We cannot settle the question even by appealing to symmetry. Obviously the right side of ({{EquationNote|7.30}}), like the left, must be unchanged if we switch {{math|'''A'''}} and {{math|'''B'''}}; and indeed it is. But if the first term on the right of ({{EquationNote|7.28}}) and of ({{EquationNote|7.29}}) were to vanish, the necessary symmetry of ({{EquationNote|7.30}}) would be maintained. And unless I'm missing something, Tai's &quot;symbolic vector&quot; method does not circumvent the problem; Tai's &quot;Lemma 2&quot; ([[#tai-95|1995]], p.&amp;#8239;53) is the Gibbs&amp;#10744;Wilson method of &quot;varying one factor at a time&quot;, written with Feynman subscripts attached to the symbolic vector instead of the del operator.{{efn|I don't overlook the fact that Tai's symbolic vector, unlike the del operator, is subject to commutative and anticommutative laws. Neither do I see how it helps.}}

For another example of the same issue, consider the following two-liner offered by Panofsky &amp;amp; Phillips ([[#panofsky-phillips-62|1962]], pp.&amp;#8239;470–71) and rightly pilloried by Tai ([[#tai-95|1995]], pp.&amp;#8239;47–8):
:&lt;math&gt;\begin{align}
  &amp;   \nabla{\times}(\mathbf{A}{\times}\mathbf{B})
   =  (\nabla{\cdot}\;\!\mathbf{B})\mathbf{A}
    - (\nabla{\cdot}    \mathbf{A})\mathbf{B}               &amp;&amp;[\mathsf{sic}] \\
  &amp;=  (\nabla{\cdot}\;\!\mathbf{B}_c ~\!\!)\mathbf{A}
    + (\nabla{\cdot}\;\!\mathbf{B}~\!\!)\mathbf{A}_c ~\!\!
    - (\nabla\mathbf{\cdot A}_c ~\!\!)\mathbf{B}
    - (\nabla\mathbf{\cdot A}~\!\!)\mathbf{B}_c \!\!\!\!\!\!\!&amp;&amp;[\mathsf{sic}].
  \end{align}&lt;/math&gt;
If the first line were right, the authors would hardly bother to continue; but evidently it isn't, because it doesn't begin by &quot;varying one factor at a time&quot;. The second line does not follow from the first and includes divergences of constants, which ought to vanish but somehow apparently do not. Let's try again, this time sticking to the rules:
:&lt;math&gt;\begin{align}
  &amp; \nabla\!\times\!(\mathbf{A}\!\times\!\mathbf{B})
  \,=\,   \nabla\!\times\!(\mathbf{A}_c \!\times\!\mathbf{B})
    \,+\, \nabla\!\times\!(\mathbf{A}\!\times\!\mathbf{B}_c) \\
  &amp;~=\,   (\nabla{\cdot}\;\!\mathbf{B})\mathbf{A}_c
        - (\mathbf{A}_c {\cdot}\nabla)\mathbf{B}
    \,+\, (\mathbf{B}_c {\cdot}\nabla)\mathbf{A}
        - (\nabla{\cdot}    \mathbf{A})\mathbf{B}_c \\
  &amp;~=\,   \mathbf{A}(\nabla{\cdot}\;\!\mathbf{B})
        - \mathbf{B}(\nabla{\cdot}    \mathbf{A})
    \,+\, (\mathbf{B\;\!\cdot}\nabla)\mathbf{A}
        - (\mathbf{A    \cdot}\nabla)\mathbf{B} \,,
  \end{align}&lt;/math&gt;
in agreement with our ({{EquationNote|67c}}). Here the first line comes from the generalized product rule, and the third is obtained from the second by rearranging terms and dropping the (now redundant) subscripts. The interesting line is the second, which is obtained from the first by expanding the formal vector triple products. But again, why must we have {{math|'''A'''&lt;sub&gt;c&lt;/sub&gt;'''&amp;sdot;'''&amp;nabla;}} and {{math|'''B'''&lt;sub&gt;c&lt;/sub&gt;'''&amp;sdot;'''&amp;nabla;,}} instead of {{math|&amp;nabla;'''&amp;sdot;A'''&lt;sub&gt;c&lt;/sub&gt;}} and {{math|&amp;nabla;'''&amp;sdot;B'''&lt;sub&gt;c&lt;/sub&gt;&amp;#8202;,}} which would make the middle two terms vanish? Again symmetry does not give an answer. The right-hand side, like the left, must change sign if we switch {{math|'''A'''}} and {{math|'''B'''&amp;#8202;}}; but the disappearance of the {{math|'''A'''&lt;sub&gt;c&lt;/sub&gt;'''&amp;sdot;'''&amp;nabla;}} and {{math|'''B'''&lt;sub&gt;c&lt;/sub&gt;'''&amp;sdot;'''&amp;nabla;}} terms would maintain the required (anti)symmetry. Funnily enough, the result would then agree with the incorrect first line given by Panofsky &amp;amp; Phillips (above). But then how would we know that it is incorrect?

The foregoing examples show that &quot;formal product&quot; arguments can be tenuous, even on their own terms. Before these examples, we might have been troubled by the omission of a general proof of the &quot;generalized&quot; product rule. After them, we might wonder whether the rule is even well defined.

I submit, however, that none of this matters. I submit that the popularity of using &quot;formal products&quot; with the del operator, in derivations of vector-analytic identities, is a reaction to the failure of early writers to use indicial notation in the Cartesian definitions of differential operators. The ensuing proliferation of terms in coordinate-based derivations led authors to seek shortcuts through &quot;formal products&quot; when more rigorous but no-less convenient shortcuts could have been taken through indicial notation, especially in combination with implicit summation. Our derivation of the gradient of a dot-product ({{EquationNote|68}}) is shorter than that of Borisenko &amp;amp; Tarapov, and even uses the right-hand sides of their identities ({{EquationNote|7.28}}) and ({{EquationNote|7.29}}), but obtains them rigorously with no ambiguity and no {{mvar|c }}subscripts. Our derivation of the curl of a cross-product ({{EquationNote|67c}}) takes six lines with a single column of &quot;&amp;equals;&quot; signs. Our subsequent formal-product derivation (not to be confused with the attempt of Panofsky &amp;amp; Phillips) seems to take only three lines; but it is only through our earlier indicial derivation that we have any confidence in our result (not to be confused with the result of Panofsky &amp;amp; Phillips). Our other indicial derivations of identities are mostly shorter than the two just mentioned. Having amassed so comprehensive a collection of identities so rigorously with so little effort, I submit that the use of formal products, Wilson subscripts, {{mvar|c }}subscripts, and Feynman subscripts for this purpose is a historical aberration, to be deciphered in other people's writings but avoided in one's own.

That being said, it is one thing to conclude, as Tai duly does, that the del-cross and del-dot notations should not be interpreted as products in derivations and proofs, and another thing to allege, as Tai also does ([[#tai-95|1995]], p.&amp;#8239;22), that&amp;#8202; {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; and {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; are &quot;not compound operators&quot; but only &quot;assemblies&quot;, or in other words that &quot;{{math|&amp;#8202;&amp;nabla; }}is not a constituent of the divergence operator nor of the curl operator.&quot; Against the latter proposition, our equations ({{EquationNote|14}}), ({{EquationNote|61o}}), and ({{EquationNote|62o}}) have been ''derived'', not merely defined, and our derivation of ({{EquationNote|14}}) is as general as we could wish. Moreover, whereas ({{EquationNote|61o}}) and ({{EquationNote|62o}}) are for Cartesian coordinates, we shall see that they have counterparts in more general coordinates.

{{cob}}

== General coordinates ==

{{cot}}

From our initial definitions of the differential operators, we derived certain identities, from which we derived expressions for the operators in Cartesian coordinates, from which we derived a comprehensive collection of identities, two of which (the multivariate chain rule, and the curl of the product of a scalar and a vector) will now be useful for expressing the operators in other coordinate systems. Cartesian coordinates are traditionally called {{math|''x'',&amp;#8239;''y'',&amp;#8239;''z'',}}&amp;#8201; which we renamed {{mvar|x&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; where&amp;#8202; {{math|''i''&amp;#8201;{{=}}&amp;#8239;1,&amp;#8202;2,&amp;#8202;3&amp;#8202;,}}&amp;#8201; respectively. The best-known 3D ''non''&amp;#8202;-Cartesian coordinate systems are the cylindrical coordinates {{math|(''&amp;rho;'',&amp;#8239;''&amp;phi;'',&amp;#8239;''z'')}} and the spherical coordinates {{math|(''r'',&amp;#8239;''&amp;theta;'',&amp;#8239;''&amp;phi;'')}}; we have already seen {{mvar|r}}&amp;#8202; in the guise of the magnitude of the position vector{{math| '''r'''}}.&amp;#8201; But now we want our coordinate system to be as general as possible&amp;mdash;with the Cartesian, cylindrical, and spherical systems and many others, and even ''classes'' of systems, as special cases.

{{cob}}

=== Natural and dual basis vectors ===

{{cot}}

We shall call our general coordinates {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}}&amp;#8202; where&amp;#8202; {{math|''i''&amp;#8201;{{=}}&amp;#8239;1,&amp;#8202;2,&amp;#8202;3&amp;#8202;}};&amp;#8201; yes, for reasons which will emerge, we shall write the coordinate index as a {{nowrap|''super''&amp;#8202;script}}. But we shall write {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; for{{math| ''{{sfrac|&amp;part;|&amp;part;u&lt;sup&gt;i&lt;/sup&gt;&amp;#8202;}}''&amp;#8202;,}}&amp;#8202; relying on context to distinguish it from the special case{{mvar| {{sfrac|&amp;part;|&amp;part;x&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202;}}.&amp;#8201; By describing the {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}}&amp;#8202; as ''coordinates''&amp;#8239; we mean two things. First, for some domain of interest, the position vector is a smooth function
:&lt;math&gt;\mathbf{r} = \mathbf{r}(u^1,u^2,u^3) \,,&lt;/math&gt;
which possesses partial derivatives w.r.t. its arguments. Second, for every position vector in the resulting range, there is only one ordered triplet&amp;#8239; {{math|(''u&lt;sup&gt;i&lt;/sup&gt;''&amp;#8202;)&amp;#8201;{{=}}&amp;#8201;(''u''&amp;sup1;,&amp;#8239;''u''&amp;sup2;,&amp;#8239;''u''&amp;sup3;),}}&amp;#8239; so that we can think of each coordinate as
:{{big|&lt;math&gt;u^i = u^i(\mathbf{r}) \,;&lt;/math&gt;}}
&amp;mdash;that is, we can think of each {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}}&amp;#8202; as a scalar field, which possesses a gradient.{{efn|Hence we want each {{math|''u&lt;sup&gt;i&lt;/sup&gt;''('''r''')}} to be, as far as possible, a ''smooth'' function. This may require some tweaking of definitions. E.g., in cylindrical coordinates, the angular coordinate {{mvar|&amp;phi;}} must be confined to some 360&amp;deg; range in order to make it unique, and we don't want it jumping from the end of the range to the beginning within the region of interest.}} (I say &quot;think of&quot; because it would seem, on its face, that the {{mvar|i&amp;#8202;}}th coordinate depends on the coordinate system and is therefore not a true scalar; but here we are treating the coordinate system itself as an object under study.)

These two properties of coordinates respectively suggest two simple ways of choosing basis vectors related to the coordinates: we shall define the '''natural basis''' vectors as
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}_i := \part_i \mathbf{r} \,,
  \qquad&lt;/math&gt;}}|{{EquationRef|80a}}}}
and the '''dual basis''' vectors as
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}^i := \nabla u^i .
  \qquad&lt;/math&gt;}}|{{EquationRef|80b}}}}
(We could ''normalize'' the natural basis vectors by dividing them by their magnitudes to obtain unit vectors; but, in the general case, we won't bother.) Just as we may think of each {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}} as a scalar field and inquire after its directional derivative or its gradient or its Laplacian, so we may think of each {{math|'''h'''&lt;sub&gt;''i''&lt;/sub&gt;}} or{{math| '''h'''&lt;sup&gt;''i''&lt;/sup&gt;}} as a vector field and inquire after its directional derivative or its curl or its divergence or its Laplacian. (That the curl of{{math|&amp;#8202; '''h'''&lt;sup&gt;''i''&lt;/sup&gt;}}&amp;#8202; is zero&amp;#8239; will be especially useful.)

In Cartesian coordinates,&amp;#8201; {{math|'''h'''&lt;sub&gt;''i''&lt;/sub&gt;}} and {{math|'''h'''&lt;sup&gt;''i''&lt;/sup&gt;}} are both equal to the unit vector{{math| '''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8239;}}; thus, in Cartesian coordinates, the natural basis vectors are their own duals.&amp;#8201; In ''general'' coordinates,&amp;#8201; {{math|'''h'''&lt;sub&gt;''i''&lt;/sub&gt;}} and {{math|'''h'''&lt;sup&gt;''i''&lt;/sup&gt;}} may differ in both direction and magnitude and are not generally unit vectors. Nevertheless, even in general coordinates, there is a simple relation between the natural and dual basis vectors. Consider the dot-product
:{{big|&lt;math&gt;
    \mathbf{h}_i \cdot \mathbf{h}^j
    = \part_i\mathbf{r} \cdot \nabla u^j \,.
  &lt;/math&gt;}}
If{{math|&amp;#8202; ''i&amp;#8239;&amp;ne;&amp;#8239;j''&amp;#8202;,}} then {{math|''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;'''r'''&amp;#8202;,}} being in a direction in which {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}} varies while each other {{mvar|u&amp;#8239;&lt;sup&gt;j&lt;/sup&gt;}} does not, is tangential to a surface of constant {{mvar|u&amp;#8239;&lt;sup&gt;j&lt;/sup&gt;}} and therefore normal to {{math|&amp;nabla;''u&amp;#8239;&lt;sup&gt;j&lt;/sup&gt;'',}} so that the dot-product is zero. But by ({{EquationNote|26g}}),
:{{big|&lt;math&gt;
    du^i = \nabla u^i \cdot d\mathbf{r} \,;
  &lt;/math&gt;}}
and if we vary {{math|'''r'''}} by varying {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}} while holding each other {{mvar|u&amp;#8239;&lt;sup&gt;j&lt;/sup&gt;}} constant, we can divide by {{mvar|du&lt;sup&gt;i&lt;/sup&gt;}} and obtain
{{NumBlk|:|{{big|&lt;math&gt;
    1 ~\!= \nabla u^i \cdot \part_i\mathbf{r}
      = \mathbf{h}^i \!\cdot \mathbf{h}_i
  \qquad&lt;/math&gt;}}[with no summation].|{{EquationRef|81i}}}}
Putting the two cases together, we have
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}_i \cdot \mathbf{h}^j =~\! \delta_i^j
  &lt;/math&gt;}}|{{EquationRef|81}}}}
where the right-hand function, known as the '''Kronecker delta''' function, is defined by
{{NumBlk|:|{{big|&lt;math&gt;\delta_i^j = \delta_{ij} = \delta^{ij}
  =~&lt;/math&gt;}}&lt;math&gt;\begin{cases}
    0 &amp;\mathsf{if}~\, i \neq j \\
    1 &amp;\mathsf{if}~\, i = j \,.
  \end{cases}&lt;/math&gt;|{{EquationRef|82}}}}
Obviously the function is symmetric: the indices {{mvar|i }}and{{mvar| j}}&amp;#8202; can be interchanged. If two lists of vectors are related so that the dot-product of the {{mvar|i&amp;#8202;}}th vector in one list and the {{mvar|j&amp;#8202;}}th in the other is{{mvar| &amp;delta;&lt;sub&gt;ij&lt;/sub&gt;&amp;#8202;}}, the two lists are described as '''reciprocal'''. Thus the triplets {{math|('''h'''&lt;sub&gt;''i''&lt;/sub&gt;)}} and {{math|('''h'''&lt;sup&gt;''i''&lt;/sup&gt;)}} are '''reciprocal bases''': the dual basis is the reciprocal of the natural basis and vice versa. Hence, taking the natural basis as a reference, the dual basis is sometimes called &quot;the&quot; reciprocal basis.

In Cartesian coordinates, ({{EquationNote|81}}) becomes
:{{big|&lt;math&gt;
    \mathbf{e}_i ~\!\!\cdot \mathbf{e}_j =~\! \delta_{ij} \,.
  &lt;/math&gt;}}
So we have a relation for general coordinates ({{EquationNote|81}}) which is just as simple as its special case for Cartesian coordinates, ''provided that we use the natural basis for one factor and the dual basis for the other''. This will be a recurring pattern.

We have deduced the reciprocity relation ({{EquationNote|81}}) from prior definitions of the natural basis {{math|('''h'''&lt;sub&gt;''i''&lt;/sub&gt;)}} and the dual basis {{math|('''h'''&lt;sup&gt;''i''&lt;/sup&gt;)}}.&amp;#8201; This result has a partial converse, in that a reciprocity relation between bases is enough to define either basis in terms of the other&amp;mdash;as we shall see later. But first we proceed to components of vector fields.

{{cob}}

=== Contravariant and covariant components ===

{{cot}}

A '''coordinate grid''' is a set of intersecting curves such that on each curve, one coordinate varies while the others are constant. If we could embed such a grid in an elastic medium, and then stretch and rotate the medium, the natural basis vectors{{math| '''h'''&lt;sub&gt;''i''&lt;/sub&gt;}} given by ({{EquationNote|80a}}) would stretch and rotate ''with the medium'' and ''with the grid''. Accordingly, the ''natural'' basis is also called the '''covariant''' basis. But according to ({{EquationNote|81}}), the dot-product of a natural basis vector and a dual basis vector is '''invariant''' (independent of the coordinate system), so that the variation of one factor ''compensates''&amp;#8202; for the variation of the other. So, as the natural basis is &quot;covariant&quot; with the coordinate grid, we say that the dual basis is '''contravariant'''. Notice that the {{nowrap|''co''&amp;#8202;variant}} factor has a {{nowrap|''sub''&amp;#8202;script}} index (easily remembered because &quot;''co''&amp;#8202; rhymes with ''low''&amp;#8239;&quot;) whereas the {{nowrap|''contra''&amp;#8202;variant}} factor has a {{nowrap|''super''&amp;#8202;script}} index, and that one kind of variation must combine with the other in order to produce an {{nowrap|''in''&amp;#8202;variant}} result; these will be recurring patterns.

A vector field {{math|'''q'''}} may be expressed in components w.r.t. the natural (covariant) basis as
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q} = q^i \mathbf{h}_i
  \qquad&lt;/math&gt;}}|{{EquationRef|83a}}}}
with summation, or in components w.r.t. the dual (contravariant) basis as
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q} = q_i \mathbf{h}^i
  \qquad&lt;/math&gt;}}|{{EquationRef|83b}}}}
with summation. If{{math| '''q'''}} is to be invariant (a true vector, existing independently of the coordinate system), the components must be contravariant in the former case and covariant in the latter, and accordingly are written with superscripts and subscripts respectively. In Cartesian coordinates, the two bases are the same, so that the components w.r.t. the two bases are also the same; that's why, in the above section headed &quot;[[#Cartesian coordinates|Cartesian coordinates]]&quot;, we got away with writing component indices as subscripts. In ''general'' coordinates, however, the basis vectors have subscripts and the components have superscripts or vice versa, so that ''the index of implicit summation appears once as a superscript and once as a subscript''.

Taking dot-products of ({{EquationNote|83a}}) with{{math| '''h'''&amp;#8239;&lt;sup&gt;''j''&lt;/sup&gt;}}, applying ({{EquationNote|81}}), and noting that only one term on the right is non-zero, we obtain
{{NumBlk|:|{{big|&lt;math&gt;
     q^j =~\! \mathbf{q} \cdot \mathbf{h}^j .
  \qquad&lt;/math&gt;}}|{{EquationRef|83c}}}}
Similarly, taking dot-products of ({{EquationNote|83b}}) with{{math| '''h'''&lt;sub&gt;''j''&lt;/sub&gt;}} yields
{{NumBlk|:|{{big|&lt;math&gt;
     q_j =~\! \mathbf{q} \cdot \mathbf{h}_j \,.
  \qquad&lt;/math&gt;}}|{{EquationRef|83d}}}}
These results depend on the reciprocity relation ({{EquationNote|81}}) but not on the earlier definitions of the bases to which that relation applies. They say:
* to find the contravariant components of a vector, take its dot-products with the contravariant basis vectors, and
* to find the covariant components of a vector, take its dot-products with the covariant basis vectors;
''or'', in terms of the bases themselves:
* to find the components of a vector w.r.t. either basis, take dot-products of that vector with the ''other'' basis.

If a particular {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}} has a particular name, such as{{mvar| &amp;theta;}} or{{mvar| &amp;phi;}}, then, if we're not using indexed summation, we may find it convenient to write that name in place of the index{{mvar| i}}&amp;#8202; in the superscript or subscript.

At the present level of generality, the basis vectors {{math|'''h'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8239;,}} unlike their Cartesian counterparts {{math|'''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8239;,}} are ''not''&amp;#8202; assumed to be uniform. One consequence of this general non-uniformity is that, although we can say&amp;#8239; {{math|'''r'''&amp;#8201;{{=}}&amp;#8201;''x&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;'''e'''&lt;sub&gt;''i''&lt;/sub&gt;}}&amp;#8239; in Cartesian coordinates and&amp;#8239; {{math|'''q'''&amp;#8201;{{=}}&amp;#8201;''q&lt;sup&gt;i&lt;/sup&gt;''&amp;#8239;'''h'''&lt;sub&gt;''i''&lt;/sub&gt;}}&amp;#8239; in general coordinates, we ''cannot'' say
:{{big|&lt;math&gt;\mathbf{r} = u^i \mathbf{h}_i \qquad&lt;/math&gt;[''sic!''&amp;#8239;]}}
in general coordinates. For example, we have seen that in spherical coordinates the position vector {{math|'''r'''}} is simply &lt;math&gt;r\mathbf{\hat{r}}&lt;/math&gt;, i.e.{{math| ''r''&amp;#8202;'''h'''&lt;sub&gt;''r''&lt;/sub&gt;&amp;#8239;}};&amp;#8202; it is ''not''&amp;#8201; {{math|''r''&amp;#8202;'''h'''&lt;sub&gt;''r''&lt;/sub&gt;&amp;#8202;+&amp;#8202;''&amp;theta;''&amp;#8202;'''h'''&lt;sub&gt;''&amp;theta;''&lt;/sub&gt;&amp;#8202;+&amp;#8202;''&amp;phi;''&amp;#8202;'''h'''&lt;sub&gt;''&amp;phi;''&lt;/sub&gt;&amp;#8202;,}} because {{mvar|&amp;theta;}} and {{mvar|&amp;phi;}} are encoded in the direction of{{math|&amp;#8202; '''h'''&lt;sub&gt;''r''&lt;/sub&gt;&amp;#8202;}}.&amp;#8201; Similarly, in cylindrical coordinates the position vector {{math|'''r'''}} is {{math|''&amp;rho;''&amp;#8202;'''h'''&lt;sub&gt;''&amp;rho;''&lt;/sub&gt;&amp;#8202;+&amp;#8239;''z''&amp;#8202;'''h'''&lt;sub&gt;''z''&lt;/sub&gt;&amp;#8239;}};&amp;#8202; it is ''not''&amp;#8201; {{math|''&amp;rho;''&amp;#8202;'''h'''&lt;sub&gt;''&amp;rho;''&lt;/sub&gt;&amp;#8202;+&amp;#8202;''&amp;phi;''&amp;#8202;'''h'''&lt;sub&gt;''&amp;phi;''&lt;/sub&gt;&amp;#8202;+&amp;#8239;''z''&amp;#8202;'''h'''&lt;sub&gt;''z''&lt;/sub&gt;&amp;#8202;,}} because {{mvar|&amp;phi;}} is encoded in the direction of{{math|&amp;#8202; '''h'''&lt;sub&gt;''&amp;rho;''&lt;/sub&gt;&amp;#8202;}}.&amp;#8201; In both examples, encoding one coordinate in the direction of another coordinate's unit vector is circular in that the said direction depends on the position vector, which is the very thing that we want to represent.

A non-uniform basis is not a ''global''&amp;#8202; basis. It cannot give a uniform representation of a uniform vector field, because the standard of representation changes; it is like having a compass whose orientation varies from place to place and&amp;#10744;or a measuring stick whose length varies from place to place. But it can serve as a '''local basis'''&amp;#8202;&amp;mdash;as in ({{EquationNote|83a}}) and ({{EquationNote|83b}}), each of which expresses a vector field at a given location in terms of a basis at that location, notwithstanding that the basis may be different at other locations. And although a local basis (as we have just seen) cannot generally represent the position vector in a non-circular manner, it ''can''&amp;#8202; represent a ''change''&amp;#8202; in the position vector. By the generality of the multivariate chain rule ({{EquationNote|75}}),
:{{big|&lt;math&gt;
    \part_t \mathbf{r} = \part_i \mathbf{r} \,\part_t u^i .
  &lt;/math&gt;}}
Multiplying by {{mvar|dt&amp;#8239;}} we get
{{NumBlk|:|{{big|&lt;math&gt;
    d\mathbf{r} = \part_i \mathbf{r} \,du^i
  &lt;/math&gt;}}|{{EquationRef|84}}}}
or, substituting from ({{EquationNote|80a}}),
{{NumBlk|:|{{big|&lt;math&gt;
    d\mathbf{r} = \mathbf{h}_i ~\!du^i .
  \qquad&lt;/math&gt;}}|{{EquationRef|85}}}}
Thus the small changes in the coordinates{{mvar| u&lt;sup&gt;i&lt;/sup&gt;}}&amp;#8202; are the components of the true vector{{math| ''d'''''r'''}} w.r.t. the ''covariant''&amp;#8202; basis. That means the changes in the coordinates must be ''contravariant''. Here at last is the explanation why we write general coordinates with superscript indices. And again the point is moot for Cartesian coordinates, for which the covariant basis is also contravariant.

Since {{mvar|du&lt;sup&gt;i&lt;/sup&gt;}}&amp;#8202; is contravariant,&amp;#8201; {{math|''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;'''r'''}}&amp;#8202; in ({{EquationNote|84}}) must be covariant in order to yield the true vector{{math| ''d'''''r'''}}. This vindicates our decision to write {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}} with a subscript. Recall, however, that {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; means{{math| ''{{sfrac|&amp;part;|&amp;part;u&lt;sup&gt;i&lt;/sup&gt;&amp;#8202;}}''&amp;#8202;}}. Thus ''the derivative w.r.t. the contravariant quantity is covariant''&amp;#8202;&amp;mdash;wherefore it is said that ''a superscript in the denominator of a derivative counts as a subscript in the derivative as a whole''.

In ({{EquationNote|85}}), the general term&amp;#8239; {{math|'''h'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8201;''du&lt;sup&gt;i&lt;/sup&gt;''}} (not the sum) is the displacement of{{math|&amp;#8202; '''r'''}} due to the small change {{mvar|du&lt;sup&gt;i&lt;/sup&gt;}} in the coordinate {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}}. The three such displacements of{{math|&amp;#8202; '''r'''}} make concurrent edges of a parallelepiped whose signed volume is
:&lt;math&gt;
  dV =~\! \mathbf{h}_1~\!du^1 \cdot~\!\mathbf{h}_2~\!du^2
          ~\!\!\times\mathbf{h}_3~\!du^3 \,;
  &lt;/math&gt;
that is,
{{NumBlk|:|&lt;math&gt;dV = J \,du^1 du^2 du^3&lt;/math&gt;|{{EquationRef|86}}}}
where
:&lt;math&gt;J := \mathbf{h}_1 ~\!\!\cdot \mathbf{h}_2 \!\times~\!\!\mathbf{h}_3&lt;/math&gt;
or, to use a standard abbreviation for the scalar triple product,
{{NumBlk|:|&lt;math&gt;
    J := [~\!\mathbf{h}_1 \mathbf{h}_2 \mathbf{h}_3] \,.
  &lt;/math&gt;|{{EquationRef|87}}}}
{{mvar|J}}&amp;#8202; is called the '''Jacobian''' of the natural (covariant) basis. We describe the basis and the associated coordinate system as '''right-handed''' if this Jacobian is ''positive'', and '''left-handed''' if this Jacobian is ''negative''. Thus the handedness depends on the standard order in which we write the vectors; e.g., the standard Cartesian basis is right-handed because we write it as{{math| ('''i''',&amp;#8202;'''j''','''k''')}} but would be left-handed if we wrote it as{{math| ('''i''','''k''',&amp;#8202;'''j''')}}.

If the covariant basis is indeed a basis, its member vectors must not be coplanar; that is, {{mvar|J }}must not be zero. Hence, if the covariant basis is to be a local basis in some region of interest, {{mvar|J }}must not go to zero anywhere in that region, and therefore must have the same sign throughout the region; that is, the handedness of the coordinate system must be the same throughout the region.

{{cob}}

=== Properties of reciprocal bases ===

{{cot}}

We have noted that formulae ({{EquationNote|83c}}) and ({{EquationNote|83d}}), for the components of a vector w.r.t. the covariant and contravariant bases, depend only on the reciprocity relation ({{EquationNote|81}}) between the bases. Now, retaining the designations &quot;covariant&quot; and &quot;contravariant&quot; for convenience, let us see what else we can deduce from that relation.

Most obviously, the reciprocity relation leads to a simple component-based expression for the dot-product of two vector fields, say {{math|'''v'''}} and{{math| '''q'''&amp;#8202;,}} provided that we use the contravariant components and covariant basis ({{EquationNote|83a}}) for one vector, and the covariant components and contravariant basis ({{EquationNote|83b}}) for the other:
:{{big|&lt;math&gt;\mathbf{v} \cdot \mathbf{q}
    =~\! v^i ~\!\mathbf{h}_i \cdot q_j \mathbf{h}^j
    =~\! v^i \,\mathbf{h}_i \!\cdot\! \mathbf{h}^j \,q_j
    =~\! v^i ~\!\delta_i^j ~\!q_j \,,
  &lt;/math&gt;}}
whence selecting the non-zero terms gives
{{NumBlk|:|{{big|&lt;math&gt;\mathbf{v} \cdot \mathbf{q}
    =~\! v^i q_i \,.
  &lt;/math&gt;}}|{{EquationRef|88a}}}}
And the two vectors, being general, can swap roles in ({{EquationNote|83a}}) and ({{EquationNote|83b}}):
{{NumBlk|:|{{big|&lt;math&gt;\mathbf{v} \cdot \mathbf{q}
    =~\! v_i q^i \,.
  &lt;/math&gt;}}|{{EquationRef|88b}}}}

The cross-product needs a bit more preparation. First we define the '''permutation symbol''' {{mvar|&amp;epsiv;&lt;sub&gt;ijk&lt;/sub&gt;}} or{{mvar| &amp;epsiv;&lt;sup&gt;ijk&lt;/sup&gt;}} (also called the '''[[w:Tullio Levi-Civita|Levi-Civita]]''' symbol) as having the value&amp;#8202; {{math|+1}} if {{math|(''i'',&amp;#8201;''j'',&amp;#8239;''k'')}} is a permutation of{{math| (1,&amp;#8202;2,&amp;#8202;3)}} in the same cyclic order,&amp;#8201; {{math|&amp;minus;1 }}if {{math|(''i'',&amp;#8201;''j'',&amp;#8239;''k'')}} is a permutation of{{math| (1,&amp;#8202;2,&amp;#8202;3)}} in the reverse cyclic order, and {{math|0}} if {{math|(''i'',&amp;#8201;''j'',&amp;#8239;''k'')}} is not a permutation, i.e. if there is at least one repeated index. To put it more formally,
{{NumBlk|:|{{big|&lt;math&gt;\epsilon_{ijk\!} = \epsilon^{ijk\!} =&lt;/math&gt;}}
  {{resize|&lt;math&gt;\begin{cases}
    +1 &amp;\mathsf{if}\,\,(i,j,k)\in\big\{(1,2,3),~\!(2,3,1),~\!(3,1,2)\big\}\\
    -1 &amp;\mathsf{if}\,\,(i,j,k)\in\big\{(3,2,1),~\!(1,3,2),~\!(2,1,3)\big\}\\
    \phantom{-}0 &amp;\mathsf{otherwise}.
  \end{cases}&lt;/math&gt;}}|{{EquationRef|89}}}}
Note that because switching any two indices changes the cyclic order, ''switching any two indices changes the sign of the permutation symbol''. Now by ({{EquationNote|81}}),&amp;#8201; {{math|'''h'''&lt;sup&gt;1&lt;/sup&gt; }}is perpendicular to both {{math|'''h'''&lt;sub&gt;2&lt;/sub&gt; }}and{{math| '''h'''&lt;sub&gt;3&lt;/sub&gt;}}. So we can say
:&lt;math&gt;\mathbf{h}_2 \!\times\!\mathbf{h}_3 =~\! \alpha_1 ~\!\mathbf{h}^1&lt;/math&gt;
where {{math|''&amp;alpha;''&lt;sub&gt;1&lt;/sub&gt; }}is a real variable to be determined. Taking dot-products with{{math| '''h'''&lt;sub&gt;1&lt;/sub&gt;}} and applying ({{EquationNote|81}}) and ({{EquationNote|87}}), we find that&amp;#8202; {{math|''&amp;alpha;''&lt;sub&gt;1&lt;/sub&gt;&amp;#8239;{{=}}&amp;#8239;''J''&amp;#8202;,}} so that
{{NumBlk|:|&lt;math&gt;
    \mathbf{h}_2 \!\times\!\mathbf{h}_3 =~\! J \mathbf{h}^1 .
  &lt;/math&gt;|{{EquationRef|90.1}}}}
By the generality of the vectors we can rotate the three indices, but the sign of the left-hand side changes if we swap the two indices on the left. All six cases are covered by
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}_i \!\times\!\mathbf{h}_j
    =~\! J \epsilon_{ijk\,} \mathbf{h}^k .
  &lt;/math&gt;}}|{{EquationRef|90a}}}}
Here we want only one term; but we need not specify &quot;no sum&quot;, because for given {{mvar|i&amp;#8202; }}and{{mvar| j}}&amp;#8201; the permutation symbol leaves only one non-zero term in the sum over{{mvar| k}}. In words, this result says that the cross-product of two covariant basis vectors, with their indices in the standard cyclic order, is the Jacobian times the contravariant basis vector with the omitted index. Similarly, or rather reciprocally,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}^i \!\times\!\mathbf{h}^j
    =~\! J' \epsilon^{ijk\,} \mathbf{h}_k \,,
  &lt;/math&gt;}}|{{EquationRef|90b}}}}
where {{mvar|J&amp;prime;}}&amp;#8202; is the Jacobian ''of the contravariant basis''.

Equations ({{EquationNote|90a}}) and ({{EquationNote|90b}}), which we have obtained from the reciprocity relation ({{EquationNote|81}}), can be solved for {{math|'''h'''&lt;sup&gt;''k''&lt;/sup&gt; }}and{{math| '''h'''&lt;sub&gt;''k''&lt;/sub&gt;}} respectively; but now we ''do'' suppress the implicit sum, because {{mvar|k}}&amp;#8202; is &quot;given&quot; instead of {{mvar|i&amp;#8202; }}and{{mvar| j&amp;#8239;}}:
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}^k = \tfrac{\,1\,}{J}~\! \mathbf{h}_i \!\times\!\mathbf{h}_j \quad
  &lt;/math&gt;}} [distinct {{math|''i'',&amp;#8201;''j'',&amp;#8239;''k''}}  in cyclic order];
  |{{EquationRef|90c}}}}
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}_k = \tfrac{1}{\,J'}~\! \mathbf{h}^i \!\times\!\mathbf{h}^j \quad
  &lt;/math&gt;}}[distinct {{math|''i'',&amp;#8201;''j'',&amp;#8239;''k''}}  in cyclic order].
  |{{EquationRef|90d}}}}
Thus ''a reciprocity relation between bases is enough to define either basis in terms of the other''&amp;mdash;as claimed above.{{efn|Our ({{EquationNote|90c}}) corresponds to [[#stratton-41|Stratton, 1941]], p.&amp;#8239;39, eqs.&amp;#8239;(9). And our ({{EquationNote|90d}}) corresponds to Stratton's subsequent eqs.&amp;#8239;(11) except that Stratton has, in our notation, {{mvar|J}} instead of{{mvar| J&amp;prime;}}; the error is noted by Tai ([[#tai-95|1995]], p.&amp;#8239;59). See also our ({{EquationNote|92}}).}} If it is not convenient to suppress an implicit sum, the last two results can instead be written
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}^k =
      \tfrac{1}{2J}~\!\epsilon^{ijk\,}\mathbf{h}_i {\times}~\!\mathbf{h}_j
  &lt;/math&gt;}}|{{EquationRef|90e}}}}
and
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}_k =
      \tfrac{1}{2J'}~\!\epsilon_{ijk\,}\mathbf{h}^i {\times}~\!\mathbf{h}^j \,,
  &lt;/math&gt;}}|{{EquationRef|90f}}}}
where the factor 2 in each denominator is needed because the right-hand side has two equal non-zero terms&amp;mdash;the sign of the permutation symbol compensating for the order of the cross-product.

Now we're ready to consider the cross-product of two vector fields. In terms of the covariant basis,
:{{big|&lt;math&gt;\begin{align}\mathbf{v} \!\times\! \mathbf{q}
     =~\! v^i \mathbf{h}_i ~\!\!\times q^j \mathbf{h}_j
    &amp;=~\! v^i \,\mathbf{h}_i {\times}~\! \mathbf{h}_j \,q^j \\
    &amp;=~\! v^i J \epsilon_{ijk~\!} \mathbf{h}^k \;\!q^j \,;
  \end{align}&lt;/math&gt;}}
i.e.,
{{NumBlk|:|{{big|&lt;math&gt;\mathbf{v} \!\times\! \mathbf{q}
    =~\! J \epsilon_{ijk\,} v^i q^j \mathbf{h}^k .
  &lt;/math&gt;}}|{{EquationRef|91a}}}}
On the right, the two components and the basis vector are contravariant, but invariance is achieved by multiplying by the covariant Jacobian (which has three covariant factors). Similarly,
{{NumBlk|:|{{big|&lt;math&gt;\mathbf{v} \!\times\! \mathbf{q}
    =~\! J' \epsilon^{ijk} v_i q_j \;\!\mathbf{h}_k .
  &lt;/math&gt;}}|{{EquationRef|91b}}}}
On the right of ({{EquationNote|91a}}) or ({{EquationNote|91b}}), the implicit triple summation has 27 terms, of which only six&amp;mdash;corresponding to the six possible permutations of the three possible indices&amp;mdash;can be non-zero. Thus the factor following the Jacobian can be recognized as the familiar determinant whose columns (or rows), in cyclic order, are the components of{{math| '''v'''&amp;#8202;,}} the components of{{math| '''q'''&amp;#8202;,}} and the three basis vectors. In Cartesian coordinates, in which the Jacobians are equal to{{math| 1}} and we don't need the co&amp;#10744;contra distinction, both equations reduce to
:{{big|&lt;math&gt;\mathbf{v} \!\times\! \mathbf{q}
    =~\! \epsilon_{ijk\,} v_i q_j \;\!\mathbf{e}_k
  &lt;/math&gt;}}
&amp;mdash;a familiar result written in a possibly unfamiliar way.

The Jacobian of the contravariant basis is
:&lt;math&gt;J' =~\! \mathbf{h}^1 \cdot \mathbf{h}^2 \!\times\!\mathbf{h}^3&lt;/math&gt;
or, if we substitute from ({{EquationNote|90c}}),
:&lt;math&gt;\begin{align}J'
  &amp;= \frac{(\mathbf{h}_2 \!\times\!\mathbf{h}_3)}{J} \cdot
     \frac{(\mathbf{h}_3 \!\times\!\mathbf{h}_1)}{J} \times
     \frac{(\mathbf{h}_1 \!\times\!\mathbf{h}_2)}{J} \\[.5ex]
  &amp;= \frac{(\mathbf{h}_2 \!\times\!\mathbf{h}_3) \cdot
           (\mathbf{h}_3 \!\times\!\mathbf{h}_1) \times
           (\mathbf{h}_1 \!\times\!\mathbf{h}_2)}
          {J^3} \,.
  \end{align}&lt;/math&gt;
In the numerator, the cross-product of cross-products can be read as a vector triple product in which the first factor is a cross-product. Expanding that triple product and noting that one term is a scalar triple product with a repeated factor, we get
:&lt;math&gt;
    J' = \frac{(\mathbf{h}_2 \!\times\!\mathbf{h}_3) \cdot J\mathbf{h}_1}{J^3}
       = \frac{\,J^2}{~J^3 ~\!} = \frac{\,1\,}{J} \,,
  &lt;/math&gt;
so that we may write
{{NumBlk|:|&lt;math&gt;J' =~\! J^{-1} &lt;/math&gt;|{{EquationRef|92}}}}
in ({{EquationNote|90b}}), ({{EquationNote|90d}}), ({{EquationNote|90f}}), and ({{EquationNote|91b}}). In words, ''the Jacobian of the reciprocal basis is the reciprocal of the Jacobian'' of the original basis. Therefore the two Jacobians have the same sign. Therefore ''a basis is right-handed if and only if its reciprocal is right-handed''. Thus the natural and dual bases of a coordinate system have the same handedness, and the handedness of either may be identified with the handedness of the coordinate system.

{{cob}}

=== The gradient, del, and advection operators ===

{{cot}}

Let {{mvar|p}}&amp;#8202; be a scalar field, and let{{mvar| s}}&amp;#8202; be arc length in the direction of the unit vector{{math| '''s&amp;#770;'''}}. By the multivariate chain rule ({{EquationNote|75}}),
:{{big|&lt;math&gt;\begin{align}\part_s p
    &amp;= \part_i p \;\part_s u^i \\
    &amp;= \part_i p \;\mathbf{\hat{s}} \cdot \nabla u^i \\
    &amp;= \part_i p \;\mathbf{\hat{s}} \cdot \mathbf{h}^i \\
    &amp;=\mathbf{\hat{s}} \cdot \mathbf{h}^i \part_i p \,.
  \end{align}&lt;/math&gt;}}
So&amp;#8202; {{math|'''h'''&lt;sup&gt;''i''&lt;/sup&gt;''&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;p''}}&amp;#8239; is the vector whose (invariant) scalar component in the direction of any{{math| '''s&amp;#770;'''}} is the directional derivative of{{mvar| p}} in that direction; that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla p = \mathbf{h}^i \part_i p \,,
  &lt;/math&gt;}}|{{EquationRef|93g}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla =~\! \mathbf{h}^i \part_i \,.
  &lt;/math&gt;}}|{{EquationRef|93o}}}}
Apart from the need to pair a superscript with a subscript, these two results look as simple as their Cartesian special cases ({{EquationNote|58g}}) and ({{EquationNote|58o}}).

If {{mvar|&amp;psi;}}&amp;#8202; is a generic field and {{math|'''q'''}} is a general vector in the direction of the same{{math| '''s'''&amp;#8202;}} then by definition ({{EquationNote|11}}),
:{{big|&lt;math&gt;\begin{align}
    \mathbf{q}\;\!{\cdot}\nabla\,\psi
    &amp;= |\mathbf{q}| \,\part_s \psi \\
    &amp;= |\mathbf{q}| \,\part_i \psi \,\part_s u^i \\
    &amp;= \part_i \psi \;|\mathbf{q}| ~\!\part_s u^i \\
    &amp;= \part_i \psi \;\mathbf{q} \cdot \nabla u^i \\
    &amp;= \part_i \psi \;\mathbf{q} \cdot \mathbf{h}^i \\
    &amp;= \part_i \psi \;q^j \mathbf{h}_j \cdot \mathbf{h}^i \\
    &amp;= \part_i \psi \;q^j \epsilon_j^i \\
    &amp;= \part_i \psi \,q^i \,;
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,\psi = q^i \part_i \psi \,,
  &lt;/math&gt;}}|{{EquationRef|94}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla =~\! q^i \part_i \,.
  &lt;/math&gt;}}|{{EquationRef|94o}}}}
These results likewise look as simple as their Cartesian special cases ({{EquationNote|64}}) and ({{EquationNote|64o}}).&amp;#8201; And by ({{EquationNote|88a}}), the {{math|'''q&amp;sdot;'''&amp;nabla;}} operator again turns out to be the formal dot-product of&amp;#8202; {{math|'''q'''}} and{{math| &amp;nabla;}}.

{{cob}}

=== The curl and divergence operators ===

{{cot}}

To express the curl of a vector field{{math| '''q'''&amp;#8202;,}} we choose the contravariant basis ({{EquationNote|83b}}) and apply identity ({{EquationNote|71c}}):
:{{big|&lt;math&gt;\begin{align}\operatorname{curl}\mathbf{q}
    &amp;= \operatorname{curl} q_j \mathbf{h}^j \\
    &amp;= q_j \operatorname{curl}\mathbf{h}^j + \nabla q_j \times \mathbf{h}^j .
  \end{align}&lt;/math&gt;}}
On the right, the first term vanishes because {{math|'''h'''&amp;#8239;&lt;sup&gt;''j''&lt;/sup&gt;}}&amp;#8202; is {{math|&amp;nabla;''u&amp;#8239;&lt;sup&gt;j&lt;/sup&gt;''}} (and the curl of a gradient is zero). Substituting from ({{EquationNote|93o}}) in the second term, we obtain
:{{big|&lt;math&gt;\operatorname{curl}\mathbf{q}
    = \mathbf{h}^i \part_i q_j \times \mathbf{h}^j
    = \mathbf{h}^i {\times}~\! \mathbf{h}^j ~\!\part_i q_j
  &lt;/math&gt;}}
or, using ({{EquationNote|90b}}),
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{curl}\mathbf{q}
    = J' \epsilon^{ijk\,} \mathbf{h}_k \part_i q_j
  &lt;/math&gt;}}|{{EquationRef|95c}}}}
or, in a more familiar form,
:&lt;math&gt;
    \operatorname{curl}\mathbf{q} \,=\, J'\,
      \begin{vmatrix}
        \mathbf{h}_1 &amp; \part_1 &amp; q_1 \\
        \mathbf{h}_2 &amp; \part_2 &amp; q_2 \\
        \mathbf{h}_3 &amp; \part_3 &amp; q_3
      \end{vmatrix} \,.
  &lt;/math&gt;
Formula ({{EquationNote|95c}}) agrees with a result obtained by Tai with his &quot;symbolic vector&quot; method.&lt;ref&gt;[[#tai-95|Tai, 1995]], p.&amp;#8239;66, eq.&amp;#8239;(9.41).&lt;/ref&gt; It is also what we would get by naively using ({{EquationNote|91b}}) to evaluate {{math|&amp;nabla;&amp;#8202;&amp;times;&amp;#8202;'''q'''&amp;#8202;}}.

But ({{EquationNote|95c}}) does not end in a subexpression for the operand{{math| '''q'''}}&amp;#8202; and therefore does not directly yield an expression for the curl ''operator''. To find this operator and the divergence operator, we return to the original definitions ({{EquationNote|4g}}), ({{EquationNote|4c}}), and ({{EquationNote|4d}}), noting that they can be combined as
{{NumBlk|:|&lt;math&gt;\nabla ~\!\!* \psi
    \,=\, \tfrac{1}{dV}\!\iint_{\delta S} (\mathbf{\hat{n}}~\!dS*\psi) \,,
  &lt;/math&gt;|{{EquationRef|96}}}}
where {{math|&amp;lowast;}} may be a null for the gradient, a cross for the curl, or a dot for the divergence.{{efn|But not dot-del for the Laplacian, as in ({{EquationNote|19}}), because we want to use an elementary product rule inside the integral.}} Recalling that the value of this expression does not depend on the shape of{{mvar| &amp;delta;S&amp;#8202;}}, let{{mvar| &amp;delta;S&amp;#8202;}} be the parallelepiped defined by the six equicoordinate surfaces at {{mvar|u&lt;sup&gt;i&lt;/sup&gt; }}and{{mvar| u&lt;sup&gt;i&lt;/sup&gt;+du&lt;sup&gt;i&lt;/sup&gt;}}, so that {{mvar|dV}}&amp;#8202; is given by ({{EquationNote|86}}). Then the contribution to the integral from the face at{{math| ''u''&amp;sup1;+''du''&amp;sup1; &amp;#8202;}}is
:&lt;math&gt;\Big[
    \big(\mathbf{h}_2 du^2 \!\times\! \mathbf{h}_3 du^3\big) * \psi
  \Big]_{u^1 + du^1}&lt;/math&gt;
where the square brackets and subscripting mean &quot;evaluated at&quot;. This can be written
:&lt;math&gt;du^2 du^3 \Big[
    (\mathbf{h}_2 {\times}~\! \mathbf{h}_3) * \psi
  \Big]_{u^1 + du^1}&lt;/math&gt;
or, by ({{EquationNote|90.1}}),
:&lt;math&gt;du^2 du^3 \big[J \mathbf{h}^1 ~\!\!* \psi \big]_{u^1 + du^1} \,.&lt;/math&gt;
Similarly, the contribution from the face at{{math| ''u''&amp;sup1;}} (where {{math|'''h'''&lt;sub&gt;1&lt;/sub&gt;}} points inward instead of outward) is
:&lt;math&gt;-du^2 du^3 \big[J \mathbf{h}^1 ~\!\!* \psi \big]_{u^1} \,.&lt;/math&gt;
The sum of the contributions from the two opposite faces can then be written
:&lt;math&gt;du^1 du^2 du^3 ~\!\part_1 \big(J \mathbf{h}^1 ~\!\!* \psi\big) ~,&lt;/math&gt;
so that when we add in the contributions from the other two pairs of opposite faces, the entire integral becomes
:{{big|&lt;math&gt;
    du^1 du^2 du^3 ~\!\part_i \big(J \mathbf{h}^i ~\!\!* \psi\big)
  &lt;/math&gt;}}
(with implicit summation over{{mvar| i}}). Substituting this and ({{EquationNote|86}}) into ({{EquationNote|96}}), we get
{{NumBlk|:|{{big|&lt;math&gt;\nabla ~\!\!* \psi
    = \tfrac{\,1\,}{J} ~\!\part_i \big(J \mathbf{h}^i ~\!\!* \psi\big) \,.
  &lt;/math&gt;}}|{{EquationRef|97}}}}
Now applying the product rule gives
{{NumBlk|:|{{big|&lt;math&gt;\nabla ~\!\!* \psi
    = \mathbf{h}^i ~\!\!* \part_i \psi
      + \tfrac{\,1\,}{J} ~\!\part_i(J \mathbf{h}^i) * \psi \,.
  &lt;/math&gt;}}|{{EquationRef|98}}}}
Here the left-hand side is {{math|&amp;nabla;&amp;#8202;&amp;lowast;''&amp;psi;''}}&amp;#8239; according to our original volume-based definition ({{EquationNote|4g}}) of the {{math|&amp;nabla; }}operator&amp;mdash;which is known to yield the curl or the divergence if&amp;#8202; {{math|&amp;lowast;}} is a cross or a dot, respectively&amp;mdash;whereas the first term on the right is what we would get for {{math|&amp;nabla;&amp;#8202;&amp;lowast;''&amp;psi;''}}&amp;#8239; by using our latest definition ({{EquationNote|93o}}) of the {{math|&amp;nabla; }}operator and allowing{{mvar| &amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; to &quot;pass by&quot; the star in the Wilsonian manner. So, if we can show that the second term on the right is zero, we shall have established the precise sense in which the del-cross and del-dot notations are valid in general coordinates. In that second term, by ({{EquationNote|90e}}),
:{{big|&lt;math&gt;\begin{align}
    \part_i(J \mathbf{h}^i)
    &amp;= \part_i \big(
         \tfrac{\,1\,}{2}~\!\epsilon^{jki}\mathbf{h}_j {\times}~\!\mathbf{h}_k
       \big) \\
    &amp;= \tfrac{\,1\,}{2}~\!\epsilon^{jki}
       \part_i \big(\mathbf{h}_j {\times}~\!\mathbf{h}_k \big) \\[.5ex]
    &amp;= \tfrac{\,1\,}{2}~\!\epsilon^{jki}
       \big(\part_i \mathbf{h}_j \!\times~\!\!\mathbf{h}_k +
            \mathbf{h}_j \!\times~\!\!\part_i \mathbf{h}_k \big) \\[.5ex]
    &amp;= \tfrac{\,1\,}{2}~\!\epsilon^{jki}
       \big(\part_i \part_j \mathbf{r} ~\!\!\times\! \mathbf{h}_k +
            \mathbf{h}_j {\times}~\! \part_i \part_k \mathbf{r} \big) \,,
  \end{align}&lt;/math&gt;}}
where the last line follows by ({{EquationNote|80a}}). But the order of partial differentiation can be switched. So, in the sum over the permutations, for each term in{{math| ''&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;&amp;part;&lt;sub&gt;j&lt;/sub&gt;''&amp;#8239;'''r'''}}&amp;#8239; there is an equal term in{{math| ''&amp;part;&lt;sub&gt;j&lt;/sub&gt;&amp;#8202;&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;'''r'''}}&amp;#8239; to which the permutation symbol attaches the opposite sign, so that the terms in{{math| ''&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;&amp;part;&lt;sub&gt;j&lt;/sub&gt;''&amp;#8239;'''r'''}}&amp;#8202; cancel. Similarly the terms in{{math| ''&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;&amp;part;&lt;sub&gt;k&lt;/sub&gt;''&amp;#8239;'''r'''}}&amp;#8202; cancel. Thus, as anticipated, the second term in ({{EquationNote|98}}) is zero and we have
{{NumBlk|:|{{big|&lt;math&gt;\nabla ~\!\!* \psi
    = \mathbf{h}^i ~\!\!* \part_i \psi \,.
  &lt;/math&gt;}}|{{EquationRef|99}}}}

If&amp;#8202; {{math|&amp;lowast;}} is a null and {{mvar|&amp;psi;}} is a scalar field {{math|''p''&amp;#8202;,}} then ({{EquationNote|99}}) becomes ({{EquationNote|93g}}) and thus (fortunately!) confirms ({{EquationNote|93o}}) as the form of the del operator in general coordinates.

Now let {{mvar|&amp;psi;}} be a ''vector'' field {{math|'''q'''&amp;#8202;}}.&amp;#8201; If{{math|&amp;#8202; &amp;lowast;}} is a cross, then ({{EquationNote|99}}) becomes
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{curl}\mathbf{q}
    = \mathbf{h}^i ~\!\!\times \part_i \mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|100c}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{curl}
    = \mathbf{h}^i ~\!\!\times \part_i \,.
  &lt;/math&gt;}}|{{EquationRef|100o}}}}
If instead {{math|&amp;lowast;}} is a dot, ({{EquationNote|99}}) becomes
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{div}\mathbf{q}
    = \mathbf{h}^i ~\!\!\cdot \part_i \mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|101d}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{div}
    = \mathbf{h}^i ~\!\!\cdot \part_i \,.
  &lt;/math&gt;}}|{{EquationRef|101o}}}}
But if we take the {{math|&amp;nabla;}} operator as given by ({{EquationNote|93o}}) and try to construct the curl and divergence operators (in the ''same''&amp;#8202; coordinates) as {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8239; respectively, we get&amp;#8202; {{math|'''h'''&lt;sup&gt;''i''&lt;/sup&gt;&amp;#8239;''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;&amp;times;}}&amp;#8239; and&amp;#8239; {{math|'''h'''&lt;sup&gt;''i''&lt;/sup&gt;&amp;#8239;''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;'''&amp;sdot;'''}}&amp;#8201; respectively [compare ({{EquationNote|61o}}) and ({{EquationNote|62o}})]; and if we then let{{mvar| &amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; &quot;pass by&quot; the cross and the dot, we get ({{EquationNote|100o}}) and ({{EquationNote|101o}}), or ({{EquationNote|100c}}) and ({{EquationNote|101d}}) if we include the operand{{math| '''q'''&amp;#8202;}}. Thus ''the del-cross and del-dot notations work in general coordinates''.

Equations ({{EquationNote|100c}}) to ({{EquationNote|101o}}) are apparently due to Tai ([[#tai-95|1995]], eqs.&amp;#8239;9.39, 9.40, 9.34, &amp;amp; 9.35, and text on p.&amp;#8239;66), who derives them, along with the corresponding form of the del operator (his eq.&amp;#8239;9.33), from volume-based definitions expressed in his &quot;symbolic vector&quot; notation. But he does not point out that the curl and divergence operators are obtainable from that del operator, as del-cross and del-dot, via the same &quot;pass by&quot; step that he condemns in the Cartesian context. Speaking of which, we should note that our equations ({{EquationNote|100c}}) to ({{EquationNote|101o}}), apart from the need to pair a superscript with a subscript, are as simple as their Cartesian special cases ({{EquationNote|59c}}), ({{EquationNote|59o}}), ({{EquationNote|60d}}), and ({{EquationNote|60o}}).

In ({{EquationNote|100c}}) and ({{EquationNote|101d}}), it goes without saying that&amp;#8202; {{math|''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;'''q'''}}&amp;#8202; must be evaluated correctly&amp;mdash;in particular, that if the operand is expressed in terms of non-uniform basis vectors, the non-uniformity must be taken into account. Formula ({{EquationNote|95c}}), for the curl, does not suffer from this complication, because it is already expressed in terms of (covariant) components w.r.t. the contravariant basis (whose non-uniformity has already been taken into account). To obtain a similarly convenient formula for the divergence, we use (contravariant) components w.r.t. the {{nowrap|''co''&amp;#8202;variant}} basis: in ({{EquationNote|97}}), if{{math|&amp;#8202; &amp;lowast;}} is a dot and {{mvar|&amp;psi;}} is a vector field{{math| '''q'''&amp;#8202;,}} we have
:{{big|&lt;math&gt;\operatorname{div}\mathbf{q}
    = \tfrac{\,1\,}{J} ~\!\part_i \big(J \mathbf{h}^i {\cdot}\, \mathbf{q}\big)
  &lt;/math&gt;}}
or, by ({{EquationNote|83c}}),
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{div}\mathbf{q}
    = \tfrac{\,1\,}{J} ~\!\part_i \big(J q^i \big) \,.
  &lt;/math&gt;}}|{{EquationRef|102d}}}}
This too agrees with Tai ([[#tai-95|1995]], p.&amp;#8239;65, eq.&amp;#8239;9.37).

{{cob}}

=== The Laplacian ===

{{cot}}

For a scalar operand, applying ({{EquationNote|101o}}) and reversing the &quot;pass by&quot;, we find that the Laplacian operator is
:{{big|&lt;math&gt;\operatorname{div}\nabla
    = \mathbf{h}^i \!\cdot \part_i \nabla
    = \mathbf{h}^i \part_i ~\!\!\cdot \nabla
    = \nabla {\cdot} \nabla = \nabla^2 .
&lt;/math&gt;}}
And by the linearity of the Laplacian, the{{math| &amp;nabla;&lt;sup&gt;2&lt;/sup&gt;}} formulation remains valid if the operand is any linear combination of scalars with uniform coefficients&amp;mdash;including a vector field, because that is expressible (even if not actually expressed) w.r.t. a uniform basis. (And if it is expressed in terms of a non-uniform basis, the non-uniformity must be taken into account in differentiations.)

In what follows, however, we shall find it convenient to take a different approach. If{{mvar| &amp;psi;}}&amp;#8202; is a scalar field, its gradient as given by ({{EquationNote|93g}}) is&amp;#8202; {{math| '''h'''&amp;#8239;&lt;sup&gt;''j''&lt;/sup&gt;''&amp;part;&lt;sub&gt;j&lt;/sub&gt;&amp;#8239;&amp;psi;''&amp;#8202;,}} of which the {{mvar|i&amp;#8202;}}th contravariant component is{{math| '''h'''&lt;sup&gt;''i''&lt;/sup&gt;'''&amp;sdot;&amp;#8239;h'''&amp;#8239;&lt;sup&gt;''j''&lt;/sup&gt;''&amp;part;&lt;sub&gt;j&lt;/sub&gt;&amp;#8239;&amp;psi;''&amp;#8202;,}} which takes the place of{{mvar| q&lt;sup&gt;i&lt;/sup&gt;}} in ({{EquationNote|102d}}), so that the divergence of the gradient of{{mvar| &amp;psi; &amp;#8202;}}is
{{NumBlk|:|{{big|&lt;math&gt;\triangle\psi
    = \tfrac{\,1\,}{J} ~\!\part_i
        \big(J \mathbf{h}^i {\cdot}~\! \mathbf{h}^j \part_j \psi \big) \,.
  &lt;/math&gt;}}|{{EquationRef|103L}}}}
This remains well-defined if{{mvar| &amp;psi;}}&amp;#8202; is a generic field (although we still need to deal with any non-uniformity of the basis in which{{mvar| &amp;psi;}}&amp;#8202; might be expressed).

{{cob}}

=== Affine coordinates ===

{{cot}}

If a basis is ''uniform'', so is its Jacobian. Hence, by ({{EquationNote|90c}}) and ({{EquationNote|90d}}), the dual (contravariant) basis is uniform if and only if the natural (covariant) basis is uniform. A coordinate system in which these bases are uniform is described as '''affine'''. In affine coordinates,
* by ({{EquationNote|80a}}), {{math|''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;'''r''' }}is uniform, so that the curves on which only one coordinate varies are straight parallel lines; and
* by ({{EquationNote|80b}}), {{math|&amp;nabla;''u&lt;sup&gt;i&lt;/sup&gt;'' }}is uniform, so that the level surfaces of each coordinate (being perpendicular to {{math|&amp;nabla;''u&lt;sup&gt;i&lt;/sup&gt;''}}) are parallel planes.
Obviously Cartesian coordinates are affine; but one can also construct affine coordinate systems in which the three vectors of each basis are not mutually perpendicular and&amp;#10744;or the coordinates have different scales or different units.

We have noted above that the correct application of the del-cross, del-dot, and del-squared notations must allow for non-uniformity of basis vectors&amp;mdash;an issue that does not arise with affine coordinates, including Cartesian coordinates. Hence, while these notations are not (as is sometimes alleged) invalid in other coordinate systems, it would be fair to say that they are safer and more convenient in affine coordinates, including Cartesian coordinates.

{{cob}}

=== Orthogonal coordinates ===

{{cot}}

We know, e.g. from ({{EquationNote|90a}}) and ({{EquationNote|90b}}), that if two bases are reciprocal, the cross-product of the {{mvar|i&amp;#8202;}}th and {{mvar|j&amp;#8202;}}th members of one basis (either one) is collinear with the {{mvar|k&amp;#8202;}}th member of the other, if {{math|''i'',&amp;#8201;''j'',&amp;#8239;''k''}}&amp;#8239; are distinct. But if the first basis is ''orthogonal'' (that is, if its three member vectors are mutually orthogonal), the same cross-product is also collinear with the {{mvar|k&amp;#8202;}}th member of the ''same'' basis, so that ''corresponding members of the two bases are collinear''. It follows that ''the natural basis of a coordinate system is orthogonal if and only if the dual basis is orthogonal''. And if the bases are orthogonal, the coordinate system itself is said to be '''orthogonal'''.

Cartesian coordinates are obviously both affine and orthogonal, and we have already implied that there is a class of coordinate systems that are affine but not orthogonal. The most widely-used class of non-Cartesian systems, however, contains the systems that are orthogonal but not affine; this class, of which the cylindrical and spherical systems are the best-known members, is the class of '''curvilinear orthogonal coordinates'''. But we shall drop the word ''curvilinear''&amp;#8202; in order to include Cartesian coordinates as a special case.

In orthogonal coordinates, expressing a member of one basis in terms of its reciprocal basis is especially simple because corresponding members of the two bases are collinear, wherefore we can say
:{{big|&lt;math&gt;
    \mathbf{h}^i =~\! \beta_i \mathbf{h}_i \,,
  &lt;/math&gt;}}
where {{mvar|&amp;beta;&lt;sub&gt;i&lt;/sub&gt;}} is a real variable to be determined (and the single index on the left-hand side means ''no summation''). Substituting this into ({{EquationNote|81i}}) gives
:{{big|&lt;math&gt;
    \beta ~\!= 1/h_i^{~2}
  &lt;/math&gt;}}
where
{{NumBlk|:|{{big|&lt;math&gt;
    h_i = \big|\mathbf{h}_i \big| \,,
  &lt;/math&gt;}}|{{EquationRef|104}}}}
so that
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}^i =~\! \mathbf{h}_i \big/ h_i^{~2} \,.
  &lt;/math&gt;}}|{{EquationRef|105}}}}
Substituting that into ({{EquationNote|83c}}) and comparing the result with ({{EquationNote|83d}}), we get
{{NumBlk|:|{{big|&lt;math&gt;
    q^i =~\! q_i \big/ h_i^{~2} \,.
  &lt;/math&gt;}}|{{EquationRef|106}}}}
Comparing ({{EquationNote|104}}) with definition ({{EquationNote|80a}}) reveals that {{mvar|h&lt;sub&gt;i&lt;/sub&gt;}} is the magnitude of{{math| ''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;'''r'''}}.&amp;#8201; Accordingly {{mvar|h&lt;sub&gt;i&lt;/sub&gt;}} is called the '''scale factor''' associated with the coordinate{{mvar| u&lt;sup&gt;i&lt;/sup&gt;&amp;#8202;}}; it is the factor by which we multiply a small change in{{mvar| u&lt;sup&gt;i&lt;/sup&gt;}} to obtain the magnitude of the consequent change in position.{{efn|Hsu ([[#hsu-84|1984]], p.&amp;#8239;171) implies that the scale factors are also called &quot;metric coefficients&quot;, and Tai ([[#tai-94|1994]], [[#tai-95|1995]]) prefers the latter term. This is loose terminology because, in general, the ''metric coefficient''&amp;#8202; is defined as
:{{math|''g&lt;sub&gt;ij&lt;/sub&gt;'' {{=}} '''h'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8239;'''&amp;sdot;&amp;#8201;h'''&lt;sub&gt;''j''&lt;/sub&gt;&amp;#8201;}}.
Hence, in the special case of orthogonal coordinates, we have {{math|''g&lt;sub&gt;ij&lt;/sub&gt;''&amp;#8201;{{=}}&amp;#8201;0}}&amp;#8202; for {{math|''i&amp;#8239;&amp;ne;&amp;#8239;j''&amp;#8202;,}}&amp;#8239; and&amp;#8202; {{math|''g&lt;sub&gt;ii&lt;/sub&gt;&amp;#8239;{{=}}&amp;#8201;h&lt;sub&gt;i&lt;/sub&gt;''&lt;sup&gt;2&lt;/sup&gt;}}&amp;#8202; [no sum]. Thus the scale factors are not special cases of the metric coefficients, but the ''square roots''&amp;#8202; of special cases of the metric coefficients (''cf''. [[#tai-95|Tai, 1995]], p.&amp;#8239;43, line 3).}}

If we now define
{{NumBlk|:|&lt;math&gt;\varsigma \,=~\! \begin{cases}
    +1 &amp;\mathsf{for~a~right{\operatorname{-}}handed~system} \\[.5ex]
    -1 &amp;\mathsf{for~a~left{\operatorname{-}}handed~system} ~,
  \end{cases}&lt;/math&gt;|{{EquationRef|107}}}}
then, due to the orthogonality,&amp;#8201; ({{EquationNote|87}}) and ({{EquationNote|92}}) are respectively reduced to
{{NumBlk|:|&lt;math&gt;
    J = \varsigma\, h_1 h_2 h_3
  &lt;/math&gt;|{{EquationRef|108}}}}
and
{{NumBlk|:|&lt;math&gt;
    J' = \frac{\,1\,}{J} = \frac{1}{\varsigma\, h_1 h_2 h_3}
       = \frac{\varsigma}{h_1 h_2 h_3}
  &lt;/math&gt;|{{EquationRef|109}}}}
&amp;mdash;although, for brevity, we shall sometimes leave things in terms of{{mvar| J}}.

At this point, we ''could''&amp;#8202; substitute ({{EquationNote|105}}) and ({{EquationNote|106}}) into earlier equations and obtain a suite of formulae for the differential operators in terms of the covariant basis and {{nowrap|''co''&amp;#8202;variant}} components! But we can avoid this confusing breach of convention by ''normalizing'' the basis vectors.

An '''orthonormal''' basis is one whose members are mutually orthogonal ''unit'' vectors. The assumption of unit vectors is introduced so late because it is more useful with orthogonality than without. If one basis consisted of unit vectors that were not all orthogonal, then the reciprocal basis vectors given by ({{EquationNote|90c}}) or ({{EquationNote|90d}}) would not all be unit vectors (because if they were, the cross-product of any two of the ''original''&amp;#8202; unit vectors would need to have the same magnitude as their scalar triple product [Jacobian], and this would require the third original unit vector to be perpendicular to the other two). But if the basis{{math| ('''h'''&lt;sub&gt;''i''&lt;/sub&gt;)}} consists of ''orthogonal''&amp;#8202; unit vectors, equation ({{EquationNote|105}}) implies that the reciprocal basis consists of the ''same'' vectors; and the converse is also true, by the symmetry of the reciprocity relations. Thus ''an orthonormal basis is its own reciprocal''. Hence, if we choose an orthonormal basis, we do not need superscripts to distinguish the reciprocal basis from the original, or to distinguish components w.r.t. the latter basis from those w.r.t. the former.

An orthonormal basis is not generally covariant, because it doesn't stretch with the coordinate grid (although it does rotate with the grid). Neither is it generally contravariant, because its reciprocal (i.e. itelf) is not generally covariant. Hence, if a ''non''&amp;#8202;-orthonormal natural or dual basis of an orthogonal coordinate system is ''normalized'' (replaced by unit vectors in the same directions), the resulting orthonormal basis is not covariant or contravariant, and components with respect thereto are not contravariant or covariant, and the new basis vectors are not given in terms of the coordinates by ({{EquationNote|80a}}) or ({{EquationNote|80b}}); the basis is therefore described as a '''non-coordinate basis'''. By default, the indices of the orthonormal basis vectors and associated components are written as subscripts, but these are not indicative of covariance. The coordinates themselves remain contravariant (e.g., if the grid dilates, the same movement in space corresponds to ''smaller'' changes in the coordinates); but, for want of covariant basis vectors to pair them with, we tend to write the coordinates with subscripts when the basis is orthonormal.

Nevertheless, it is convenient to have one basis instead of two. Moreover, the components of a vector w.r.t. an orthonormal basis are '''physical components''': they have the same dimension (same units) as the represented vector, and they are the components that we would have in mind if we wanted to ''measure'' the &quot;components&quot; in the directions of the basis vectors. Hence an orthonormal basis is called a '''physical basis'''. Accordingly, it is indeed common practice to normalize the basis vectors of orthogonal coordinate systems. This together with the prevalence of such coordinate systems helps to account for the familiarity of subscripts as indices, and for the jarring unfamiliarity of superscript indices when general (possibly non-orthogonal) coordinates are encountered for the first time.

To normalize the covariant basis, let{{math| '''h&amp;#770;'''&lt;sub&gt;''i''&lt;/sub&gt;}} (as usual) be the unit vector in the direction of{{math|&amp;#8202; '''h'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8202;.}} Then, by ({{EquationNote|104}}),
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}_i =~\! h_i \mathbf{\hat{h}}_i
  &lt;/math&gt;}}|{{EquationRef|110}}}}
(again with no summation, due to the single index on the left). Hence ({{EquationNote|105}}) becomes:
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}^i =~\! \mathbf{\hat{h}}_i \big/ h_i \,.
  &lt;/math&gt;}}|{{EquationRef|111}}}}
A vector field {{math|'''q'''}} is expressed in components w.r.t. the basis{{math| ('''h&amp;#770;'''&lt;sub&gt;''i''&lt;/sub&gt;)}} as
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q} = \hat{q_i} \;\!\mathbf{\hat{h}}_i
  \qquad&lt;/math&gt;}}|{{EquationRef|112}}}}
(with summation), where
{{NumBlk|:|{{big|&lt;math&gt;
    \hat{q_i} =~\! \mathbf{q} \!\cdot\! \mathbf{\hat{h}}_i \,.
  \qquad&lt;/math&gt;}}|{{EquationRef|113}}}}
(Here the hat on{{math| ''q''&amp;#770;&lt;sub&gt;''i''&lt;/sub&gt;}} is needed to distinguish the coefficient of{{math| '''h&amp;#770;'''&lt;sub&gt;''i''&lt;/sub&gt;}} from the coefficient of{{math| '''h'''&lt;sup&gt;''i''&lt;/sup&gt;,}} and indicates that{{math| ''q''&amp;#770;&lt;sub&gt;''i''&lt;/sub&gt;}} is the coefficient of a unit vector&amp;mdash;''not'' that{{math| ''q''&amp;#770;&lt;sub&gt;''i''&lt;/sub&gt;}} has unit magnitude.) Taking{{mvar| q&lt;sub&gt;i&lt;/sub&gt;}} as given by ({{EquationNote|83d}}) and applying ({{EquationNote|110}}) and ({{EquationNote|113}}), we get
{{NumBlk|:|{{big|&lt;math&gt;
    q_i =~\! h_i \hat{q_i} \,,
  \qquad&lt;/math&gt;}}|{{EquationRef|114}}}}
whence ({{EquationNote|106}}) gives
{{NumBlk|:|{{big|&lt;math&gt;
    q^i =~\! \hat{q_i} \big/ h_i \,.
  \qquad&lt;/math&gt;}}|{{EquationRef|115}}}}

Substituting ({{EquationNote|110}}) into ({{EquationNote|85}}), we find that the components of{{math| ''d'''''r'''}} with respect to{{math| '''h&amp;#770;'''&lt;sub&gt;''i''&lt;/sub&gt;}}&amp;#8202; are not simply{{math| ''du&lt;sup&gt;i&lt;/sup&gt;''}}, but{{math| ''h&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;du&lt;sup&gt;i&lt;/sup&gt;''}}, with no sum. That there is no sum here is another reason not to write the coordinates with superscripts.

We can now re-express dot- and cross-products w.r.t. the orthonormal basis{{math| ('''h&amp;#770;'''&lt;sub&gt;''i''&lt;/sub&gt;)}}. If we apply ({{EquationNote|114}}) and ({{EquationNote|115}}) in ({{EquationNote|88a}}) or ({{EquationNote|88b}}), the scale factors cancel and we are left with
{{NumBlk|:|{{big|&lt;math&gt;\mathbf{v} \cdot \mathbf{q}
    =~\! \hat{v_i} \hat{q_i} \,,
  &lt;/math&gt;}}|{{EquationRef|116}}}}
as if the coordinates were Cartesian. And if we apply ({{EquationNote|108}}), ({{EquationNote|115}}), and ({{EquationNote|111}}) in ({{EquationNote|91a}}), the product of the scale factors cancels and we are left with
:{{big|&lt;math&gt;\mathbf{v} \!\times\! \mathbf{q} =~\!
    \varsigma~\!\epsilon_{ijk\,} \hat{v_i} \hat{q_j} \;\!\mathbf{\hat{h}}_k \,,
  &lt;/math&gt;}}
again as if the coordinates were Cartesian, except that the handedness symbol {{mvar|&amp;varsigma;}}&amp;#8202; gives a change of sign for left-handed coordinates. The last result is confirmed by applying ({{EquationNote|109}}), ({{EquationNote|114}}), and ({{EquationNote|110}}) in ({{EquationNote|91b}}). It can also be written
{{NumBlk|:|&lt;math&gt;\mathbf{v} \!\times\! \mathbf{q} \,=\,
    \varsigma\, \begin{vmatrix}
      \hat{v_1} &amp; \hat{q_1} &amp; \mathbf{\hat{h}}_1 \\
      \hat{v_2} &amp; \hat{q_2} &amp; \mathbf{\hat{h}}_2 \\
      \hat{v_3} &amp; \hat{q_3} &amp; \mathbf{\hat{h}}_3
    \end{vmatrix} \,.
  &lt;/math&gt;|{{EquationRef|117}}}}

We can similarly re-express the first-order differential operators. Applying ({{EquationNote|111}}) in ({{EquationNote|93o}}), ({{EquationNote|101o}}), and ({{EquationNote|100o}}) gives respectively
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla = \tfrac{1}{\,h_{\scriptstyle i}}\;\!\mathbf{\hat{h}}_i \part_i \,,
  &lt;/math&gt;}}|{{EquationRef|118}}}}
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{div}
      = \tfrac{1}{\,h_{\scriptstyle i}}\;\!\mathbf{\hat{h}}_i ~\!\!\cdot \part_i \,,
  &lt;/math&gt;}}|{{EquationRef|119}}}}
and
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}
      = \tfrac{1}{\,h_{\scriptstyle i}}\;\!\mathbf{\hat{h}}_i ~\!\!\times \part_i \,.
  &lt;/math&gt;}}|{{EquationRef|120}}}}
And applying ({{EquationNote|115}}) in ({{EquationNote|94o}}) and ({{EquationNote|102d}}) gives
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla
      = \tfrac{1}{\,h_{\scriptstyle i}} \;\!\hat{q_i} \part_i
  &lt;/math&gt;}}|{{EquationRef|121}}}}
and
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{div}\mathbf{q}
    = \tfrac{\,1\,}{J} ~\!\part_i \Big(\tfrac{J}{\,h_{\scriptstyle i}} ~\!\hat{q_i} \Big) \,.
  &lt;/math&gt;}}|{{EquationRef|122}}}}
And applying ({{EquationNote|109}}), ({{EquationNote|110}}), and ({{EquationNote|114}}) in ({{EquationNote|95c}}) gives
:{{big|&lt;math&gt;\operatorname{curl}\mathbf{q}
    = \tfrac{\,1\,}{J} ~\!\epsilon_{ijk\,}
      h_k \mathbf{\hat{h}}_k ~\!\part_i \big(h_j \hat{q_j} \big)
  &lt;/math&gt;}}
or, in determinant form,
{{NumBlk|:|&lt;math&gt;
    \operatorname{curl}\mathbf{q} \,=\, \frac{\varsigma}{h_1 h_2 h_3}\,
      \begin{vmatrix}
        h_1 \mathbf{\hat{h}}_1 &amp; \part_1 &amp; h_1 \hat{q_1} \\
        h_2 \mathbf{\hat{h}}_2 &amp; \part_2 &amp; h_2 \hat{q_2} \\
        h_3 \mathbf{\hat{h}}_3 &amp; \part_3 &amp; h_3 \hat{q_3}
      \end{vmatrix} \,.
  &lt;/math&gt;|{{EquationRef|123}}}}

For the Laplacian, applying ({{EquationNote|111}}) twice in ({{EquationNote|103L}}) gives
:{{big|&lt;math&gt;\triangle\psi =
    \tfrac{\,1\,}{J} ~\!\part_i \Big(
      \tfrac{J}{h_i h_j} (\mathbf{\hat{h}}_i {\cdot}~\! \mathbf{\hat{h}}_j)
      ~\!\part_j \psi
    \Big) \,,
  &lt;/math&gt;}}
where the parenthesized dot-product is simply{{mvar| &amp;delta;&lt;sub&gt;ij&lt;/sub&gt;&amp;#8202;}}.&amp;#8201; Selecting the non-zero terms, we are left with
{{NumBlk|:|{{big|&lt;math&gt;\triangle\psi =
    \tfrac{\,1\,}{J} ~\!\part_i \Big(
      \tfrac{J}{\,h_{\scriptstyle i}^{~2}} ~\!\part_i \psi
    \Big) \,.
  &lt;/math&gt;}}|{{EquationRef|124}}}}

Working entirely within the coordinates{{math| ''u&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;,}} we can use equations ({{EquationNote|118}}) and ({{EquationNote|121}}) to ({{EquationNote|124}}) provided that we know the scale factors in terms of{{mvar| u&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;}} and, in the case of ({{EquationNote|121}}) or ({{EquationNote|124}}) for a ''vector'' field{{math| ''&amp;psi;''&amp;#8202;,}} provided that we know the derivatives w.r.t.{{mvar| u&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;}} of the basis vectors, in terms of{{mvar| u&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;}}.&amp;#8201; And we can find the scale factors in terms of{{mvar| u&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; if we know the Cartesian coordinates{{mvar| x&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; in terms of{{mvar| u&lt;sub&gt;i&lt;/sub&gt;}}.&amp;#8201; For then the position vector can be written
:{{big|{{math|'''r''' {{=}} ''x&lt;sub&gt;j&lt;/sub&gt;''&amp;#8239;'''e'''&lt;sub&gt;''j''&lt;/sub&gt; ,}}}}
whence
:{{big|{{math|'''h'''&lt;sub&gt;''i''&lt;/sub&gt; {{=}} ''&amp;part;&lt;sub&gt;u&lt;sub&gt;i&lt;/sub&gt;&lt;/sub&gt;''&amp;#8239;'''r''' {{=}} ''&amp;part;&lt;sub&gt;u&lt;sub&gt;i&lt;/sub&gt;&lt;/sub&gt;&amp;#8201;x&lt;sub&gt;j&lt;/sub&gt;''&amp;#8201;'''e'''&lt;sub&gt;''j''&lt;/sub&gt; ,}}}}
so that the scale factors can be found from
:{{big|{{math|''h&lt;sub&gt;i&lt;/sub&gt;''&lt;sup&gt;2&lt;/sup&gt; {{=}} &amp;sum;&lt;sub&gt;&amp;#8202;''j'' &lt;/sub&gt;(''&amp;part;&lt;sub&gt;u&lt;sub&gt;i&lt;/sub&gt;&lt;/sub&gt;&amp;#8201;x&lt;sub&gt;j&lt;/sub&gt;'')&lt;sup&gt;2&lt;/sup&gt;.}}}}

In ({{EquationNote|121}}) to ({{EquationNote|123}}), the hat on{{math| ''q''&amp;#770;&lt;sub&gt;''i''&lt;/sub&gt;}} was needed because we treated the orthonormal basis as a special case, having used a hatless{{mvar| q&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; in less special cases; the hat would not have been needed if we had assumed an orthonormal basis at the outset. In more elementary introductions to curvilinear orthogonal coordinates, the basis vectors are indeed chosen as unit vectors and consequently as orthonormal vectors. Hence, if the coordinates are called &lt;math&gt;u,v,w\,&lt;/math&gt; and the respective basis vectors are called &lt;math&gt;\mathbf{e}_u , \mathbf{e}_v , \mathbf{e}_w&lt;/math&gt; (understood to be unit vectors), the components of the vector{{math| '''q'''}} w.r.t. that basis are called &lt;math&gt;q_u , q_v , q_w ~\!,\,&lt;/math&gt; with no hats. In this notation, in which sums are written out longhand without numerical indices, it is convenient also to write out the Jacobian in full, in order to exploit cancellations of scale factors. If the Jacobian appears in both a numerator inside parentheses and a denominator outside, the handedness symbol{{mvar| &amp;varsigma;}}&amp;#8202; also cancels. Thus the equations numbered ({{EquationNote|116}}) to ({{EquationNote|124}}) can be rewritten as, respectively,
{{NumBlk||&lt;math&gt;\begin{align}
    \mathbf{f} \cdot \mathbf{q} ~\!&amp;=~\!
      f_u q_u + f_v q_v + f_w q_w \\[1ex]
    \mathbf{f} ~\!\!\times\! \mathbf{q} ~\!&amp;=~\!
      \varsigma\, \begin{vmatrix}
        f_u &amp; q_u &amp; \mathbf{e}_u \\
        f_v &amp; q_v &amp; \mathbf{e}_v \\
        f_w &amp; q_w &amp; \mathbf{e}_w
      \end{vmatrix} \\[1ex]
    \nabla &amp;=
      \tfrac{1}{~\!h_{\scriptstyle u}}\;\!\mathbf{e}_u \part_u +
      \tfrac{1}{~\!h_{\scriptstyle v}}\;\!\mathbf{e}_v \part_v +
      \tfrac{1}{~\!h_{\scriptstyle w}}\;\!\mathbf{e}_w \part_w \\[.5ex]
    \operatorname{div} &amp;=
      \tfrac{1}{~\!h_{\scriptstyle u}} \mathbf{e}_u ~\!\!\cdot \part_u +
      \tfrac{1}{~\!h_{\scriptstyle v}} \mathbf{e}_v ~\!\!\cdot \part_v +
      \tfrac{1}{~\!h_{\scriptstyle w}} \mathbf{e}_w ~\!\!\cdot \part_w \\[.5ex]
    \operatorname{curl} ~\!&amp;=
      \tfrac{1}{~\!h_{\scriptstyle u}} \mathbf{e}_u ~\!\!\times \part_u +
      \tfrac{1}{~\!h_{\scriptstyle v}} \mathbf{e}_v ~\!\!\times \part_v +
      \tfrac{1}{~\!h_{\scriptstyle w}} \mathbf{e}_w ~\!\!\times \part_w\\[.5ex]
    \mathbf{q}\;\!{\cdot}\nabla &amp;=
      \tfrac{1}{~\!h_{\scriptstyle u}} \;\!q_u \part_u +
      \tfrac{1}{~\!h_{\scriptstyle v}} \;\!q_v \part_v +
      \tfrac{1}{~\!h_{\scriptstyle w}} \;\!q_w \part_w \\[.5ex]
    \operatorname{div}\mathbf{q} ~\!&amp;= \tfrac{1}{h_u h_v h_w} \Big(
        \part_u (h_v h_w q_u) + \part_v (h_w h_u q_v) + \part_w (h_u h_v q_w)
      \Big) \\[.5ex]
    \operatorname{curl}\mathbf{q} ~\!&amp;=~\! \frac{\varsigma}{h_u h_v h_w}\,
      \begin{vmatrix}
        h_u \mathbf{e}_u &amp; \part_u &amp; h_u q_u \\
        h_v \mathbf{e}_v &amp; \part_v &amp; h_v q_v \\
        h_w \mathbf{e}_w &amp; \part_w &amp; h_w q_w
      \end{vmatrix} \\[.5ex]
    \triangle\psi ~\!&amp;=
      \tfrac{1}{h_{\scriptstyle u\;\!}h_{\scriptstyle v\;\!}h_{\scriptstyle w}\!}
      \bigg\{\!
        \tfrac{\part}{\part u\!}
          \Big(\!\tfrac{h_v h_w}{h_u\;\!}\tfrac{\part\psi}{\part u}\!\Big)
        \!+~\!\! \tfrac{\part}{\part v\!}
          \Big(\!\tfrac{h_w h_u}{\;\!h_v}\tfrac{\part\psi}{\part v}\!\Big)
        \!+~\!\! \tfrac{\part}{\part w\!}
          \Big(\!\tfrac{h_u h_v}{\;\!h_w}\tfrac{\part\psi}{\part w}\!\Big)
      \!\bigg\} .
  \end{align}&lt;/math&gt;|{{EquationRef|125}}}}
Only in the cross-product and the curl does the handedness factor{{mvar| &amp;varsigma;}}&amp;#8202; make any difference. If the system is right-handed&amp;mdash;as is also often assumed at the outset&amp;mdash;this factor is replaced by{{math| 1}}.

For some readers, equation group ({{EquationNote|125}}) will announce a return to familiar territory. For the writer, it offers a convenient place to stop.

{{cob}}

== Appendix: Mathematizing Huygens' principle ==

{{cot}}

If a wavelike disturbance originating ''outside''&amp;#8202; a region{{math| ''V'',}} bounded by a surface{{math| ''S''&amp;#8202;,}} enters the region, it must do so through the surface{{mvar| S}}.&amp;#8201; Unless we believe in &quot;action at a distance&quot;, we must conclude that ''the behavior of the wave function throughout the region is fully determined by its behavior on the bounding surface''. That reasoning, being qualitative, does not tell us precisely what aspects of the behavior at the boundary determine the behavior throughout the region, or how. In this appendix, we shall answer these questions using tools of vector analysis. The aim is to express the wave function in the region{{mvar| V}} as a surface integral, over the bounding surface{{math| ''S''&amp;#8202;,}} of an integrand related to the wave function incident at a general point on that surface.

'''[[w:Huygens' principle|Huygens' principle]]''' asserts not only that the behavior of the wave function throughout the region (containing no sources) is determined by the behavior at the boundary, but also that the behavior at the boundary is equivalent to a distribution of sources over the boundary, so that the wave function throughout the region is ''as if''&amp;#8202; the original sources outside the region ('''primary sources''') were ''replaced''&amp;#8202; by sources distributed over the boundary ('''secondary sources''').{{efn|Notice that the desired secondary sources are ''not''&amp;#8202; segments of the moving wavefronts, but segments of a stationary surface influenced by the passing waves. Compare Huygens' original statement: &quot;that ''each particle of matter''&amp;#8202; in which a wave spreads, ought not to communicate its motion only to the next particle which is in the straight line drawn from the luminous point, but that it also imparts some of it necessarily to all the others which touch it and which oppose themselves to its movement. So it arises that around each particle there is made a wave of which ''that particle''&amp;#8202; is the centre&quot; ([[#huygens-1690-thompson|Huygens, 1690, tr.&amp;#8239;Thompson]], p.&amp;#8239;19; my emphasis). Huygens chooses secondary sources on the same primary wavefront at the same time for the purpose of constructing the &quot;continuation&quot; of the wavefront (the same wavefront at a later time) in the same medium (''ibid.'', pp.&amp;#8239;19,&amp;#8239;50–51), but ''not''&amp;#8202; for the purpose of constructing a wavefront reflected or refracted at an interface between two media; for the latter purpose, he chooses secondary sources at various points on the reflecting or refracting surface, although the primary wavefront reaches those points at various times (''ibid.'', pp.&amp;#8239;23–4,&amp;#8239;35–7,&amp;#8239;etc.).}} We shall find that by appropriately arranging the integrand for the wave function inside the region, we can indeed recognize the distribution of boundary sources that would generate the wave function.

{{cob}}

=== Hints ===

{{cot}}

Let {{math|''&amp;psi;''('''r''',&amp;#8239;''t'')}} be the primary wave function, and let {{math|'''r&amp;prime;'''}} be the position of the observation point ('''field point''') at a distance{{mvar| s}}&amp;#8202; from position{{math| '''r'''}}. If the surface integrand is the wave function at{{math| '''r&amp;prime;'''}} due to a secondary source-strength density, it will not only be related to the primary wave function{{mvar| &amp;psi;}} at a general point{{math| '''r'''}} on{{mvar| S}}, but will also be delayed by the propagation time from {{math|'''r'''}} to{{math| '''r&amp;prime;'''}}, and attenuated in accordance with the propagation distance{{mvar| s}}. Hence the integrand (or at least the dominant term thereof) will be proportional to
{{NumBlk|:|{{big|{{math|{{sfrac|&amp;#8239;''s''&amp;#8239;}}&amp;#8202;''&amp;psi;''('''r''', ''t&amp;#8201;&amp;minus;&amp;#8201;s''&amp;#10744;''c'') .}}}}|{{EquationRef|126}}}}
But the necessary operations may cause{{mvar| &amp;psi;}} to be replaced by, e.g., one of its derivatives or a linear combination of its derivatives; such a replacement would be &quot;proportional&quot; to{{mvar| &amp;psi;}} in the requisite sense.

Of course we would like our distribution of secondary sources to be valid for an arbitrarily shaped boundary{{mvar| S}}. This preference will be easier to satisfy if the distribution of secondary sources, by itself, produces a zero wave function outside{{mvar| V}}&amp;#8202;&amp;mdash;in other words, ''no backward secondary waves''&amp;#8202;&amp;mdash;because in that case, even if {{mvar|S}}&amp;#8202; is concave outward, the wave function inside{{mvar| V}}&amp;#8239; will not be complicated by &quot;backward&quot; waves generated at one point on{{mvar| ''S''}}&amp;#8202; and entering{{mvar| V}}&amp;#8202; through another point on{{mvar| S}}.&amp;#8201; Accordingly, we would like our surface integral to be equal to the ''volume''&amp;#8202; integral over{{mvar| V}}&amp;#8239; of
{{NumBlk|:|{{big|{{math|''&amp;psi;''('''r''',&amp;#8239;''t'')&amp;#8239;''&amp;delta;''('''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;''') ,}}}}|{{EquationRef|127}}}}
because that volume integral will be {{math| ''&amp;psi;''('''r&amp;prime;''',&amp;#8239;''t'')}}&amp;#8202; if {{math|'''r&amp;prime;'''}} is inside{{mvar| V}} (where the secondary waves are forward), but zero if it is outside (where any secondary waves are backward).

Relating the volume integral of ({{EquationNote|127}}) to the surface integral of ({{EquationNote|126}}) would seem to require a surface-to-volume '''integral identity''' involving two different fields. Some promising identities are available; but, as we shall see, they tend to treat the two fields symmetrically, and they get simpler if the two fields have more properties in common. We might therefore seek fields with more in common than ({{EquationNote|126}}) and ({{EquationNote|127}}). In the first factor in ({{EquationNote|127}}),&amp;#8201; {{mvar|t}} can be replaced by&amp;#8202; {{math|''t&amp;#8202;&amp;minus;&amp;#8202;s''&amp;#10744;''c''}}&amp;#8202; because the second factor is zero for non-zero{{mvar| s}}. Thus the second factor in ({{EquationNote|127}}), by selecting the time, makes the primary wave function{{math| ''&amp;psi;''('''r''',&amp;#8239;''t'')}} equivalent to the second factor in ({{EquationNote|126}}). That primary wave function is of course a solution of the wave equation in{{mvar| V}}. So, if the factor
:{{big|{{math|''&amp;delta;''('''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;''')}}}}
in ({{EquationNote|127}}) can be replaced by solution of the wave equation with an equivalent &quot;selecting&quot; property, and especially if that solution includes the factor {{math|1&amp;#10744;''s''}} in ({{EquationNote|126}}), perhaps we can pick that solution and the primary wave function as the two fields to substitute into the integral identity. The &quot;solution&quot; that suggests itself is
{{NumBlk|:|{{big|{{math|{{sfrac|&amp;#8239;''s''&amp;#8239;}}&amp;#8202;''&amp;delta;''(''t&amp;#8201;+&amp;#8201;s''&amp;#10744;''c'') ,}}}}|{{EquationRef|128}}}}
where the defining properties of {{math|''&amp;delta;''(''t'')}} are that its integral over all ''time'' is zero and, ideally, that it is zero except at {{math|''t''&amp;#8239;{{=}}&amp;#8202;0}}; but, to ensure that {{math|''&amp;delta;''(''t'')}} is sufficiently differentiable for our purposes, we shall allow it to be a smooth function which is zero except within a negligibly short interval around{{math| ''t''&amp;#8239;{{=}}&amp;#8202;0}}. Solution ({{EquationNote|128}}) describes an ''incoming'' spherical wave converging on{{math| '''r&amp;prime;'''}} [recall the discussion of ({{EquationNote|54a}}) above], which is appropriate because, for an observer at{{math| '''r&amp;prime;'''}}, the secondary waves are incoming; to put it more precisely, the delta function selects the time{{math| ''t&amp;#8239;{{=}}&amp;#8202;&amp;minus;s''&amp;#10744;''c''}}, which is the time of emission of the secondary waves that affect the wave function at{{math| '''r&amp;prime;'''}} at{{math| ''t''&amp;#8239;{{=}}&amp;#8202;0}} (which is a general time, because the origin of{{mvar| t}} is arbitrary). Moreover, as{{math| ''t''&amp;rightarrow;&amp;#8202;0&lt;sup&gt;&amp;minus;&lt;/sup&gt;}}, the temporal delta function in ({{EquationNote|128}}) looks like the spatial delta function in ({{EquationNote|127}}).

So let us tentatively pick the primary wave function {{math| ''&amp;psi;''('''r''',&amp;#8239;''t'')}} and the auxiliary wave function ({{EquationNote|128}}) as the two fields to be related by the integral identity&amp;mdash;which we must now choose.

{{cob}}

=== Green's identities ===

{{cot}}

If&lt;math&gt;~u&lt;/math&gt; and&lt;math&gt;~v&lt;/math&gt; are scalar fields, then by identity ({{EquationNote|71d}}),
:&lt;math&gt;\mathrm{div}(u~\!\nabla v)
    \equiv u~\!\triangle v + \nabla u \cdot~\!\! \nabla v \,.
  &lt;/math&gt;
Integrating both sides over a volume{{math| ''V''}} enclosed by a surface{{math| ''S''&amp;#8202;,}} and applying the divergence theorem on the left, we get
:&lt;math&gt;\iint_S u~\!\nabla v \cdot \mathbf{\hat{n}}~\!dS \,\equiv
    \iiint_V \big(u~\!\triangle v + \nabla u \cdot~\!\! \nabla v \big)~\!dV \,,
  &lt;/math&gt;
where &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is the unit normal to{{mvar| S}}&amp;#8202; pointing out of{{mvar| V}}. This integral equation is called '''[[w:George Green (mathematician)|Green]]'s first identity'''. Switching the roles of&lt;math&gt;~u&lt;/math&gt; and&lt;math&gt;~v&lt;/math&gt; yields a second integral equation, which can be subtracted from the first to obtain
:&lt;math&gt;
    \iint_S \!\big(u~\!\nabla v - v~\!\nabla u \big) \cdot \mathbf{\hat{n}}~\!dS
    \,\equiv \iiint_V \!\big(u~\!\triangle v - v~\!\triangle u \big)~\!dV \,;
  &lt;/math&gt;
this is '''Green's second identity'''. If {{mvar|n}} is the normal distance from{{mvar| S}} (positive outside{{math| ''V'',}} negative inside), then, by relation ({{EquationNote|9g}}) between the gradient and the directional derivative, we can rewrite Green's second identity in the alternative form
{{NumBlk|:|&lt;math&gt;
    \iint_S \!\big(u~\!\part_n v - v~\!\part_n u\big)~\!dS
    \,\equiv \iiint_V \!\big(u~\!\triangle v - v~\!\triangle u\big)~\!dV \,,
  &lt;/math&gt;|{{EquationRef|129}}}}
which remains meaningful if one of the two operands is a ''generic'' field. And indeed, by the linearity of the various operators, the identity remains valid in that case (which is not always pointed out).

{{cob}}

=== Kirchhoff's integral theorem ===

{{cot}}

Now, as foreshadowed above,&lt;ref&gt;The following demonstration of the Kirchhoff integral theorem is indebted to Stratton ([[#stratton-41|1941]], pp.&amp;#8239;424–8), especially as regards the choice of the &quot;auxiliary&quot; wave function &lt;math&gt;v&lt;/math&gt; (my nomenclature) and the insight that the time origin is arbitrary (p.&amp;#8239;427). However, Stratton's treatment does not consider the case with {{math|'''r&amp;prime;'''}} outside{{mvar| V}}, handles the &quot;inside{{mvar|&amp;#8201;V&amp;#8239;}}&quot; case differently, and takes a less heuristic approach, without introductory &quot;hints&quot;.&lt;/ref&gt; let us see what happens if we put
{{NumBlk|:|&lt;math&gt;u = \psi(\mathbf{r},t)&lt;/math&gt;|{{EquationRef|130u}}}}
and
{{NumBlk|:|&lt;math&gt;v = \tfrac{\,1\,}{s}~\!\delta\big(t+s/c\big)&lt;/math&gt;|{{EquationRef|130v}}}}
in ({{EquationNote|129}}). As &lt;math&gt;u&lt;/math&gt; satisfies the wave equation in{{mvar| V}}, we have
{{NumBlk|:|&lt;math&gt;\triangle u
  = \tfrac{1}{c^2} \ddot{u} \,.&lt;/math&gt;|{{EquationRef|131u}}}}
With &lt;math&gt;v&lt;/math&gt;&amp;#8201; we need to be more careful, because &lt;math&gt;v&lt;/math&gt; is undefined at{{math| '''r&amp;prime;'''}}, which may be inside{{mvar| V}}.  By rule ({{EquationNote|54}}), the D'Alembertian of &lt;math&gt;v&lt;/math&gt; (with the {{math|&amp;#9744;}} operator written out in full) is
:&lt;math&gt;\triangle v - \tfrac{1}{c^2} \ddot{v}
  = -4\pi \delta(t)\,\delta(\mathbf{r}{-}\mathbf{r}') \,,&lt;/math&gt;
whence
{{NumBlk|:|&lt;math&gt;\triangle v = \tfrac{1}{c^2} \ddot{v}
  - 4\pi \delta(t)\,\delta(\mathbf{r}{-}\mathbf{r}') \,.
  &lt;/math&gt;|{{EquationRef|131v}}}}
Substituting ({{EquationNote|131u}}) and ({{EquationNote|131v}}) into ({{EquationNote|129}}) gives
:&lt;math&gt;\begin{align}
  &amp;\iint_S \big(u~\!\part_n v - v~\!\part_n u\big)~\!dS \\
  &amp;= \tfrac{1}{c^2}\!\iiint_V (u\ddot{v} - v\ddot{u})~\!dV
     - \!\iiint_V\!4\pi u~\!\delta(t)~\!\delta(\mathbf{r}{-}\mathbf{r}')~\!dV \\
  &amp;= \tfrac{1}{c^2}\!
     \iiint_V \tfrac{\part}{\part t} \big(u\dot{v} - v\dot{u}\big) ~\!dV
     - \!\iiint_V \!4\pi\psi(\mathbf{r},t)~\!\delta(t)
                  ~\!\delta(\mathbf{r}{-}\mathbf{r}') ~\!dV \,,
  \end{align}&lt;/math&gt;
using ({{EquationNote|130u}}) in the last term. In that term we may now set{{math| '''r'''}} to{{math| '''r&amp;prime;'''}} [because {{math|''&amp;delta;''('''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;''')}} is zero elsewhere] and then take the {{math|'''r'''}}-independent factor outside the volume integral, obtaining
:&lt;math&gt;\begin{align}
  \iint_S &amp;\big(u~\!\part_n v - v~\!\part_n u\big)~\!dS \\
  &amp;= \tfrac{1}{c^2}\!
     \iiint_V \tfrac{\part}{\part t} \big(u\dot{v} - v\dot{u}\big) ~\!dV
     - \,4\pi\psi(\mathbf{r}',t)~\!\delta(t)
       ~\!\operatorname{if}(\mathbf{r}'{\in}~\!V) \,,
  \end{align}&lt;/math&gt;
where {{math|if()}} is an ad-hoc function taking the value{{math|&amp;#8201;1}} if its argument is true ({{math|'''r&amp;prime;''' }}is in{{mvar| V&amp;#8202;}}), and{{math| 0}}&amp;#8202; if its argument is false. Then, to eliminate{{math| ''&amp;delta;''(''t'')}}, we integrate w.r.t.{{mvar| t}}&amp;#8202; over all time, obtaining
{{NumBlk|:|&lt;math&gt;\begin{align}
    \iint\limits_{S\;} &amp;\int_{-\infty}^{\infty}
      \!\!\big(u~\!\part_n v - v~\!\part_n u\big)
    dt\;dS \\
    &amp;= \tfrac{1}{c^2}\!\iiint_V\!(u\dot{v}-v\dot{u})\Big|_{-\infty}^{\infty} dV
       - \,4\pi\psi(\mathbf{r}',0)~\!\operatorname{if}(\mathbf{r}'{\in}~\!V)\,.
  \end{align}&lt;/math&gt;|{{EquationRef|132}}}}
In the remaining volume integral, substituting for &lt;math&gt;v&lt;/math&gt; from ({{EquationNote|130v}}), we have
{{NumBlk|:|&lt;math&gt;\begin{align}(u\dot{v}-v\dot{u})\Big|_{-\infty}^{\infty}
    &amp;= \Big(\tfrac{\,u\,}{s}~\!\delta'\!(t+s/c)
            - \tfrac{\,\dot{u}\,}{s}~\!\delta(t+s/c)\Big)
       \bigg|_{t\to-\infty}^{t\to\infty} \\
    &amp;= ~\!0
  \end{align}&lt;/math&gt;|{{EquationRef|133}}}}
because the expression in the big parentheses is zero except where {{math|''t&amp;#8201;&amp;#8776;&amp;#8201;&amp;minus;s''&amp;#10744;''c''}}, and{{mvar| s}} is finite in{{mvar| V}}.  So the volume integral in ({{EquationNote|132}}) vanishes, and what remains is
{{NumBlk|:|&lt;math&gt;
    \iint_{\!S} \!\textstyle\Big\{\!
      \int_{-\infty}^{\infty} \!u~\!\part_n v \,dt
      - \!\int_{-\infty}^{\infty} \!v~\!\part_n u \,dt
    \Big\} ~\!dS
    = - 4\pi\psi(\mathbf{r}'\!,0) ~\!\operatorname{if}(\mathbf{r}'{\in}~\!V)
  &lt;/math&gt;|{{EquationRef|134}}}}
&amp;mdash;which is what we wanted: a surface integral over{{mvar| S}}, equal (up to a scale factor) to the primary wave function at{{math| '''r&amp;prime;'''}} if {{math|'''r&amp;prime;'''}} is inside{{mvar| V}}, but zero if it is outside.

It remains to put the surface integral into a more convenient form, by substituting from ({{EquationNote|130u}}) and ({{EquationNote|130v}}) and simplifying. The second inner time-integral is
:&lt;math&gt;\begin{align}
    \int_{-\infty}^{\infty} \!v~\!\part_n u \,dt
    &amp;= \!\int_{-\infty}^{\infty}\!
         \tfrac{\,1\,}{s}~\!\delta\big(t+s/c\big)~\!\part_n\psi(\mathbf{r},t)
       \,dt \\[.5ex]
    &amp;= \!\int_{-\infty}^{\infty}\!
         \tfrac{\,1\,}{s}~\!\delta\big(t+s/c\big)~\!\part_n\psi(\mathbf{r},-s/c)
       \,dt \\
    &amp;= \tfrac{\,1\,}{s}~\!\part_n\psi(\mathbf{r},-s/c)\!\int_{-\infty}^{\infty}
         \!\delta\big(t+s/c\big)
       \,dt \,,
  \end{align}&lt;/math&gt;
i.e.
{{NumBlk|:|&lt;math&gt;\textstyle
    \int_{-\infty}^{\infty} \!v~\!\part_n u \,dt
    = \frac{\,1\,}{s}~\!\frac{\part\psi}{\part n}\big(\mathbf{r},-s/c\big) \,,
  &lt;/math&gt;|{{EquationRef|135}}}}
where the differentiation w.r.t.{{mvar| n}} does ''not'' account for the variation of{{mvar| s}} with{{mvar| n}}, because the delta function selects{{mvar| t}} (and hence{{mvar| s}}) ''after'' the spatial differentiation. For the other time-integral in ({{EquationNote|134}}), however, it's the other way around: we differentiate a function of{{mvar| s}}, treating {{mvar|s}} as a function of{{mvar| n}} (and of two other coordinates which are also parameters of the surface{{mvar| S}}), using the chain rule and the product rule:
:&lt;math&gt;\begin{align}
    \int_{-\infty}^{\infty} \!u~\!\part_n v \,dt
    &amp;= \!\int_{-\infty}^{\infty}\!
         \psi(\mathbf{r},t)\,
         \part_n\!\Big(\!\tfrac{\,1\,}{s}~\!\delta(t+s/c)\Big)
       \,dt \\[.5ex]
    &amp;= \!\int_{-\infty}^{\infty}\!
         \psi(\mathbf{r},t)\,
         \part_s\!\Big(\!\tfrac{\,1\,}{s}~\!\delta(t+s/c)\Big)~\!
         \tfrac{\part s}{\part n}
       \,dt \\[.5ex]
    &amp;= \!\int_{-\infty}^{\infty}\!
         \psi(\mathbf{r},t)
         \Big(\tfrac{1}{cs}~\!\delta'\!(t+s/c)
              -\tfrac{1}{s^2}~\!\delta(t+s/c)\Big)
         \tfrac{\part s}{\part n}
       \,dt \\[.5ex]
    &amp;= \!\int_{-\infty}^{\infty}\!
         \tfrac{1}{cs}~\!\tfrac{\part s}{\part n}~\!\psi(\mathbf{r},t)
         ~\delta'\!(t+s/c)
       \,dt \\[.5ex]
    &amp;~~~~~- \int_{-\infty}^{\infty}\!
              \tfrac{1}{s^2}~\!\tfrac{\part s}{\part n}~\!\psi(\mathbf{r},t)
              \,\delta(t+s/c)
            \,dt \,.
  \end{align}&lt;/math&gt;
Expanding the first integral by parts, and processing the delta function in the second integral in the usual manner, we get
:&lt;math&gt;\begin{align}
    \int_{-\infty}^{\infty} \!u~\!\part_n v \,dt =\;
    &amp;\Big(\tfrac{1}{cs}~\!\tfrac{\part s}{\part n}~\!\psi(\mathbf{r},t)
         ~\delta(t+s/c)\Big)\bigg|_{-\infty}^{\infty} \\
    &amp; - \!\int_{-\infty}^{\infty}\!
          \tfrac{1}{cs}~\!\tfrac{\part s}{\part n}~\!\dot{\psi}(\mathbf{r},t)
          ~\delta(t+s/c)
        \,dt \\[.5ex]
    &amp; - \tfrac{1}{s^2}~\!\tfrac{\part s}{\part n}~\!\psi(\mathbf{r},-s/c) \,,
  \end{align}&lt;/math&gt;
in which we can now process the remaining delta functions to obtain
:&lt;math&gt;\textstyle\int_{-\infty}^{\infty} \!u~\!\part_n v \,dt \,=\,
    0 - \tfrac{1}{cs}~\!\tfrac{\part s}{\part n}~\!\dot{\psi}(\mathbf{r},-s/c)
      - \tfrac{1}{s^2}~\!\tfrac{\part s}{\part n}~\!\psi(\mathbf{r},-s/c) \,.
  &lt;/math&gt;
Substituting this and ({{EquationNote|135}}) into ({{EquationNote|134}}), renaming the (arbitrary) time origin as time{{mvar| t}}, and multiplying through by{{math| &amp;minus;1}}, we get the desired result:
{{NumBlk|:|&lt;math&gt;\begin{align}
    \iint_{S} \!\Big\{\!
    &amp; \tfrac{1}{cs} \tfrac{\part s}{\part n}
      ~\!\dot{\psi}\big(\mathbf{r},t{-}\tfrac{s}{c}\big)
      + \tfrac{1}{s^2} \tfrac{\part s}{\part n}
        ~\!\psi\big(\mathbf{r},t{-}\tfrac{s}{c}\big)
      + \tfrac{\,1\,}{s}~\!
        \tfrac{\part\psi}{\part n}\big(\mathbf{r},t{-}\tfrac{s}{c}\big)
    \Big\} ~\!dS \\
    &amp;=\, 4\pi\psi(\mathbf{r}',t) ~\!\operatorname{if}(\mathbf{r}'{\in}~\!V)\,.
  \end{align}&lt;/math&gt;|{{EquationRef|136}}}}
This is more usually written
:&lt;math&gt;
    \iint_{S} \!\Big\{\!
      \tfrac{1}{cs} \tfrac{\part s}{\part n} ~\!\big[\dot{\psi}\big]
      + \tfrac{1}{s^2} \tfrac{\part s}{\part n} ~\![\psi]
      + \tfrac{\,1\,}{s} \Big[\tfrac{\part\psi}{\part n}\Big]
    \Big\} ~\!dS
    \,=\, 4\pi\psi(\mathbf{r}',t) ~\!\operatorname{if}(\mathbf{r}'{\in}~\!V)\,,
  &lt;/math&gt;
where the square brackets indicate that the contents are to be ''delayed'' (or, in older literature, &quot;retarded&quot;) by the propagation time from {{math|'''r'''}} to{{math| '''r&amp;prime;'''}}&amp;mdash;that is, delayed by{{math| ''s''&amp;#10744;''c''}}&amp;#8202; relative to the default arguments{{math| ('''r''',&amp;#8239;''t'')}}.

It is common to write &lt;math&gt;-\tfrac{\part}{\part n}\big(\!\tfrac{\,1\,}{s}\!\big)&lt;/math&gt;&amp;#8202; instead of&amp;#8201; &lt;math&gt;\tfrac{1}{s^2}\tfrac{\part s}{\part n}&lt;/math&gt; (reversing the chain rule), so that the last result becomes
{{NumBlk|:|&lt;math&gt;
    \iint_{S} \!\Big\{\!
      \tfrac{1}{cs} \tfrac{\part s}{\part n} ~\!\big[\dot{\psi}\big]
      - [\psi]~\!\tfrac{\part}{\part n}\big(\!\tfrac{\,1\,}{s}\!\big)
      + \tfrac{\,1\,}{s} \Big[\tfrac{\part\psi}{\part n}\Big]
    \Big\} ~\!dS
    \,=\, 4\pi\psi(\mathbf{r}',t) ~\!\operatorname{if}(\mathbf{r}'{\in}~\!V)\,.
  &lt;/math&gt;|{{EquationRef|137}}}}

Although we derived ({{EquationNote|137}}) by supposing that {{mvar|V &amp;#8202;}}is a ''finite''&amp;#8202; region, we can extend the result to an infinite region by adding another sheet to the bounding surface{{mvar| S}}&amp;#8202; in such a way that (i) the region becomes finite, but (ii) the additional sheet makes no contribution to the surface integral. The simplest way to do this is to suppose that the additional sheet is at such a large distance that the disturbance has not reached it yet! Alternatively, we can consider how the wave function decays with distance.&lt;ref&gt;[[#baker-copson-39|Baker &amp;amp; Copson, 1939]], pp.&amp;#8239;37–8.&lt;/ref&gt; By such methods we can apply ({{EquationNote|137}}) not only to the region inside a closed surface, but also (e.g.) to the region outside a closed surface, or the region on one side of an infinite open surface.

Although we derived ({{EquationNote|137}}) by supposing, as usual in this paper, that&amp;#8202; &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; points out of{{mvar| V}}&amp;#8239; and that {{mvar|n &amp;#8202;}}is measured out of{{mvar| V}}, this has the arguably counterintuitive implication that&amp;#8202; &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is typically against the direction of propagation&amp;mdash;''directly'' against it in the simplest case, in which {{mvar|V}}&amp;#8202; is the exterior of a sphere with a monopole source at its center.

So, in the following formal statement of our result, let us drop the symbol {{mvar|V}}&amp;#8202; and define {{mvar|n}}&amp;#8202; as being measured out of the region containing the sources, and consequently ''into'' the region that satisfies the homogeneous wave equation, ''changing the signs''&amp;#8202; on the left side of ({{EquationNote|137}}).

'''[[w:Gustav Kirchhoff|Kirchhoff]]'s integral theorem''':  If
* the wave function {{mvar|&amp;psi;}}&amp;#8202; satisfies the wave equation (with speed{{mvar| c}}) in a region{{mvar| R}}&amp;#8202; bounded by a surface{{mvar| S}}&amp;#8202; (with all sources consequently on the other side of{{mvar| S&amp;#8202;}}), and
* {{mvar|s}}&amp;#8202; is the distance of the general point at position{{math| '''r'''}}&amp;#8202; from the observation point at position{{math| '''r&amp;prime;''',&amp;#8202;}} and
* quantities in square brackets are to be delayed by{{math| ''s''&amp;#10744;''c''&amp;#8202;,}} and
* {{mvar|n}}&amp;#8202; is the normal coordinate measured from the general point on{{mvar| S}}&amp;#8202; ''into''{{mvar| R}}&amp;#8202; [contrary to the usual direction for a named region, and contrary to the convention we have used above!],
then the expression
{{NumBlk|:|&lt;math&gt;
    \tfrac{1}{4\pi} \!\iint_{S} \!\Big\{\!
      [\psi]~\!\tfrac{\part}{\part n}\big(\!\tfrac{\,1\,}{s}\!\big)
      - \tfrac{1}{cs} \tfrac{\part s}{\part n} ~\!\big[\dot{\psi}\big]
      - \tfrac{\,1\,}{s} \Big[\tfrac{\part\psi}{\part n}\Big]
    \Big\} ~\!dS
  &lt;/math&gt;|{{EquationRef|138}}}}
is equal to the wave function at{{math| '''r&amp;prime;'''}}&amp;#8202; if{{math| '''r&amp;prime;'''}}&amp;#8202; is inside{{math| ''R''&amp;#8202;,}} but zero if it is outside.&lt;ref&gt;[[#born-wolf-02|Born &amp;amp; Wolf, 2002]], pp.&amp;#8239;420–21, eq.&amp;#8239;(13).&amp;#8201; ''Cf''. Baker &amp;amp; Copson ([[#baker-copson-39|1939]], p.&amp;#8239;37) and Miller ([[#miller-91|1991]], eq.&amp;#8239;2), who use {{mvar|r}}&amp;#8202; instead of{{mvar| s}}&amp;#8202; (among other notational differences). Baker &amp;amp; Copson, in their last equation on p.&amp;#8239;40, give the opposite sign because on this occasion they measure the normal coordinate&lt;math&gt;~\nu&lt;/math&gt; ''out'' of the region.&lt;/ref&gt;

The above derivation does not assume sinusoidal time-dependence at any stage. An alternative approach&lt;ref&gt;E.g., [[#baker-copson-39|Baker &amp;amp; Copson, 1939]], pp.&amp;#8239;36–7; [[#born-wolf-02|Born &amp;amp; Wolf, 2002]], pp.&amp;#8239;420–21.&lt;/ref&gt; is to derive the special case for sinusoidal time-dependence (due to [[w:Hermann von Helmholtz|Helmholtz]]) from Green's identities, and then generalize the time-dependence; this method has the advantage of being more readily applicable to ''dispersive''&amp;#8202; media (in which {{mvar|c &amp;#8202;}}is frequency-dependent), but the disadvantages of depending on complex numbers and on the premise that a general function of time can be expressed as a sum of sinusoids. Helmholtz's integrand is a sinusoidal version of our expression ({{EquationNote|139}}) below. That expression, and thence the Kirchhoff integral, can be obtained in a far more elementary manner, albeit with some loss of rigor, by ''assuming'' (instead of justifying) the form of the wave function due to a monopole source. From this, together with considerations of causality and superposition, we can work out the required distribution of secondary sources and then expresses the wave function as a surface integral.&lt;ref&gt;[[#putland-22|Putland, 2022&amp;ndash;]].&lt;/ref&gt; In the present paper, however, we argue in the other direction: from the integral to the secondary sources.

{{cob}}

=== Monopole and dipole secondary sources ===

{{cot}}

If the dependence on{{math| '''r'''}} is taken as implicit, the integrand inside the braces in ({{EquationNote|138}}) can be written out as
:&lt;math&gt;\begin{align}
    \psi&amp;\big(t\!-\!s/c\big)\,\part_n \tfrac{\,1\,}{s}
    - \tfrac{1}{cs}~\!\part_n s \,\psi'\big(t\!-\!s/c\big)
    - \tfrac{\,1\,}{s}~\!\tfrac{\part\psi}{\part n}\big(t\!-\!s/c\big) \\
    &amp;= \psi\big(t\!-\!s/c\big)\,\part_n \tfrac{\,1\,}{s}
       + \tfrac{\,1\,}{s}~\!\psi'\big(t\!-\!s/c\big)~\!
         \big({-}1/c\big)~\!\part_n s
       - \tfrac{\,1\,}{s}~\!\tfrac{\part\psi}{\part n}\big(t\!-\!s/c\big) \\
    &amp;= \psi\big(t\!-\!s/c\big)\,\part_n \tfrac{\,1\,}{s}
       + \tfrac{\,1\,}{s}~\!\part_n \psi\big(t\!-\!s/c\big)
       - \tfrac{\,1\,}{s}~\!\tfrac{\part\psi}{\part n}\big(t\!-\!s/c\big)
  \end{align}&lt;/math&gt;
or, recognizing the first two terms as the derivative of a product,
{{NumBlk|:|&lt;math&gt;
    \tfrac{\part}{\part n} \Big(\tfrac{\,1\,}{s}~\!\psi\big(t\!-\!s/c\big)\Big)
    - \tfrac{\,1\,}{s}~\!\tfrac{\part\psi}{\part n}\big(t\!-\!s/c\big) \,,
  &lt;/math&gt;|{{EquationRef|139}}}}
where {{mvar|{{sfrac|&amp;part;|&amp;part;n}}}} in the first term accounts for the variation of {{mvar|s}}&amp;#8202; through{{mvar| n}}, but {{mvar|{{sfrac|&amp;part;&amp;psi;|&amp;part;n}}}} in the second term does not [see remarks after ({{EquationNote|135}}) above]. Let{{mvar| h}} be a ''small''&amp;#8202; change in{{math| ''n''&amp;#8202;,}} from&amp;#8201; {{mvar|n&amp;#8201;{{=}}&amp;#8201;&amp;minus;h}}&amp;#8201; to&amp;#8201; {{math|''n''&amp;#8201;{{=}}&amp;#8201;0}}. Treating{{mvar| h}} simply as a constant, the integrand can be written
{{NumBlk|:|&lt;math&gt;
    h~\!\tfrac{\part}{\part n} \bigg(\frac{\,1\,}{s}~\!\frac{\psi\big(t\!-\!s/c\big)}{h}\bigg)
    - \tfrac{\,1\,}{s}~\!\tfrac{\part\psi}{\part n}\big(t\!-\!s/c\big) \,,
  &lt;/math&gt;|{{EquationRef|140}}}}
where the second term (including the minus sign) is recognizable as the contribution to the wave function from a monopole source with strength{{mvar| &amp;minus;{{sfrac|&amp;part;&amp;psi;|&amp;part;n}}&amp;#8202;}}.&lt;ref&gt;''Reminder&amp;#8202;:''&amp;#8201; There are rival definitions of the &quot;strength&quot; of a monopole source; see the text and footnote under equation ({{EquationNote|54}}) above.&lt;/ref&gt; Similarly, in the first term, the expression in the big parentheses is the contribution from a monopole source with strength{{math| ''&amp;psi;''&amp;#10744;''h''&amp;#8202;}}; but the operator {{mvar|h{{sfrac|&amp;part;|&amp;part;n}}}} gives the change in that contribution due to{{mvar| n}}&amp;#8202; increasing from {{mvar|&amp;minus;h}}&amp;#8202; to{{math| 0&amp;#8202;,}}&amp;#8201; i.e. the change in that contribution due to moving the said monopole from&amp;#8201; {{mvar|n&amp;#8201;{{=}}&amp;#8201;&amp;minus;h}}&amp;#8201; to&amp;#8201; {{math|''n''&amp;#8201;{{=}}&amp;#8201;0&amp;#8202;,}}&amp;#8201; i.e. the whole contribution due to the combination of a monopole with strength{{math| &amp;minus;''&amp;psi;''&amp;#10744;''h''}}&amp;#8202; at&amp;#8201; {{mvar|n&amp;#8201;{{=}}&amp;#8201;&amp;minus;h}}&amp;#8201; and a monopole with strength{{math| ''&amp;psi;''&amp;#10744;''h''}}&amp;#8202; at&amp;#8201; {{math|''n''&amp;#8201;{{=}}&amp;#8201;0}}. This combination is called a '''dipole''' (or ''doublet'')&lt;ref&gt;The term ''doublet'', which seems to be older, is used by Baker &amp;amp; Copson ([[#baker-copson-39|1939]]), Born &amp;amp; Wolf ([[#born-wolf-02|2002, p.&amp;#8239;421]]), and Larmor ([[#larmor-1904|1904]]).&lt;/ref&gt; with strength{{mvar| &amp;psi;}}&amp;#8202; in the normal ({{mvar|n}}) direction.

According to ({{EquationNote|138}}), the expression ({{EquationNote|140}}) is to be scaled by {{math|{{sfrac|4''&amp;pi;''}}}}&amp;#8202; and integrated over the surface{{mvar| S}}. Thus the secondary source distribution can be described as a monopole distribution of strength density {{math|&amp;minus;{{sfrac|4''&amp;pi;''}}{{sfrac|''&amp;part;&amp;psi;''|''&amp;part;n''}}}}&amp;#8239; plus a normal dipole distribution of strength density{{math| {{sfrac|''&amp;psi;''|4''&amp;pi;''}}&amp;#8202;,}} where &quot;strength density&quot; means strength per unit area. This description is well known.&lt;ref&gt;E.g., [[#born-wolf-02|Born &amp;amp; Wolf, 2002]], p.&amp;#8239;421.&lt;/ref&gt;

The implication is not that the specified secondary sources really exist, or even that they ''could''&amp;#8202; exist, but only that the wave function in the region{{mvar| R}}&amp;#8202; is ''as if''&amp;#8202; it had been generated by the specified secondary sources (which would also give a null wave function outside the region). We should note, however, that a monopole contribution of the form ({{EquationNote|48}}) ''can''&amp;#8202; really exist, even for a vector wave function, notwithstanding that it requires not only the magnitude but also the direction of the vector to be independent of the direction of propagation. That requirement might seem to exclude electromagnetic waves, for which the electric and magnetic fields are transverse to the direction of propagation and therefore not independent of it. But it is possible to describe such waves in terms of an electric scalar potential and a magnetic vector potential, such that the contribution to the latter from a current element has the same direction as the current element for all directions of propagation.&lt;ref&gt;[[#stratton-41|Stratton, 1941]], pp.&amp;#8239;428–30.&lt;/ref&gt;

{{cob}}

=== Spatiotemporal-dipole secondary sources ===

{{cot}}

The &quot;dipole&quot; discussed so far is a ''spatial''&amp;#8202; dipole, in which the constituent monopoles differ only in sign and by a small spatial displacement. In the Helmholtz–Kirchhoff integrand ({{EquationNote|139}}), the source per unit area (scaled by{{math| 4''&amp;pi;''}}) comprises a monopole of strength{{mvar| &amp;minus;{{sfrac|&amp;part;&amp;psi;|&amp;part;n}}}} (second term) and a ''spatial'' dipole of strength{{mvar| &amp;psi;}}&amp;#8202; in the {{mvar|n }}direction (first term). In that dipole, let the monopole with strength{{math| &amp;minus;''&amp;psi;''&amp;#10744;''h''}}&amp;#8202; at&amp;#8201; {{mvar|n&amp;#8201;{{=}}&amp;#8201;&amp;minus;h}}&amp;#8202; be called the ''inverted''&amp;#8202; monopole, and let the monopole with strength{{math| ''&amp;psi;''&amp;#10744;''h''}}&amp;#8202; at&amp;#8201; {{math|''n''&amp;#8201;{{=}}&amp;#8201;0}} be called the ''uninverted''&amp;#8202; monopole.

If there is only a '''single monopole primary source''', this combination of a monopole and a spatial dipole is exactly equivalent to a modified dipole in which the inverted monopole has a certain fixed delay, and a certain fixed attenuation, relative to the uninverted monopole.&lt;ref&gt;The derivation of this &quot;generalized spatiotemporal dipole&quot; (GSTD) was first given in ver.&amp;#8239;0.3 of [[#putland-22|Putland, 2022&amp;ndash;]] (&amp;sect;&amp;#8239;3.7). It was included in earlier versions of the present paper, but is now more conveniently available in a much smaller document ([[#putland-25|Putland, 2025]]).&lt;/ref&gt;

If, in addition, the surface {{mvar|S}}&amp;#8202; coincides with a primary wavefront, the required &quot;fixed delay&quot; is simply{{math| ''h''&amp;#10744;''c''&amp;#8202;}}, i.e. the propagation time from the uninverted monopole to the inverted one. If the primary wavefronts are plane (for a general{{mvar| S&amp;#8202;}}), the inverted monopole should be unattenuated. If {{mvar|S}}&amp;#8202; coincides with a primary wavefront ''and''&amp;#8202; is plane (a large-{{mvar|r}} approximation), the modified dipole reduces to what D.A.B.&amp;#8239;Miller called a '''spatiotemporal dipole''',&lt;ref&gt;[[#miller-91|Miller, 1991]].&lt;/ref&gt; in which the only modification of the spatial dipole is the delay{{math| ''h''&amp;#10744;''c''}}.

{{cob}}

=== Application to diffraction by an aperture ===

{{cot}}

Suppose that the primary sources are partly obstructed by an opaque baffle with an aperture in it. What is the wave function that propagates beyond the baffle? Let us choose a surface{{mvar| S}}&amp;#8202; consisting of two segments, namely {{mvar|S&lt;sub&gt;a&lt;/sub&gt; }}spanning the aperture, and {{mvar|S&lt;sub&gt;b&lt;/sub&gt; }}on the side of the baffle facing away from the sources (the dark side or quiet side of the baffle). The obvious way to proceed is to suppose that the baffle simply eliminates the secondary sources on{{mvar| S&lt;sub&gt;b&lt;/sub&gt;}} while leaving the secondary sources on{{mvar| S&lt;sub&gt;a&lt;/sub&gt;}} unchanged (as if the baffle were not there). The result, as far as the wave function in{{mvar| R}} (beyond the baffle) is concerned, is simply that the integral is taken over {{mvar|S&lt;sub&gt;a&lt;/sub&gt; }}only.

Integrating over the aperture alone is indeed the standard answer, but there are various other ways of explaining it. Some explanations, including the famously inconsistent one offered by Kirchhoff himself, are discussed in [[#putland-22|Putland, 2022&amp;ndash;]] (&amp;sect;&amp;#8239;2.2 and Appendices A &amp;amp; B) and the references therein.

{{cob}}

== Acknowledgment ==

This learning resource uses images from ''Wikimedia Commons''.

== Notes ==

{{cot}}

{{notelist|30em}}

{{cob}}

== Citations ==

{{cot}}

{{reflist|19em}}

{{cob}}

== References ==

{{cot}}

&lt;div style=&quot;font-size: 111%&quot;&gt;

{{refbegin|indent=yes}}

*&lt;span id=&quot;axler-95&quot;&gt;S.J. Axler, 1995, &quot;Down with Determinants!&quot;&amp;#8201; ''American Mathematical Monthly'', vol.&amp;#8239;102, no.&amp;#8239;2 (Feb.&amp;#8239;1995), pp.&amp;#8239;139–54; [https://www.jstor.org/stable/2975348 jstor.org/stable/2975348].&amp;#8201; (Author's preprint, with different pagination: [https://www.researchgate.net/publication/265273063_Down_with_Determinants researchgate.net/publication/265273063_Down_with_Determinants].)&lt;/span&gt;
*&lt;span id=&quot;axler-23-&quot;&gt;S.J. Axler, 2023–, ''Linear Algebra Done Right'', 4th Ed., Springer; [https://linear.axler.net/ linear.axler.net] (open access).&lt;/span&gt;
*&lt;span id=&quot;baker-copson-39&quot;&gt;B.B. Baker and E.T. Copson, 1939, ''The Mathematical Theory of&amp;#8202; Huygens' Principle'', Oxford; 3rd Ed.&amp;#8201;(same pagination, with addenda), New York: Chelsea, 1987, [https://archive.org/details/mathematicaltheo0000bake archive.org/details/mathematicaltheo0000bake].&lt;/span&gt;
*&lt;span id=&quot;borisenko-tarapov-68&quot;&gt;A.I. Borisenko and I.E.&amp;#8239;Tarapov (tr.&amp;#8239;&amp;amp; ed. R.A.&amp;#8239;Silverman), 1968, ''Vector and Tensor Analysis with Applications'', Prentice-Hall; reprinted New York: Dover, 1979, [https://archive.org/details/vectortensoranal0000bori archive.org/details/vectortensoranal0000bori].&lt;!-- Typo on p.180: First cross in equation before (4.93) should be &quot;=&quot;. --&gt;&lt;/span&gt;
*&lt;span id=&quot;born-wolf-02&quot;&gt;M.&amp;#8201;Born and E.&amp;#8239;Wolf, 2002, ''Principles of Optics'', 7th Ed., Cambridge, 1999 (reprinted with corrections, 2002).&lt;/span&gt;
*&lt;span id=&quot;broyden-75&quot;&gt;C.G. Broyden, 1975, ''Basic Matrices'', London: Macmillan.&lt;/span&gt;
*&lt;span id=&quot;feynman-63&quot;&gt;R.P. Feynman, R.B. Leighton, &amp;amp; M.&amp;#8239;Sands, 1963 etc., ''The Feynman Lectures on Physics'', California Institute of Technology; [http://www.feynmanlectures.caltech.edu/ feynmanlectures.caltech.edu].&lt;/span&gt;
*&lt;span id=&quot;fletcher-74&quot;&gt;N.H. Fletcher, 1974, &quot;Adiabatic assumption for wave propagation&quot;, ''American Journal of Physics'', vol.&amp;#8239;42, no.&amp;#8239;6 (June 1974), pp.&amp;#8239;487–9; [https://doi.org/10.1119/1.1987757 doi.org/10.1119/1.1987757].&lt;/span&gt;
*&lt;span id=&quot;gibbs-1881-4&quot;&gt;J.W. Gibbs, 1881–84, &quot;Elements of Vector Analysis&quot;, privately printed New Haven: Tuttle, Morehouse &amp;amp; Taylor, 1881 (&amp;sect;&amp;sect;&amp;#8239;1–101), 1884 (&amp;sect;&amp;sect;&amp;#8239;102–189, etc.), [https://archive.org/details/elementsvectora00gibb archive.org/details/elementsvectora00gibb]; published in ''The Scientific Papers of J.&amp;#8239;Willard Gibbs'' (ed. H.A.&amp;#8239;Bumstead &amp;amp; R.G.&amp;#8239;Van Name), New York: Longmans, Green, &amp;amp; Co., 1906, vol.&amp;#8239;2, [https://archive.org/details/scientificpapers02gibbuoft archive.org/details/scientificpapers02gibbuoft], pp.&amp;#8239;17–90.&lt;/span&gt;
*&lt;span id=&quot;hsu-84&quot;&gt;H.P. Hsu, 1984, ''Applied Vector Analysis'', Harcourt Brace Jovanovich; [https://archive.org/details/appliedvectorana00hsuh archive.org/details/appliedvectorana00hsuh].&lt;/span&gt;
*&lt;span id=&quot;huygens-1690-thompson&quot;&gt;C. Huygens, 1690, tr. S.P.&amp;#8239;Thompson, ''Treatise on Light'', University of Chicago Press, 1912 / [https://gutenberg.org/files/14725/14725-h/14725-h.htm gutenberg.org/files/14725/14725-h/14725-h.htm], 2005. (See also &quot;Errata in various editions of Huygens' ''Treatise on Light''&amp;#8239;&quot;, ''www.grputland.com'' or ''grputland.blogspot.com'', June 2016.)&lt;/span&gt;
*&lt;span id=&quot;katz-79&quot;&gt;V.J. Katz, 1979, &quot;The history of Stokes' theorem&quot;, ''Mathematics Magazine'', vol.&amp;#8239;52, no.&amp;#8239;3 (May 1979), pp.&amp;#8239;146–56; [https://www.jstor.org/stable/2690275 jstor.org/stable/2690275].&lt;/span&gt;
*&lt;span id=&quot;kemin-et-al-00&quot;&gt;S. Kemin, X.&amp;#8239;Zhenting, T.&amp;#8239;Jinsheng, &amp;amp; H.&amp;#8239;Xuemei, 2000, &quot;The comprehension, some problems and suggestions to symbolic vector method and some defenses for Gibbs' symbol&quot;, ''Applied Mathematics and Mechanics'' (English Ed.), vol.&amp;#8239;21, no.&amp;#8239;5 (May 2000), pp.&amp;#8239;603–6; [https://doi.org/10.1007/BF02459044 doi.org/10.1007/BF02459044].&lt;/span&gt;
*&lt;span id=&quot;kemmer-77&quot;&gt;N. Kemmer, 1977, ''Vector Analysis: A physicist's guide to the mathematics of fields in three dimensions'', Cambridge; [https://archive.org/details/isbn_0521211581 archive.org/details/isbn_0521211581].&lt;/span&gt;
*&lt;span id=&quot;kreyszig-62-&quot;&gt;E. Kreyszig, 1962 etc., ''Advanced Engineering Mathematics'', New York: Wiley;&amp;#8201; 5th Ed., 1983;&amp;#8201; 6th Ed., 1988;&amp;#8201; 9th Ed., 2006;&amp;#8201; 10th Ed., 2011.&lt;/span&gt;
*&lt;span id=&quot;larmor-1904&quot;&gt;J. Larmor, 1904, &quot;On the mathematical expression of the principle of&amp;#8202; Huygens&quot; (read 8 Jan.&amp;#8239;1903), ''Proceedings of the London Mathematical Society'', Ser.&amp;#8239;2, vol.&amp;#8239;1 (1904), pp.&amp;#8239;1–13.&lt;!-- Listed as &quot;Issue 1&quot;; only issue for that volume. --&gt;&lt;/span&gt;
*&lt;span id=&quot;miller-91&quot;&gt;D.A.B. Miller, 1991, &quot;Huygens's wave propagation principle corrected&quot;, ''Optics Letters'', vol.&amp;#8239;16, no.&amp;#8239;18 (15 Sep.&amp;#8239;1991), pp.&amp;#8239;1370–72; [http://ee.stanford.edu/~dabm/146.pdf stanford.edu/~dabm/146.pdf].&lt;/span&gt;
*&lt;span id=&quot;moon-spencer-65&quot;&gt;P.H. Moon and D.E.&amp;#8239;Spencer, 1965, ''Vectors'', Princeton, NJ: Van Nostrand.&lt;/span&gt;
*&lt;span id=&quot;panofsky-phillips-62&quot;&gt;W.K.H. Panofsky and M.&amp;#8239;Phillips, 1962, ''Classical Electricity and Magnetism'', 2nd Ed., Addison-Wesley; reprinted Mineola, NY: Dover, 2005.&lt;/span&gt;
*&lt;span id=&quot;putland-22&quot;&gt;G.R. Putland, 2022&amp;ndash;, &quot;Consistent derivation of Kirchhoff's integral theorem and diffraction formula and the Maggi-Rubinowicz transformation using high-school math&quot; (working paper), [https://doi.org/10.5281/zenodo.7205781 doi.org/10.5281/zenodo.7205781] (Creative Commons).&lt;/span&gt;
*&lt;span id=&quot;putland-25&quot;&gt;G.R. Putland, 2025, &quot;Exact formulation of Huygens' principle in terms of generalized spatiotemporal-dipole secondary sources&quot;, [https://doi.org/10.48550/arXiv.2510.20825 doi.org/10.48550/arXiv.2510.20825] (Creative Commons).&lt;/span&gt;
*&lt;span id=&quot;rocci-20&quot;&gt;A. Rocci, 2020, &quot;Back to the roots of vector and tensor calculus: Heaviside versus Gibbs&quot; (online 10 Nov.&amp;#8239;2020), ''Archive for History of Exact Sciences'', vol.&amp;#8239;75, no.&amp;#8239;4 (July 2021), pp.&amp;#8239;369–413. (Author's preprint, with different pagination: [https://arxiv.org/abs/2010.09679 arxiv.org/abs/2010.09679].)&lt;/span&gt;
*&lt;span id=stratton-41&gt;J.A. Stratton, 1941, ''Electromagnetic Theory'', New York: McGraw-Hill; [https://archive.org/details/electromagnetict0000juli archive.org/details/electromagnetict0000juli].&lt;/span&gt;
*&lt;span id=&quot;tai-94&quot;&gt;C.-T. Tai, 1994, &quot;A survey of the improper use of &amp;nabla; in vector analysis&quot; (Technical Report RL&amp;#8239;909), Dept.&amp;#8201;of Electrical Engineering &amp;amp; Computer Science, University of Michigan; [https://deepblue.lib.umich.edu/handle/2027.42/7869 hdl.handle.net/2027.42/7869].&lt;/span&gt;
*&lt;span id=&quot;tai-95&quot;&gt;C.-T. Tai, 1995, &quot;A historical study of vector analysis&quot; (Technical Report RL&amp;#8239;915), Dept.&amp;#8201;of Electrical Engineering &amp;amp; Computer Science, University of Michigan; [https://deepblue.lib.umich.edu/handle/2027.42/7868 hdl.handle.net/2027.42/7868].&lt;/span&gt;
*&lt;span id=&quot;tai-fang-91&quot;&gt;C.-T. Tai and N.&amp;#8239;Fang, 1991, &quot;A systematic treatment of vector analysis&quot;, ''{{serif|IEEE}} Transactions on Education'', vol.&amp;#8239;34, no.&amp;#8239;2 (May 1991), pp.&amp;#8239;167–74; [https://doi.org/10.1109/13.81596 doi.org/10.1109/13.81596].&lt;/span&gt;
*&lt;span id=&quot;wilson-1901&quot;&gt;E.B.&amp;#8201;Wilson, 1901, ''Vector Analysis: A text-book for the use of students of mathematics and physics'' (&quot;Founded upon the lectures of J.&amp;#8239;Willard Gibbs&amp;hellip;&quot;), New York: Charles Scribner's Sons; 12th printing, Yale University Press, 1958, [https://archive.org/details/vectoranalysiste0000gibb archive.org/details/vectoranalysiste0000gibb].&lt;/span&gt;
*&lt;span id=&quot;wrede-spiegel-10&quot;&gt;R.C.&amp;#8201;Wrede and M.R.&amp;#8239;Spiegel, 2010, ''Advanced Calculus'', 3rd Ed., New York: McGraw-Hill (Schaum's Outlines); [https://archive.org/details/schaumsoutlinesa0000wred archive.org/details/schaumsoutlinesa0000wred].&lt;/span&gt;

{{refend}}

&lt;/div&gt;

{{cob}}

== Further reading ==

{{cot}}

M.J. Crowe, &quot;A History of Vector Analysis&quot; (address at the University of Louisville, Autumn term, 2002), [https://www.researchgate.net/publication/244957729_A_History_of_Vector_Analysis researchgate.net/publication/244957729_A_History_of_Vector_Analysis] (including much discussion of quaternions).

P. Lynch, &quot;Matthew O'Brien: An inventor of vector analysis&quot;, ''Bulletin of the Irish Mathematical Society'', No.&amp;#8239;74 (Winter 2014), pp.&amp;#8239;81–8; [https://doi.org/10.33232/BIMS.0074.81.88 doi.org/10.33232/BIMS.0074.81.88].

{{cob}}

[[Category:Mathematics]]
[[Category:Calculus]]
[[Category:Vectors]]
[[Category:Vector calculus]]
[[Category:Multivariable calculus]]
[[Category:Applied mathematics]]
[[Category:Mathematical physics]]
[[Category:Waves]]
[[Category:Coordinate systems]]</text>
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&lt;!-- Categories below this line; interwikis at Wikidata --&gt;
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      <text bytes="6607" sha1="aav3zclyy04so8wp2dlktwtezqhxbvb" xml:space="preserve">{{for2|tagging wrappable segments within nowrap-scoped text|Template:Wrap|a wrapper template generator|Module:Template wrapper}}
{{Documentation subpage}}
{{Lua|Module:Params}}

This template is a doc page banner that flags a wrapper template.

== Usage ==

'''Wikitext (see {{Tl|Family name footnote}}):'''

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Family name explanation
	| except =&lt;/nowiki&gt; type -&gt; footnote // reftype -&gt; &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{reftype|efn}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;
}}&lt;/nowiki&gt;|lang=wikitext}}

'''Effect:'''

{{ {{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}|nocat=yes|Family name explanation
	| except = type -&gt; footnote // reftype -&gt; &lt;nowiki&gt;{{{reftype|efn}}}&lt;/nowiki&gt;
}}

&lt;div style=&quot;width: 100%; height: 2em;&quot;&gt;&lt;/div&gt;

'''Wikitext (see {{Tl|Bts}}):'''

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Station link
	| except =&lt;/nowiki&gt; MBTA // &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{1|}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt; // &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{2|}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt; // &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{3|}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;
	| passing-through =
}}&lt;/nowiki&gt;|lang=wikitext}}

or, equivalently,

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Station link
	| except =&lt;/nowiki&gt;
		1 -&gt; MBTA //
		2 -&gt; &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{1|}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt; //
		3 -&gt; &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{2|}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt; //
		4 -&gt; &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{3|}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;
	| passing-through =
}}&lt;/nowiki&gt;|lang=wikitext}}

'''Effect:'''

{{ {{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}|nocat=yes|Station link
	| except =
		1 -&gt; MBTA //
		2 -&gt; &lt;nowiki&gt;{{{1|}}}&lt;/nowiki&gt; //
		3 -&gt; &lt;nowiki&gt;{{{2|}}}&lt;/nowiki&gt; //
		4 -&gt; &lt;nowiki&gt;{{{3|}}}&lt;/nowiki&gt;
	| passing-through =
}}

&lt;div style=&quot;width: 100%; height: 2em;&quot;&gt;&lt;/div&gt;

'''Wikitext (see {{Tl|Don't ping}}):'''

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Please ping
	| except =&lt;/nowiki&gt; 1 -&gt; no&lt;nowiki&gt;
	| omitting =&lt;/nowiki&gt; tps // nw // big // thanks // cond&lt;nowiki&gt;
	| passing-through =
}}&lt;/nowiki&gt;|lang=wikitext}}

'''Effect:'''

{{ {{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}|nocat=yes|Please ping
	| except = 1 -&gt; no
	| omitting = tps // nw // big // thanks // cond
	| passing-through =
}}

&lt;div style=&quot;width: 100%; height: 2em;&quot;&gt;&lt;/div&gt;

'''Wikitext (see {{Tl|Sfnlinka}}):'''

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Sfnlink
	| except =&lt;/nowiki&gt; article -&gt; *&lt;nowiki&gt;
	| omitting =&lt;/nowiki&gt; rev // lang&lt;nowiki&gt;
	| passing-through =&lt;/nowiki&gt; p // page // pp // pages // loc // nb // text&lt;nowiki&gt;
}}&lt;/nowiki&gt;|lang=wikitext}}

'''Effect:'''

{{ {{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}|nocat=yes|Sfnlink
	| except = article -&gt; *
	| omitting = rev // lang
	| passing-through = p // page // pp // pages // loc // nb // text
}}

&lt;div style=&quot;width: 100%; height: 2em;&quot;&gt;&lt;/div&gt;

'''Wikitext (see {{Tl|Tanakhverse}}):'''

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Bibleverse
	| except =&lt;/nowiki&gt; 3 -&gt; &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{3|HE}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;
	| omitting =&lt;/nowiki&gt; 4 // 5&lt;nowiki&gt;
}}&lt;/nowiki&gt;|lang=wikitext}}

'''Effect:'''

{{ {{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}|nocat=yes|Bibleverse
	| except = 3 -&gt; &lt;nowiki&gt;{{{3|HE}}}&lt;/nowiki&gt;
	| omitting = 4 // 5
}}

&lt;div style=&quot;width: 100%; height: 2em;&quot;&gt;&lt;/div&gt;

'''Wikitext:'''

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Debug}}&lt;/nowiki&gt;|lang=wikitext}}

'''Effect:'''

{{ {{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}|nocat=yes|Debug}}

&lt;div style=&quot;width: 100%; height: 2em;&quot;&gt;&lt;/div&gt;

== Template data ==

&lt;templatedata&gt;{
	&quot;description&quot;: &quot;This documentation box informs that a template is a wrapper of another template.&quot;,
	&quot;params&quot;: {
		&quot;1&quot;: {
			&quot;label&quot;: &quot;Template&quot;,
			&quot;description&quot;: &quot;Name of the wrapped template&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: true,
			&quot;example&quot;: &quot;Please ping&quot;
		},
		&quot;except&quot;: {
			&quot;label&quot;: &quot;Bound parameters&quot;,
			&quot;description&quot;: &quot;The list of pre-assigned parameters; use ‘//’ to separate them and ‘-&gt;’ to assign them a value—see ‘|separator=’ and ‘|setter=’ to set other strings&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: false,
			&quot;example&quot;: &quot;1 -&gt; no&quot;
		},
		&quot;omitting&quot;: {
			&quot;label&quot;: &quot;Undefined parameters&quot;,
			&quot;description&quot;: &quot;The list of parameter names left undefined; use ‘//’ to separate them—see ‘|separator=’ to set another string&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: false,
			&quot;example&quot;: &quot;tps // nw // big // thanks // cond&quot;
		},
		&quot;passing-through&quot;: {
			&quot;label&quot;: &quot;Parameters passed through&quot;,
			&quot;description&quot;: &quot;The list of parameter names left available to use; use ‘//’ to separate them—see ‘|separator=’ to set another string; you can provide a blank parameter to signal that no parameters are left available&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: false,
			&quot;example&quot;: &quot;lorem // ipsum // foo // bar&quot;
		},
		&quot;separator&quot;: {
			&quot;label&quot;: &quot;Separator string&quot;,
			&quot;description&quot;: &quot;The string to use as separator between parameters&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: false,
			&quot;default&quot;: &quot;//&quot;,
			&quot;example&quot;: &quot;;&quot;
		},
		&quot;setter&quot;: {
			&quot;label&quot;: &quot;Setter string&quot;,
			&quot;description&quot;: &quot;The string to use as value setter&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: false,
			&quot;default&quot;: &quot;-&gt;&quot;,
			&quot;example&quot;: &quot;:&quot;
		},
		&quot;table-class&quot;: {
			&quot;label&quot;: &quot;Table class&quot;,
			&quot;description&quot;: &quot;The ‘class=’ HTML attribute assigned to the table of managed parameters&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: false,
			&quot;default&quot;: &quot;wikitable&quot;,
			&quot;example&quot;: &quot;wikitable mw-collapsible mw-collapsed&quot;
		},
		&quot;nocat&quot;: {
			&quot;label&quot;: &quot;No category&quot;,
			&quot;description&quot;: &quot;Disable automatic categorization into ‘Category:Templates calling Example’ or ‘Category:Wrapper templates’ (depending on the case)&quot;,
			&quot;type&quot;: &quot;boolean&quot;,
			&quot;required&quot;: false,
			&quot;default&quot;: &quot;no&quot;,
			&quot;example&quot;: &quot;yes&quot;
		}
	}
}&lt;/templatedata&gt;

== See also ==
* [[:Category:Wrapper templates]]
* {{ml|Template wrapper|wrap}}
* {{mfl|params|concat_and_call}}

&lt;includeonly&gt;{{sandbox other||
&lt;!-- Categories below this line --&gt;
[[Category:Documentation header templates]]
[[Category:Wrapper templates|!]]
}}&lt;/includeonly&gt;</text>
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    <revision>
      <id>2818570</id>
      <parentid>2818569</parentid>
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        <username>Grufo</username>
        <id>1192007</id>
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      <text bytes="6364" sha1="ljr3xpc5kkyqvh845pvhwlmxcb37hii" xml:space="preserve">{{Documentation subpage}}
{{Lua|Module:Params}}

This template is a doc page banner that flags a wrapper template.

== Usage ==

'''Wikitext (see {{Tl|Family name footnote}}):'''

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Family name explanation
	| except =&lt;/nowiki&gt; type -&gt; footnote // reftype -&gt; &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{reftype|efn}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;
}}&lt;/nowiki&gt;|lang=wikitext}}

'''Effect:'''

{{ {{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}|nocat=yes|Family name explanation
	| except = type -&gt; footnote // reftype -&gt; &lt;nowiki&gt;{{{reftype|efn}}}&lt;/nowiki&gt;
}}

&lt;div style=&quot;width: 100%; height: 2em;&quot;&gt;&lt;/div&gt;

'''Wikitext (see {{Tl|Bts}}):'''

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Station link
	| except =&lt;/nowiki&gt; MBTA // &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{1|}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt; // &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{2|}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt; // &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{3|}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;
	| passing-through =
}}&lt;/nowiki&gt;|lang=wikitext}}

or, equivalently,

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Station link
	| except =&lt;/nowiki&gt;
		1 -&gt; MBTA //
		2 -&gt; &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{1|}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt; //
		3 -&gt; &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{2|}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt; //
		4 -&gt; &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{3|}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;
	| passing-through =
}}&lt;/nowiki&gt;|lang=wikitext}}

'''Effect:'''

{{ {{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}|nocat=yes|Station link
	| except =
		1 -&gt; MBTA //
		2 -&gt; &lt;nowiki&gt;{{{1|}}}&lt;/nowiki&gt; //
		3 -&gt; &lt;nowiki&gt;{{{2|}}}&lt;/nowiki&gt; //
		4 -&gt; &lt;nowiki&gt;{{{3|}}}&lt;/nowiki&gt;
	| passing-through =
}}

&lt;div style=&quot;width: 100%; height: 2em;&quot;&gt;&lt;/div&gt;

'''Wikitext (see {{Tl|Don't ping}}):'''

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Please ping
	| except =&lt;/nowiki&gt; 1 -&gt; no&lt;nowiki&gt;
	| omitting =&lt;/nowiki&gt; tps // nw // big // thanks // cond&lt;nowiki&gt;
	| passing-through =
}}&lt;/nowiki&gt;|lang=wikitext}}

'''Effect:'''

{{ {{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}|nocat=yes|Please ping
	| except = 1 -&gt; no
	| omitting = tps // nw // big // thanks // cond
	| passing-through =
}}

&lt;div style=&quot;width: 100%; height: 2em;&quot;&gt;&lt;/div&gt;

'''Wikitext (see {{Tl|Sfnlinka}}):'''

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Sfnlink
	| except =&lt;/nowiki&gt; article -&gt; *&lt;nowiki&gt;
	| omitting =&lt;/nowiki&gt; rev // lang&lt;nowiki&gt;
	| passing-through =&lt;/nowiki&gt; p // page // pp // pages // loc // nb // text&lt;nowiki&gt;
}}&lt;/nowiki&gt;|lang=wikitext}}

'''Effect:'''

{{ {{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}|nocat=yes|Sfnlink
	| except = article -&gt; *
	| omitting = rev // lang
	| passing-through = p // page // pp // pages // loc // nb // text
}}

&lt;div style=&quot;width: 100%; height: 2em;&quot;&gt;&lt;/div&gt;

'''Wikitext (see {{Tl|Tanakhverse}}):'''

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Bibleverse
	| except =&lt;/nowiki&gt; 3 -&gt; &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{3|HE}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;
	| omitting =&lt;/nowiki&gt; 4 // 5&lt;nowiki&gt;
}}&lt;/nowiki&gt;|lang=wikitext}}

'''Effect:'''

{{ {{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}|nocat=yes|Bibleverse
	| except = 3 -&gt; &lt;nowiki&gt;{{{3|HE}}}&lt;/nowiki&gt;
	| omitting = 4 // 5
}}

&lt;div style=&quot;width: 100%; height: 2em;&quot;&gt;&lt;/div&gt;

'''Wikitext:'''

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Debug}}&lt;/nowiki&gt;|lang=wikitext}}

'''Effect:'''

{{ {{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}|nocat=yes|Debug}}

&lt;div style=&quot;width: 100%; height: 2em;&quot;&gt;&lt;/div&gt;

== Template data ==

&lt;templatedata&gt;{
	&quot;description&quot;: &quot;This documentation box informs that a template is a wrapper of another template.&quot;,
	&quot;params&quot;: {
		&quot;1&quot;: {
			&quot;label&quot;: &quot;Template&quot;,
			&quot;description&quot;: &quot;Name of the wrapped template&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: true,
			&quot;example&quot;: &quot;Please ping&quot;
		},
		&quot;except&quot;: {
			&quot;label&quot;: &quot;Bound parameters&quot;,
			&quot;description&quot;: &quot;The list of pre-assigned parameters; use ‘//’ to separate them and ‘-&gt;’ to assign them a value—see ‘|separator=’ and ‘|setter=’ to set other strings&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: false,
			&quot;example&quot;: &quot;1 -&gt; no&quot;
		},
		&quot;omitting&quot;: {
			&quot;label&quot;: &quot;Undefined parameters&quot;,
			&quot;description&quot;: &quot;The list of parameter names left undefined; use ‘//’ to separate them—see ‘|separator=’ to set another string&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: false,
			&quot;example&quot;: &quot;tps // nw // big // thanks // cond&quot;
		},
		&quot;passing-through&quot;: {
			&quot;label&quot;: &quot;Parameters passed through&quot;,
			&quot;description&quot;: &quot;The list of parameter names left available to use; use ‘//’ to separate them—see ‘|separator=’ to set another string; you can provide a blank parameter to signal that no parameters are left available&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: false,
			&quot;example&quot;: &quot;lorem // ipsum // foo // bar&quot;
		},
		&quot;separator&quot;: {
			&quot;label&quot;: &quot;Separator string&quot;,
			&quot;description&quot;: &quot;The string to use as separator between parameters&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: false,
			&quot;default&quot;: &quot;//&quot;,
			&quot;example&quot;: &quot;;&quot;
		},
		&quot;setter&quot;: {
			&quot;label&quot;: &quot;Setter string&quot;,
			&quot;description&quot;: &quot;The string to use as value setter&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: false,
			&quot;default&quot;: &quot;-&gt;&quot;,
			&quot;example&quot;: &quot;:&quot;
		},
		&quot;table-class&quot;: {
			&quot;label&quot;: &quot;Table class&quot;,
			&quot;description&quot;: &quot;The ‘class=’ HTML attribute assigned to the table of managed parameters&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: false,
			&quot;default&quot;: &quot;wikitable&quot;,
			&quot;example&quot;: &quot;wikitable mw-collapsible mw-collapsed&quot;
		},
		&quot;nocat&quot;: {
			&quot;label&quot;: &quot;No category&quot;,
			&quot;description&quot;: &quot;Disable automatic categorization into ‘Category:Templates calling Example’ or ‘Category:Wrapper templates’ (depending on the case)&quot;,
			&quot;type&quot;: &quot;boolean&quot;,
			&quot;required&quot;: false,
			&quot;default&quot;: &quot;no&quot;,
			&quot;example&quot;: &quot;yes&quot;
		}
	}
}&lt;/templatedata&gt;

&lt;includeonly&gt;{{sandbox other||
&lt;!-- Categories below this line --&gt;
[[Category:Documentation header templates]]
[[Category:Wrapper templates|!]]
}}&lt;/includeonly&gt;</text>
      <sha1>ljr3xpc5kkyqvh845pvhwlmxcb37hii</sha1>
    </revision>
    <revision>
      <id>2818572</id>
      <parentid>2818570</parentid>
      <timestamp>2026-07-19T19:54:07Z</timestamp>
      <contributor>
        <username>Grufo</username>
        <id>1192007</id>
      </contributor>
      <comment>/* See also */</comment>
      <origin>2818572</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6414" sha1="osekb7m4wxkvskl4x04l1n188lkqe84" xml:space="preserve">{{Documentation subpage}}
{{Lua|Module:Params}}

This template is a doc page banner that flags a wrapper template.

== Usage ==

'''Wikitext (see {{Tl|Family name footnote}}):'''

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Family name explanation
	| except =&lt;/nowiki&gt; type -&gt; footnote // reftype -&gt; &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{reftype|efn}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;
}}&lt;/nowiki&gt;|lang=wikitext}}

'''Effect:'''

{{ {{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}|nocat=yes|Family name explanation
	| except = type -&gt; footnote // reftype -&gt; &lt;nowiki&gt;{{{reftype|efn}}}&lt;/nowiki&gt;
}}

&lt;div style=&quot;width: 100%; height: 2em;&quot;&gt;&lt;/div&gt;

'''Wikitext (see {{Tl|Bts}}):'''

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Station link
	| except =&lt;/nowiki&gt; MBTA // &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{1|}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt; // &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{2|}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt; // &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{3|}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;
	| passing-through =
}}&lt;/nowiki&gt;|lang=wikitext}}

or, equivalently,

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Station link
	| except =&lt;/nowiki&gt;
		1 -&gt; MBTA //
		2 -&gt; &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{1|}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt; //
		3 -&gt; &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{2|}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt; //
		4 -&gt; &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{3|}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;
	| passing-through =
}}&lt;/nowiki&gt;|lang=wikitext}}

'''Effect:'''

{{ {{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}|nocat=yes|Station link
	| except =
		1 -&gt; MBTA //
		2 -&gt; &lt;nowiki&gt;{{{1|}}}&lt;/nowiki&gt; //
		3 -&gt; &lt;nowiki&gt;{{{2|}}}&lt;/nowiki&gt; //
		4 -&gt; &lt;nowiki&gt;{{{3|}}}&lt;/nowiki&gt;
	| passing-through =
}}

&lt;div style=&quot;width: 100%; height: 2em;&quot;&gt;&lt;/div&gt;

'''Wikitext (see {{Tl|Don't ping}}):'''

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Please ping
	| except =&lt;/nowiki&gt; 1 -&gt; no&lt;nowiki&gt;
	| omitting =&lt;/nowiki&gt; tps // nw // big // thanks // cond&lt;nowiki&gt;
	| passing-through =
}}&lt;/nowiki&gt;|lang=wikitext}}

'''Effect:'''

{{ {{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}|nocat=yes|Please ping
	| except = 1 -&gt; no
	| omitting = tps // nw // big // thanks // cond
	| passing-through =
}}

&lt;div style=&quot;width: 100%; height: 2em;&quot;&gt;&lt;/div&gt;

'''Wikitext (see {{Tl|Sfnlinka}}):'''

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Sfnlink
	| except =&lt;/nowiki&gt; article -&gt; *&lt;nowiki&gt;
	| omitting =&lt;/nowiki&gt; rev // lang&lt;nowiki&gt;
	| passing-through =&lt;/nowiki&gt; p // page // pp // pages // loc // nb // text&lt;nowiki&gt;
}}&lt;/nowiki&gt;|lang=wikitext}}

'''Effect:'''

{{ {{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}|nocat=yes|Sfnlink
	| except = article -&gt; *
	| omitting = rev // lang
	| passing-through = p // page // pp // pages // loc // nb // text
}}

&lt;div style=&quot;width: 100%; height: 2em;&quot;&gt;&lt;/div&gt;

'''Wikitext (see {{Tl|Tanakhverse}}):'''

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Bibleverse
	| except =&lt;/nowiki&gt; 3 -&gt; &lt;&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;{{{3|HE}}}&lt;/nowiki&gt;&lt;/&lt;nowiki /&gt;nowiki&gt;&lt;nowiki&gt;
	| omitting =&lt;/nowiki&gt; 4 // 5&lt;nowiki&gt;
}}&lt;/nowiki&gt;|lang=wikitext}}

'''Effect:'''

{{ {{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}|nocat=yes|Bibleverse
	| except = 3 -&gt; &lt;nowiki&gt;{{{3|HE}}}&lt;/nowiki&gt;
	| omitting = 4 // 5
}}

&lt;div style=&quot;width: 100%; height: 2em;&quot;&gt;&lt;/div&gt;

'''Wikitext:'''

{{#tag:syntaxhighlight|&lt;nowiki&gt;{{&lt;/nowiki&gt;{{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}&lt;nowiki&gt;|Debug}}&lt;/nowiki&gt;|lang=wikitext}}

'''Effect:'''

{{ {{#ifeq:{{SUBPAGENAME}}|doc|{{BASEPAGENAME}}|{{PAGENAME}}}}|nocat=yes|Debug}}

&lt;div style=&quot;width: 100%; height: 2em;&quot;&gt;&lt;/div&gt;

== Template data ==

&lt;templatedata&gt;{
	&quot;description&quot;: &quot;This documentation box informs that a template is a wrapper of another template.&quot;,
	&quot;params&quot;: {
		&quot;1&quot;: {
			&quot;label&quot;: &quot;Template&quot;,
			&quot;description&quot;: &quot;Name of the wrapped template&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: true,
			&quot;example&quot;: &quot;Please ping&quot;
		},
		&quot;except&quot;: {
			&quot;label&quot;: &quot;Bound parameters&quot;,
			&quot;description&quot;: &quot;The list of pre-assigned parameters; use ‘//’ to separate them and ‘-&gt;’ to assign them a value—see ‘|separator=’ and ‘|setter=’ to set other strings&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: false,
			&quot;example&quot;: &quot;1 -&gt; no&quot;
		},
		&quot;omitting&quot;: {
			&quot;label&quot;: &quot;Undefined parameters&quot;,
			&quot;description&quot;: &quot;The list of parameter names left undefined; use ‘//’ to separate them—see ‘|separator=’ to set another string&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: false,
			&quot;example&quot;: &quot;tps // nw // big // thanks // cond&quot;
		},
		&quot;passing-through&quot;: {
			&quot;label&quot;: &quot;Parameters passed through&quot;,
			&quot;description&quot;: &quot;The list of parameter names left available to use; use ‘//’ to separate them—see ‘|separator=’ to set another string; you can provide a blank parameter to signal that no parameters are left available&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: false,
			&quot;example&quot;: &quot;lorem // ipsum // foo // bar&quot;
		},
		&quot;separator&quot;: {
			&quot;label&quot;: &quot;Separator string&quot;,
			&quot;description&quot;: &quot;The string to use as separator between parameters&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: false,
			&quot;default&quot;: &quot;//&quot;,
			&quot;example&quot;: &quot;;&quot;
		},
		&quot;setter&quot;: {
			&quot;label&quot;: &quot;Setter string&quot;,
			&quot;description&quot;: &quot;The string to use as value setter&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: false,
			&quot;default&quot;: &quot;-&gt;&quot;,
			&quot;example&quot;: &quot;:&quot;
		},
		&quot;table-class&quot;: {
			&quot;label&quot;: &quot;Table class&quot;,
			&quot;description&quot;: &quot;The ‘class=’ HTML attribute assigned to the table of managed parameters&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;required&quot;: false,
			&quot;default&quot;: &quot;wikitable&quot;,
			&quot;example&quot;: &quot;wikitable mw-collapsible mw-collapsed&quot;
		},
		&quot;nocat&quot;: {
			&quot;label&quot;: &quot;No category&quot;,
			&quot;description&quot;: &quot;Disable automatic categorization into ‘Category:Templates calling Example’ or ‘Category:Wrapper templates’ (depending on the case)&quot;,
			&quot;type&quot;: &quot;boolean&quot;,
			&quot;required&quot;: false,
			&quot;default&quot;: &quot;no&quot;,
			&quot;example&quot;: &quot;yes&quot;
		}
	}
}&lt;/templatedata&gt;

== See also ==
* [[:Category:Wrapper templates]]

&lt;includeonly&gt;{{sandbox other||
&lt;!-- Categories below this line --&gt;
[[Category:Documentation header templates]]
[[Category:Wrapper templates|!]]
}}&lt;/includeonly&gt;</text>
      <sha1>osekb7m4wxkvskl4x04l1n188lkqe84</sha1>
    </revision>
  </page>
  <page>
    <title>Template:Wrapper</title>
    <ns>10</ns>
    <id>308967</id>
    <revision>
      <id>2818566</id>
      <parentid>2658807</parentid>
      <timestamp>2026-07-19T19:49:05Z</timestamp>
      <contributor>
        <username>Grufo</username>
        <id>1192007</id>
      </contributor>
      <comment>Modernize it</comment>
      <origin>2818566</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2902" sha1="m9kbvocbk36zsb83120qa9nglnua3ox" xml:space="preserve">&lt;includeonly&gt;{{#if:{{{1|}}}
	| {{Mbox 
		| type = style
		| image = [[File:Symbol template class.svg|45px|alt=|link=]]
		| text = '''This is a customized wrapper for the {{Tl|1={{{1}}}}} template,''' which means it's built upon the other template's code. {{#ifeq:{{{passing-through|+}}}|{{{passing-through|-}}}
			| {{#invoke:params|new|
				recalling|passing-through|
				reinterpreting|passing-through|trim_all|splitter_string|{{{separator|//}}}|setter_string||
				with_value_not_matching||strict|
				mixing_names_and_values|$@|&lt;code style{{=}}&quot;white-space: preserve nowrap; word-break: keep-all;&quot;&gt;&amp;#124;&lt;span style{{=}}&quot;color: #767600;&quot;&gt;$@&lt;/span&gt;&amp;#61;&lt;/code&gt;|
				setting|h/i/l/s/f/n|The following parameters from {{Tl|1={{{1}}}}} will work here: |, | and |, and |.|Since all parameters are already managed, no parameters from {{Tl|1={{{1}}}}} will work here.|
				all_sorted|
				list_values|
			}}
			| {{#if:{{{except|}}}{{{omitting|}}}
				| Parameters from {{Tl|1={{{1}}}}} will work here.
				| All parameters from {{Tl|1={{{1}}}}} will work here.
			}}
		}}{{#invoke:params|new|
			recalling|except|
			reinterpreting|except|trim_all|splitter_string|{{{separator|//}}}|setter_string|{{{setter|-&gt;}}}|
			snapshotting|entering_substack|
				with_value_matching||strict|
				detaching_substack|
				mapping_by_mixing|''(empty string)''|
			leaving_substack|
			mapping_by_magic|#tag|values_only_as|2|let|1|syntaxhighlight|let|lang|wikitext|let|inline|true|
			flushing|
			mapping_by_mixing|&lt;tr&gt;&lt;td&gt;&lt;code style{{=}}&quot;white-space: preserve nowrap; word-break: keep-all;&quot;&gt;&amp;#124;&lt;span style{{=}}&quot;color: #767600;&quot;&gt;$#&lt;/span&gt;&amp;#61;&lt;/code&gt;&lt;/td&gt;&lt;td&gt;$@&lt;/td&gt;&lt;/tr&gt;|
			entering_substack|new|
				recalling|omitting|
				reinterpreting|omitting|trim_all|splitter_string|{{{separator|//}}}|setter_string||
				mixing_names_and_values|$@|&lt;tr&gt;&lt;td&gt;&lt;code style{{=}}&quot;white-space: preserve nowrap; word-break: keep-all;&quot;&gt;&lt;s&gt;&amp;#124;&lt;span style{{=}}&quot;color: #767600;&quot;&gt;$@&lt;/span&gt;&amp;#61;&lt;/s&gt;&lt;/code&gt;&lt;/td&gt;&lt;td&gt;''(undefined)''&lt;/td&gt;&lt;/tr&gt;|
			merging_substack|
			setting|h/f| {{#ifeq:{{{passing-through|+}}}|{{{passing-through|-}}}
				| The {{#if:{{{passing-through|}}}|other p|p}}arameters passed are managed as follows:
				| However, the following are exceptions:
			}}&lt;tabl{{#if:{{{table-class|/}}}
				| e class{{=}}&quot;{{{table-class|wikitable}}}&quot;
				| e
			}} style{{=}}&quot;margin-left: auto; margin-right: auto;&quot;&gt;&lt;tr&gt;&lt;th&gt;Parameter passed to {{Tl|1={{{1}}}}}&lt;/th&gt;&lt;th&gt;Value&lt;/th&gt;&lt;/tr&gt;|&lt;/table&gt;|
			all_sorted|
			list_values
		}}
	}}{{#if:{{yesno|{{{nocat|}}}}}
		|
		| {{Sandbox other
			| 1 =
			| 2 = {{#ifexist:Category:Templates calling {{{1}}}
				| [[Category:Templates calling {{{1}}}|{{#ifeq:{{First word|{{PAGENAME}}}}|Infobox|{{Remove first word|{{PAGENAME}}}}}}]]
				| [[Category:Wrapper templates]]
			}}
		}}
	}}
	| {{Error|Error: Missing template name.}}
}}&lt;/includeonly&gt;&lt;noinclude&gt;{{Documentation}}&lt;/noinclude&gt;</text>
      <sha1>m9kbvocbk36zsb83120qa9nglnua3ox</sha1>
    </revision>
  </page>
  <page>
    <title>Template:Admin backlog/doc</title>
    <ns>10</ns>
    <id>308968</id>
    <revision>
      <id>2818568</id>
      <parentid>2658808</parentid>
      <timestamp>2026-07-19T19:50:26Z</timestamp>
      <contributor>
        <username>Grufo</username>
        <id>1192007</id>
      </contributor>
      <comment>Parameter info</comment>
      <origin>2818568</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6925" sha1="6t0zo709v6yiwjwkszy2dkmjye4i549" xml:space="preserve">{{Documentation subpage}}
&lt;!-- Please place categories where indicated at the bottom of this page and interwikis at Wikidata (see [[Wikipedia:Wikidata]]) --&gt;
{{Wrapper|Backlog
	| except =
		admin -&gt; yes //
		category -&gt; //
		remove -&gt; no
}}

This is the {{tl|admin backlog}} message box.

It can be put at the top of pages and categories that have a backlog which require the attention of [[Wikiversity:Administrators|administrators]]. A backlog is a list of things that need to be done and that have not been done for some time.

This template puts the pages into [[:Category:Administrative backlog]], where admins can see where work needs to be done.

== Usage ==

Usually this template is used without any parameters, then it is always visible until it is manually removed. Like this:
{{indent|5}}{{tlc|admin backlog}}

Which renders like this:

{{admin backlog|demospace=category|category=&lt;nowiki /&gt;}}

This template automatically shows the appropriate style depending on what kind of page it is shown on. The style above is for &quot;other&quot; pages such as &quot;Wikiversity:&quot; pages, and the style shown in the examples below is for category pages.

By adding &lt;code&gt;disabled=yes&lt;/code&gt;, the box and accompanying category disappear from the page. This is functionally equivalent to removing the backlog notice.

== Autoreport ==

Often it is more efficient to handle several cases in a backlog at a time, since then we are up to speed with what needs to be done. Thus we often don't want to know about a backlog until it has reached some size.

When this template is placed on a category page it can count the number of items in the category (pages + images + subcategories). Then it can be set to automatically only report when the number of items is above some limit. The autoreport limit can be set to whatever value you prefer. Like this:
{{indent|5}}{{tlc|admin backlog|10}}

If the category has 10 or more items this template will render like this:

{{admin backlog|0|10|demospace=category|category=&lt;nowiki /&gt;}}

But if the category has less than 10 items then this template will not report the page (not categorise it into [[:Category:Administrative backlog]]), and will instead render like this:

{{admin backlog|10|demospace=category|category=&lt;nowiki /&gt;}}

Note that MediaWiki only parses the code when the category page is re-rendered. That is, when someone views the page ''and'' it is more than one week since it was last re-rendered. Thus, it can take a week before this template changes when the number of items have changed. (But if no one visits the page then it can take forever.) To get an immediate change you can [[Wikiversity:Purge|purge]] the page, for instance by clicking the &lt;small&gt;({{purge|recount}})&lt;/small&gt; button in the template.

You can display a different number from the number actually used by placing the display number after the number which the auto-detection system uses. This option is designed to deal with a situation where a category has a few permanent items (such as subcategories), which shouldn't be counted. For example, [[:Category:Requests for unblock]] has 3 subcategories which don't count towards the backlog, so it has the following header:
{{indent|5}}{{tlc|admin backlog|13|10}}

==Silence==
To prevent any output when there is no backlog, use {{tlc|admin backlog|2=silent=yes}}

==Bot updates==

In some cases, a bot automatically adds and removes this template from a page. In this instance, the bot parameter should be used as follows (using &quot;RFC bot&quot; as an example):
{{indent|5}}{{tlc|admin backlog|2=bot=RFC bot}}

Which renders like this:

{{admin backlog|bot=RFC bot|demospace=category|category=&lt;nowiki /&gt;}}
==TemplateData==
&lt;templatedata&gt;
{
	&quot;params&quot;: {
		&quot;1&quot;: {
			&quot;label&quot;: &quot;Backlog threshold&quot;,
			&quot;description&quot;: &quot;The number of items pending before the category/page/etc. is considered backlogged. For example, setting the threshold to 1 means the process is considered backlogged if it has any items in the queue.&quot;,
			&quot;type&quot;: &quot;number&quot;,
			&quot;suggestedvalues&quot;: [
				&quot;1&quot;,
				&quot;25&quot;,
				&quot;100&quot;
			]
		},
		&quot;2&quot;: {
			&quot;label&quot;: &quot;Displayed backlog threshold &quot;,
			&quot;description&quot;: &quot;The number displayed as the backlog threshold. Useful if a backlog category has members that should not be counted towards the backlog (e.g. subcategories or example pages). See documentation for more information.&quot;,
			&quot;type&quot;: &quot;number&quot;,
			&quot;suggestedvalues&quot;: [
				&quot;1&quot;,
				&quot;10&quot;
			]
		},
		&quot;disabled&quot;: {
			&quot;description&quot;: &quot;If true, disables the template. This is functionally equivalent to removing the template.&quot;,
			&quot;example&quot;: &quot;yes&quot;,
			&quot;type&quot;: &quot;boolean&quot;,
			&quot;default&quot;: &quot;no&quot;
		},
		&quot;silent&quot;: {
			&quot;description&quot;: &quot;If true, hides the template entirely when there is no backlog. This parameter is ignored if not used on a category page&quot;,
			&quot;example&quot;: &quot;yes&quot;,
			&quot;type&quot;: &quot;boolean&quot;,
			&quot;default&quot;: &quot;no&quot;
		},
		&quot;backloglink&quot;: {
			&quot;description&quot;: &quot;The destination of the backlog wikilink.&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;default&quot;: &quot;:Category:Administrative backlog&quot;,
			&quot;label&quot;: &quot;Backlog link&quot;
		},
		&quot;demospace&quot;: {
			&quot;description&quot;: &quot;For use in testing. Makes the template function as if it were placed in the given namespace.&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;suggestedvalues&quot;: [
				&quot;Category&quot;,
				&quot;Wikipedia&quot;
			]
		},
		&quot;page&quot;: {
			&quot;description&quot;: &quot;The type of location of the backlog. Defaults to \&quot;This category\&quot; if used in that namespace. Otherwise, defaults to \&quot;This page\&quot;.&quot;,
			&quot;type&quot;: &quot;string&quot;,
			&quot;default&quot;: &quot;This page&quot;
		},
		&quot;bot&quot;: {
			&quot;description&quot;: &quot;The name of the bot which updates this notice, without the User: prefix.&quot;,
			&quot;example&quot;: &quot;RMCD bot&quot;,
			&quot;type&quot;: &quot;wiki-user-name&quot;
		},
		&quot;auto&quot;: {
			&quot;label&quot;: &quot;Automatically updated?&quot;,
			&quot;description&quot;: &quot;If true, says that the notice will automatically hide itself. Note: this only affects the message on the template; it does NOT facilitate automatically removing the notice.&quot;,
			&quot;example&quot;: &quot;yes&quot;,
			&quot;type&quot;: &quot;boolean&quot;,
			&quot;default&quot;: &quot;no&quot;
		},
		&quot;remove&quot;: {
			&quot;description&quot;: &quot;If true, asks editors to remove the message entirely when the backlog is cleared. Otherwise, the template will ask editors to change the template to {{no admin backlog}}.&quot;,
			&quot;example&quot;: &quot;yes&quot;,
			&quot;type&quot;: &quot;boolean&quot;,
			&quot;default&quot;: &quot;no&quot;
		},
		&quot;_debug&quot;: {
			&quot;label&quot;: &quot;debug&quot;,
			&quot;description&quot;: &quot;If true, wraps the result in nowiki tags to allow for easier debugging&quot;,
			&quot;example&quot;: &quot;yes&quot;,
			&quot;type&quot;: &quot;boolean&quot;,
			&quot;default&quot;: &quot;no&quot;
		}
	},
	&quot;description&quot;: &quot;A banner to keep track of backlogs that require administrators' attention&quot;
}
&lt;/templatedata&gt;
== See also ==
* {{tl|backlog}} – For tagging pages that have a backlog that can be handled by regular editors (by non admins).
* {{tl|editprotected}} – For requesting assistance from an admin to edit a fully protected page.
* {{tl|no admin backlog}}

&lt;includeonly&gt;{{Sandbox other||
&lt;!-- Categories below this line, please; interwikis at Wikidata --&gt;
[[Category:Backlog templates|{{PAGENAME}}]]
}}&lt;/includeonly&gt;</text>
      <sha1>6t0zo709v6yiwjwkszy2dkmjye4i549</sha1>
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    <title>WikiJournal Preprints/Cut the coordinates! (or Vector Analysis Done Fast)</title>
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        <id>238563</id>
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      <comment>Bot: Fixing double redirect from [[WikiJournal Preprints/Coordinates Last: Vector Analysis Done Fast]] to [[Coordinates Last: Vector Analysis Done Fast]]</comment>
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  <page>
    <title>User:Dc.samizdat/Golden chords of the 120-cell</title>
    <ns>2</ns>
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    <revision>
      <id>2818533</id>
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      <text bytes="86230" sha1="ppvs9l8vfwxmk2oqgner1d6sy3ohaqh" 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). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 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;
|}

== 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;. Their 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}+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;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;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.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &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, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &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 &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; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &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 twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &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 revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
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 tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &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 intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &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 &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form a 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four different ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three different ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four different ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[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 of Clifford parallel great squares, also isocline chords in great 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 great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;6&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chord
! colspan=&quot;3&quot; |Isocline chord
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The 24 edge chords lie in invariant planes of the rotation. The 24 isocline chords form the rotation's Clifford {24}-gon and also lie in invariant planes of the rotation. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°.  

...  

{{Clear}}

== 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 (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &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)=\phi^{-1} \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. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&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{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&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 \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.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short edge chord
! Section
! colspan=&quot;3&quot; |Long isocline chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&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: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&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: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&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; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{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; |
|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]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{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; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{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 [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell 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 [[w:120-cell#Concentric_hulls|polyhedral section 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 at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, 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;. 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]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell 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. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

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.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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>ppvs9l8vfwxmk2oqgner1d6sy3ohaqh</sha1>
    </revision>
    <revision>
      <id>2818537</id>
      <parentid>2818533</parentid>
      <timestamp>2026-07-19T17:36:14Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="86287" sha1="3clzs5if13f2nz9vrgy41qqiribptcf" 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). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 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;
|}

== 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;. Their 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}+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;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;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.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &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, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &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 &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; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &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 twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &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 revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
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 tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &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 intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &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 &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form a 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four different ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three different ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four different ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[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 of Clifford parallel great squares, also isocline chords in great 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 great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;6&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chord
! colspan=&quot;3&quot; |Isocline chord
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The edge chords form the rotation's edge polygons over which vertices circle as the edge polygon moves orthogonally like a coin flipping. The isocline chords form the rotation's stationary Clifford polygons over which vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°.  

{{Clear}}

== 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 (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &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)=\phi^{-1} \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. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&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{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&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 \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.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short edge chord
! Section
! colspan=&quot;3&quot; |Long isocline chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&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: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&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: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&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; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{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; |
|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]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{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; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{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 [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell 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 [[w:120-cell#Concentric_hulls|polyhedral section 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 at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, 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;. 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]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell 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. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

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.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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>3clzs5if13f2nz9vrgy41qqiribptcf</sha1>
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    <revision>
      <id>2818538</id>
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        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="86581" sha1="odtbro92x1srr0153w3gwpcpbaj8u3m" 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). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 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;
|}

== 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;. Their 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}+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;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;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.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &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, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &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 &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; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &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 twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &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 revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
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 tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &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 intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &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 &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form a 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four different ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three different ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four different ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[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 of Clifford parallel great squares, also isocline chords in great 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 great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chord
!Invariant planes
! colspan=&quot;3&quot; |Isocline chord
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The edge chords form the rotation's edge polygons over which vertices circle as the edge polygon moves orthogonally like a coin flipping. The isocline chords form the rotation's stationary Clifford polygons over which vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°.  

{{Clear}}

== 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 (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &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)=\phi^{-1} \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. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&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{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&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 \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.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short edge chord
! Section
! colspan=&quot;3&quot; |Long isocline chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&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: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&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: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&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; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{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; |
|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]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{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; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{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 [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell 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 [[w:120-cell#Concentric_hulls|polyhedral section 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 at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, 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;. 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]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell 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. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

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.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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>odtbro92x1srr0153w3gwpcpbaj8u3m</sha1>
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    <revision>
      <id>2818539</id>
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      <text bytes="86643" sha1="awnxflor61hs3f9n2fr86qvjxrn9jtx" 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). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 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;
|}

== 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;. Their 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}+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;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;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.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &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, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &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 &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; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &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 twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &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 revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
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 tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &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 intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &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 &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form a 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four different ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three different ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four different ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[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 of Clifford parallel great squares, also isocline chords in great 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 great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chord
!Invariant planes
! colspan=&quot;3&quot; |Isocline chord
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The edge chords form the rotation's edge polygons over which vertices circle as the edge polygon moves orthogonally like a coin flipping. The isocline chords form the rotation's stationary Clifford polygons over which vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°.  

{{Clear}}

== 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 (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &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)=\phi^{-1} \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. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&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{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&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 \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.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short edge chord
! Section
! colspan=&quot;3&quot; |Long isocline chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&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: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&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: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&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; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{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; |
|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]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{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; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{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 [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell 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 [[w:120-cell#Concentric_hulls|polyhedral section 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 at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, 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;. 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]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell 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. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

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.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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>awnxflor61hs3f9n2fr86qvjxrn9jtx</sha1>
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    <revision>
      <id>2818540</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="86644" sha1="s7hqaw4cvp0tdn4tu3gcowrglv601ol" 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). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 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;
|}

== 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;. Their 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}+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;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;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.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &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, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &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 &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; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &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 twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &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 revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
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 tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &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 intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &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 &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form a 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four different ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three different ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four different ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[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 of Clifford parallel great squares, also isocline chords in great 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 great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chord
!Invariant planes
! colspan=&quot;3&quot; |Isocline chord
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The edge chords form the rotation's edge polygons over which vertices circle as the edge polygon moves orthogonally like a coin flipping. The isocline chords form the rotation's stationary Clifford polygons over which vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°.  

{{Clear}}

== 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 (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &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)=\phi^{-1} \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. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&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{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&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 \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.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short edge chord
! Section
! colspan=&quot;3&quot; |Long isocline chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&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: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&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: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&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; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{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; |
|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]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{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; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{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 [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell 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 [[w:120-cell#Concentric_hulls|polyhedral section 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 at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, 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;. 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]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell 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. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

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.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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>s7hqaw4cvp0tdn4tu3gcowrglv601ol</sha1>
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    <revision>
      <id>2818542</id>
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        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="86696" sha1="pcju20rfgzhmrau9jlyr6wtwu8v00f8" 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). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 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;
|}

== 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;. Their 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}+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;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;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.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &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, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &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 &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; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &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 twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &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 revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
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 tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &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 intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &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 &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form a 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four different ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three different ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four different ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[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 of Clifford parallel great squares, also isocline chords in great 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 great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chords
!Invariant planes
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The edge chords form the rotation's edge polygons over which vertices circle as the edge polygon tilts orthogonally. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices. The isocline chords form the rotation's stationary Clifford polygons over which vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°.  

{{Clear}}

== 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 (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &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)=\phi^{-1} \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. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&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{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&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 \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.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short edge chord
! Section
! colspan=&quot;3&quot; |Long isocline chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&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: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&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: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&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; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{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; |
|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]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{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; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{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 [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell 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 [[w:120-cell#Concentric_hulls|polyhedral section 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 at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, 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;. 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]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell 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. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

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.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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="86709" sha1="qkxgzuitvkjf9ahacfwy9sneauuzzw5" 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). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 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;
|}

== 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;. Their 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}+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;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;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.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &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, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &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 &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; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &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 twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &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 revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
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 tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &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 intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &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 &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form a 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four different ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three different ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four different ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[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 of Clifford parallel great squares, also isocline chords in great 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 great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chords
!Invariant planes
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. The edge chords form the rotation's edge polygons over which vertices circle as the edge polygon tilts orthogonally. The isocline chords form the rotation's stationary Clifford polygons over which vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices.  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &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 (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &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)=\phi^{-1} \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. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&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{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&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 \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.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short edge chord
! Section
! colspan=&quot;3&quot; |Long isocline chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&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: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&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: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&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; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{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; |
|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]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{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; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{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 [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell 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 [[w:120-cell#Concentric_hulls|polyhedral section 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 at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, 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;. 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]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell 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. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

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.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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>qkxgzuitvkjf9ahacfwy9sneauuzzw5</sha1>
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    <revision>
      <id>2818550</id>
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        <username>Dc.samizdat</username>
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      <text bytes="86677" sha1="2ijezoubcuv353ng4mu1zs8zkefjpwq" 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). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 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;
|}

== 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;. Their 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}+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;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;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.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &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, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &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 &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; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &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 twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &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 revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
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 tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &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 intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &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 &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[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 of Clifford parallel great squares, also isocline chords in great 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 great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chords
!Invariant planes
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. The edge chords form the rotation's edge polygons over which vertices circle as the edge polygon tilts orthogonally. The isocline chords form the rotation's stationary Clifford polygons over which vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices.  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &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 (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &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)=\phi^{-1} \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. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&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{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&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 \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.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short edge chord
! Section
! colspan=&quot;3&quot; |Long isocline chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&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: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&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: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&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; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{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; |
|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]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{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; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{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 [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell 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 [[w:120-cell#Concentric_hulls|polyhedral section 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 at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, 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;. 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]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell 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. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

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.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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>2818551</id>
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      <text bytes="86724" sha1="agnk7gqxhvvjz36ssx239y9u6kvkwhq" 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). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 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;
|}

== 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;. Their 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}+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;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;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.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &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, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &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 &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; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &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 twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &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 revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
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 tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &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 intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &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}. The unit-radius planar {12}-gon has these distinct 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 &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[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 of Clifford parallel great squares, also isocline chords in great 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 great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chords
!Invariant planes
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. The edge chords form the rotation's edge polygons over which vertices circle as the edge polygon tilts orthogonally. The isocline chords form the rotation's stationary Clifford polygons over which vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices.  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &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 (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &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)=\phi^{-1} \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. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&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{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&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 \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.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short edge chord
! Section
! colspan=&quot;3&quot; |Long isocline chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&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: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&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: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&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; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{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; |
|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]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{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; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{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 [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell 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 [[w:120-cell#Concentric_hulls|polyhedral section 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 at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, 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;. 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]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell 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. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

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.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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>agnk7gqxhvvjz36ssx239y9u6kvkwhq</sha1>
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    <revision>
      <id>2818553</id>
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        <username>Dc.samizdat</username>
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      <text bytes="86552" sha1="q2ihlo9604b12ned6584fh85dd1csm7" 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). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 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;
|}

== 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;. Their 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}+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;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;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.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &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, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &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 &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; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &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 twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &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 revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
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 tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &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 intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &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}. The unit-radius planar {12}-gon has these distinct 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 &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[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 of Clifford parallel great squares, also isocline chords in great 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 great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chords
!Invariant planes
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. The edge chords form the rotation's edge polygons over which vertices circle as the edge polygon tilts orthogonally. The isocline chords form the rotation's stationary Clifford polygons over which vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices.  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &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 (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &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)=\phi^{-1} \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. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&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{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&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 \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.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Edge chords
! Section
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&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: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&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: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short edge chord and long isocline chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&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; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{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; |
|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]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{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; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{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 [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell 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 [[w:120-cell#Concentric_hulls|polyhedral section 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 at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, 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;. 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]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell 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. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

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.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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>q2ihlo9604b12ned6584fh85dd1csm7</sha1>
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    <revision>
      <id>2818556</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="86557" sha1="t229mvcvp6icc67f15exyqizrj5z7kv" 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). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 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;
|}

== 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;. Their 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}+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;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;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.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &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, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &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 &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; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &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 twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &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 revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
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 tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &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 intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &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}. The unit-radius planar {12}-gon has these distinct 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 &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[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 of Clifford parallel great squares, also isocline chords in great 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 great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chords
!Invariant planes
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. The edge chords form the rotation's edge polygons over which vertices circle as the edge polygon tilts orthogonally. The isocline chords form the rotation's stationary Clifford polygons over which vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices.  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &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 (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &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)=\phi^{-1} \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. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&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{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&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 \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.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Edge chords
! Section
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&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: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&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: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short edge chord and long isocline chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Edge chords
! Section
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{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; |
|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]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{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; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{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 [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell 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 [[w:120-cell#Concentric_hulls|polyhedral section 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 at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, 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;. 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]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell 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. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

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.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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>t229mvcvp6icc67f15exyqizrj5z7kv</sha1>
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    <revision>
      <id>2818558</id>
      <parentid>2818556</parentid>
      <timestamp>2026-07-19T19:33:15Z</timestamp>
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        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* The 5-cell 4-simplex */</comment>
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      <text bytes="86540" sha1="sufqpcwuqzh6wi0vgwcb2fu7ol8b9ct" 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). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 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;
|-
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|-
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|-
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|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
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|-
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|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
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|&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;
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|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
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|&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;
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|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
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|-
|&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;
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|&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;
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|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
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|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
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|&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;
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|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
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|-
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|&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;
|}

== 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;. Their 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}+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;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;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.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &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, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &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 &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; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &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 twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &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 revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
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 tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &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 intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &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}. The unit-radius planar {12}-gon has these distinct 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 &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[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 of Clifford parallel great squares, also isocline chords in great 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 great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chords
!Invariant planes
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. The edge chords form the rotation's edge polygons over which vertices circle as the edge polygon tilts orthogonally. The isocline chords form the rotation's stationary Clifford polygons over which vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices.  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &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 (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &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)=\phi^{-1} \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. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&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{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&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 \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.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Edge chords
! Section
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&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: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&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: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short edge chord and long isocline chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Edge chords
! Section
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{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; |
|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]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{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; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{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 [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &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 [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &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 at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, 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;. 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]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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>sufqpcwuqzh6wi0vgwcb2fu7ol8b9ct</sha1>
    </revision>
    <revision>
      <id>2818560</id>
      <parentid>2818558</parentid>
      <timestamp>2026-07-19T19:36:00Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Conclusions */</comment>
      <origin>2818560</origin>
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      <format>text/x-wiki</format>
      <text bytes="86511" sha1="ejdmtbhrbia2oelqm90d8zxafsu4hfz" 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). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 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;
|}

== 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;. Their 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}+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;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;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.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &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, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[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 {8/1} 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 the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &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 &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; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &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 twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &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 revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
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 tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &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 the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &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 intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &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}. The unit-radius planar {12}-gon has these distinct 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 &lt;math&gt;r_1&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5&lt;/math&gt; &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[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 of Clifford parallel great squares, also isocline chords in great 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 great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt; &lt;/small&gt;chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chords
!Invariant planes
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. The edge chords form the rotation's edge polygons over which vertices circle as the edge polygon tilts orthogonally. The isocline chords form the rotation's stationary Clifford polygons over which vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices.  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &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 (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &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)=\phi^{-1} \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. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&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{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&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 \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.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Edge chords
! Section
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&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: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&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: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short edge chord and long isocline chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Edge chords
! Section
! colspan=&quot;3&quot; |Isocline chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{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]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{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; |
|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]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{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; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{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; |
|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; |
|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]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{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 [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &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 [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &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 at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, 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;. 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]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &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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    <title>Intuitive Calculus</title>
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        <username>Atcovi</username>
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      <comment>relinquishing this to the wild</comment>
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      <text bytes="15582" sha1="auclbfbn99npjiyo5nkj790vhktonsc" xml:space="preserve">{{mathematics}}'''&lt;u&gt;Book&lt;/u&gt;''': ''Infinite Powers'' by Steven Strogatz (ISBN#: 1328879984){{tertiary}}
{{Notes}}
{{juststarted}}
== Notes ==
[[File:Parts of Parabola.svg|thumb|A diagram of a parabola.]]

=== 4/11/2026 (Archimedes and the method of exhaustion) ===
* Archimedes and figuring out the ''quadratic'' (or computation of the area) of a parabolic segment. This is just basically spamming smaller triangles into a [[parabola]] to equal one big triangle (&lt;math display=&quot;inline&quot;&gt;=1&lt;/math&gt;) in order to figure out the area.

Total area of a parabolic segment from Archimedes findings: &lt;math display=&quot;inline&quot;&gt;1&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/16&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/64&lt;/math&gt; ← geometric series.

^each term is &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; of the term preceding it as the daughter triangles always contribute a total of 1 quarter as much area as their parents do. 

Archimedes proved that &lt;math display=&quot;inline&quot;&gt;a = 4/3&lt;/math&gt; through a '''double reductio ad absurdum'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=36}}&lt;/ref&gt; using the '''method of exhaustion''', an analytical way of finding a result&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=102}}&lt;/ref&gt;.

=== 5/2/2026 (Johannes Kepler) ===

==== [[w:Johannes_Kepler|Johannes Kepler]] ====

# '''[[w:Elliptic orbit|Elliptical orbits]]'''
#*'''Ellipse''': Plane curve where the sum of distances from any point on the curve to two fixed points (foci) is constant. For example, a circle is a type of ellipse. A circle is a set of points where distance from a given point (aka its center) is constant. Kepler stated that all planets follow an elliptical orbit. 
# '''[https://www.socratica.com/pages/keplers-second-law-of-motion Equal Areas in Equal Times]'''
#*'''Formula''': Time (P&lt;sub&gt;1&lt;/sub&gt; → P&lt;sub&gt;2&lt;/sub&gt;) = Time (P&lt;sub&gt;3&lt;/sub&gt; → P&lt;sub&gt;4&lt;/sub&gt;) [their sectors have equal areas]
# '''Third Law and the Sacred Frenzy'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=84}}&lt;/ref&gt;
#*&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt;&lt;sup&gt;3&lt;/sup&gt;
#**&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt; = how long it takes for a planet to go around the sun just once.
#**&lt;math display=&quot;inline&quot;&gt;A&lt;/math&gt; = avg. of the planet's nearest and farthest distance from the sun.

=== 5/14/2026 (Calculus definitions, introduction to adequality) ===
* '''[[w:Differential_calculus|Differential calculus]]:''' cuts complicated problems into infinitely many simpler pieces. Ex, derivatives. 
* '''[[w:Integral_calculus|Integral calculus]]''': puts the pieces back together again to solve the original problem. Ex, integrals. 
[[File:Tangent function animation.gif|thumb|The derivative at different points of a differentiable function. In this case, the derivative is equal to &lt;math&gt;\sin \left(x^2\right) + 2x^2 \cos\left(x^2\right)&lt;/math&gt;.&lt;ref&gt;{{Cite journal|date=2026-04-13|title=Derivative|url=https://en.wikipedia.org/w/index.php?title=Derivative&amp;oldid=1348562692|journal=Wikipedia|language=en}}&lt;/ref&gt;]]
[[File:Cartesian-coordinate-system.svg|thumb|This is known as a ''Cartesian coordinate system''.|left]]
* '''[[w:Analytical_geometry|Analytical geometry]]''': Also known as Cartesian geometry, is geometry using a coordinate system (pictured towards the left). Analytical geometry is used in physics, engineering, and aviation. &quot;Analysis&quot; in analytic geometry is meant to be understood as a way of ''figuring out'' the results rather than proving the results&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=101}}&lt;/ref&gt;. 
==== Adequality ====
''See pages 103 to 107, which provide a breakdown of [[w:Pierre_de_Fermat|Pierre de Fermat]] and his concept of adequality.''

Pierre de Fermat's concept of adequality (meaning ''approximate equality''&lt;ref&gt;{{Cite journal|date=2024-09-18|title=Number Theory: An Approach Through History from Hammurapi to Legendre|url=https://en.wikipedia.org/w/index.php?title=Number_Theory:_An_Approach_Through_History_from_Hammurapi_to_Legendre&amp;oldid=1246411217|journal=Wikipedia|language=en}}&lt;/ref&gt;) was a way of finding the maxima, minima, tangents, and other problems in calculus. For example, two nearly equal values, [let's say] ''a'' and ''b'' at the maximum of a parabola, are used to find the maxima of a parabola through a small 'nudge' in the variable&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=106}}&lt;/ref&gt;.

Fermat's ideas eventually led to the concept of derivatives (illustrated towards the right) in modern calculus. 
=== 5/16/2026 (continuation of Fermat's adequality) ===
[[File:Week 9 Fermat and Adequality Proto-Calculus Notes - Part 1.jpg|thumb|438x438px|'''Figure 1.''' Written statements [in all caps] are as follows (from the top-down): 1. WHAT IS THE MAXIMUM VALUE? 2. TWO NEARBY X-VALUES, X&lt;sub&gt;1&lt;/sub&gt; AND X&lt;sub&gt;2&lt;/sub&gt;, PRODUCE ALMOST THE SAME OUTPUT; l = left side, r = right side in the hill diagram]]
==== What does b - (x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;) = 0 represent? ====
b = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;

Reference the hill diagram in '''Figure 1''' (you may have to open the file and zoom in). X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; represent two nearby points on both sides of the &quot;hill&quot; which both produce almost the same output.  

For both of the values, adding both X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; would equal &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt; (the total length). B = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt; would come out to B = 2x, with '''x = b/2''' (where the maximum is). This is the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would ideally give the highest value for &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (see below).
==== Purpose of bx - x&lt;sup&gt;2&lt;/sup&gt; = c? ====
What is the purpose of the equation (see https://youtube.com/AOKoo_nQSts?si=1RfOYMAHm-Ll5sVT&amp;t [minute 4:17] for context/writing of this equation): &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;?

If we take a line (total = &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt;), and make a cut at some point in the line (and designate the cut 'mark' as &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;), how could we figure out &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (output produced by the equation, &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;)? 

&lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; represents a portion of the line, while &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; represents the remaining portion of the line. The product of both &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt;. The goal is to find the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would produce the highest &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; value.

=== 5/20/2026 [Fermet's Theorem] ===

* Pages 107 to 113 detail Fermat's concept of adequality and other mathematical findings led to the decompression of fingerprint files for the FBI in the 1990s. Read [https://www.osti.gov/servlets/purl/400027 this] for more about the FBI's decision to digitalize fingerprint files and the process behind it.
* ''[expand upon Fermat's optimization? Use the PDF?]''
* '''Fermet's Theorem =''' If a real-valued function, &lt;math&gt;f(x)&lt;/math&gt;, is differentiable&lt;ref&gt;function has a well-defined, smooth slope at every single point&lt;/ref&gt; in an interval &lt;math&gt;(a, b)&lt;/math&gt; and &lt;math&gt;f(x)&lt;/math&gt; has a maximum OR minimum at &lt;math&gt;c&lt;/math&gt; ∈ &lt;math&gt;(a, b)&lt;/math&gt;, then &lt;math display=&quot;inline&quot;&gt;f'(c)&lt;/math&gt; = &lt;math display=&quot;inline&quot;&gt;0&lt;/math&gt;&lt;ref&gt;{{Cite web|url=https://old.maa.org/press/periodicals/convergence/fermat-s-method-for-finding-maxima-and-minima-a-mini-primary-source-project-for-calculus-1-students|title=Fermat’s Method for Finding Maxima and Minima: A Mini-Primary Source Project for Calculus 1 Students {{!}} Mathematical Association of America|website=old.maa.org|access-date=2026-05-21}}&lt;/ref&gt;.
** Explanation of ∈: essentially &quot;belongs to/inside/a member of.&quot; For example, &lt;math&gt;c&lt;/math&gt; ∈ &lt;math&gt;(a, b)&lt;/math&gt; → &quot;the number c&lt;math&gt;&lt;/math&gt; is inside the interval between &lt;math&gt;a&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt;&quot;.

=== 5/23/2026 [Logarithms] ===
''[insert logarithms introduction/lesson]''

log(''a'' x ''b'') = log ''a'' + log ''b''

Multiply two numbers together, take the log = answer is the SUM of their individual logs. Logarithms are like an &quot;undo&quot; tool. They &quot;undo&quot; the mathematical operations done by exponential functions, and the relationship between logarithms and exponential functions is reciprocal. 

* ''e''  = 2.71828... similar to π in circles&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=136}}&lt;/ref&gt;. See [https://simple.wikipedia.org/wiki/E_(mathematical_constant) e (mathematical constant)] (simple-wiki) &amp; [[w:Natural logarithm]] (wikipedia). The rate of change of ''e''&lt;sup&gt;x&lt;/sup&gt; is ''e''&lt;sup&gt;x&lt;/sup&gt;. The rate of exponential growth is proportional to the function's current level&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=137}}&lt;/ref&gt;. An example to illustrate this is the following: as a microphone picks up a noise that increases in volume (perhaps the source of the sound is moving closer to the microphone), the loudspeaker amplifies the noise at a constant, exponential rate ''in proportional'' (NOT equal) to the noise it is picking up through the microphone.

=== 5/27/2026 [Derivatives] ===
[[File:2020-03-25 00 08 15 A Five Cheese Pizza Hot Pocket after being heated in the Franklin Farm section of Oak Hill, Fairfax County, Virginia.jpg|thumb|When looking at how many ''more'' calories I will consume per infinitesimally small bite of the hot pocket, we are assessing the derivative of the hot pocket's calories. 

Yes, this may not be practical, but hopefully bringing food into the 'equation' will help you understand the concept of derivatives better.]]

* What is the definition of a '''derivative'''? Essentially the rate of change: ''dy/dx''. An example of a derivative is [[PlanetPhysics/Acceleration|acceleration]]. Another example of a derivative is the following question: how many calories will I consume per bite of a hot pocket (each bite being infinitesimally small)?

The question posed by the book is as follows: ''how do we define the slope when the slope keeps changing?''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=143}}&lt;/ref&gt;

Shifting our mindset from [[Speak Math Now!|algebra]]: In calculus, the rate of change is ''not'' constant, as the IV changes (and is therefore regarded as a '''function'''). We go from Δy/Δx [set rate of change] → ''dy/dx'' [infinitesimally tiny, varied changes]. 

So instead of thinking of the hourly rate for a cashier as a set number (let's say $16/hr), we should think of the $16/hr as a ''constant'' function. This is going to pay off in calculus as we deal with rates of changes that are not always 'set in stone', or constant. For example, measuring a horse's total speed in a [[w:Horse_racing|horse race]] is not going to be a constant, set number - it will be a function with a constantly changing rate. For this specific example:

* '''x''' = time
* '''y''' = speed
* '''dy/dx''' = rate of change of horse's speed with respect to time (think of it as: &quot;rate of change of [y] in respect to [x]&quot;).

=== 6/6/2026 [Definite Integrals &amp; Area Function] ===
{{NOTE|'''TO-DO [6/6/2026]''': provide example problem of area function modeling of YT video, do the work, take a pic and upload.}}''background info...''

A '''definite''' '''integral''', in calculus, is the generalized area under the curved function&lt;ref&gt;{{Cite web|url=https://openstax.org/books/calculus-volume-1/pages/5-2-the-definite-integral|title=5.2 The Definite Integral - Calculus Volume 1 {{!}} OpenStax|last=Strang|first=Gilbert|last2=Herman|first2=Edwin “Jed”|date=2016-03-30|website=openstax.org|language=English|access-date=2026-06-06}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-3/a/definite-integral-as-the-limit-of-a-riemann-sum|title=Khan Academy|website=www.khanacademy.org|language=en|access-date=2026-06-06}}&lt;/ref&gt;.

Overview (page 184):

* &quot;finding the rate of change/derivative of a known function&quot; = differentiation.
* &quot;inferring an unknown function from its rate of change&quot; = integration.

Make sure to apply these principles on your understanding of differential calculus and integration calculus. 

* '''Area function''' (calculus) - Accumulated area under a curved line on an ''xy'' graph from point ''a'' to point ''x'' [upper bound that can be moved as opposed to point ''b'']. [https://www.youtube.com/watch?v=6WfetaviTrQ YT video]. 
** ''What is the area problem?'' &quot;Predicting the relationship between anything that changes at a changing rate and how much that thing builds up over time&quot;&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=184}}&lt;/ref&gt;.
== Wikipedia/Study Links ==
[[w:Archimedes|'''Archimedes''']]
* [[w:Approximations_of_pi|approximations of pi]]
* quadrature (computation of area) of a parabolic segment
* [[w:Archimedes_Palimpsest|''Archimedes Palimpsest'']]
* [https://math.nyu.edu/Archimedes/Lever/LeverLaw.html Archimedes' Law of the Lever]
'''[[w:Pierre_de_Fermat|Pierre de Fermat]]'''

* [https://old.maa.org/sites/default/files/images/upload_library/46/Barnett_TRIUMPHS_MiniPSPs/MiniPSP_FermatsMethod_2023_02_20.pdf ''Fermat’s Method for Finding Maxima and Minima'']- Kenneth M Monks (2023)
'''Other'''

* [[w:Glossary_of_mathematical_symbols|Glossary of mathematical symbols]]

== See Also ==
* [[User:Addemf/sandbox/Who Invented Calculus?]]

== References/Sources ==
{{reflist}}
[[Category:Atcovi's Work]]
[[Category:Calculus]]
[[Category:Quotes]]</text>
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      <comment>I have added the link to the article in the ARAL.</comment>
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      <text bytes="575" sha1="l9ijzj689u9wtmu0fqozj0md9pszp1h" xml:space="preserve">Welcome to a dialogical space dedicated to discussions about a 2026 essay titled &quot;''(Re)imagining (an)other wor(l)d(s) through undisciplinary research.'' ''Reflections on epistemic, existential, sustainable and solidarity, E2-S2-based, work''&quot;, published in ''Annual Review of Applied Linguistics'', ARAL 2026 issue on the theme: '''Interdisciplinarity in Applied Linguistics''' (the official journal of the ''American Association for Applied Linguistics''). See Doi: 10.1017/S0267190526100348 and https://www.cambridge.org/core/journals/annual-review-of-applied-linguistics.</text>
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  <page>
    <title>Plurilingualism in marginalized contexts</title>
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      <contributor>
        <username>Filomena Capucho</username>
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      <comment>Just a few formal corrections in language</comment>
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      <text bytes="9794" sha1="49i586df3tkb5mt1vppeg7gb79svngn" xml:space="preserve">== Starting activity ==
Watch [https://www.youtube.com/watch?v=ITThmEXn1zc this video] about theatre and art activities in prison.

Now, ask yourself a few questions:

* What makes the prison environment particularly marginalized?
* Why are theatre and other artistic activities important in the lives of inmates?
* What other approaches could have the same effect?
* Are there other contexts where the marginalization of social groups is just as significant? Which ones?

== Objectives ==
By the end of this section, you should be able to…

* identify socially marginalized contexts
* characterize these contexts from a social and linguistic perspective
* analyze the needs for plurilingual education in these contexts
* describe concrete practices aimed at developing plurilingualism in marginalized contexts
* outline the individual and social benefits of these practices

== Keywords ==
Marginalized contexts, human dignity, inclusion, plurilingual communication, intercomprehension, translanguaging, carceral environment 

== Prerequisites ==
You may  want to check out the pages on [[theories and models of plurilingualism]], [[intercomprehension]], and [[translanguaging]]. 

== Introduction ==
The Common European Framework of Reference for Languages (CEFR) highlights the risk of exclusion faced by people who do not speak the “useful” languages in an increasingly interconnected society. For example, linguistic minorities, such as speakers of Catalan, Basque, or Galician in Spain, have often been stigmatized or silenced in the past, and revitalizing their languages remains a challenge. In disadvantaged neighbourhoods, people from immigrant backgrounds or precarious circumstances face a double disadvantage: their native language is devalued, and access to learning dominant languages is limited. Marginalized contexts, such as prisons, depopulated rural areas, or communities where people with disabilities live, also illustrate these inequalities. In prison, for example, inmates have few educational opportunities tailored to their language, which exacerbates their exclusion. Similarly, immigrants and refugees face obstacles in having their credentials recognized or accessing essential information in their language. Finally, power dynamics between languages create hierarchies: institutional languages (such as English, Spanish or French) are privileged, while minority languages are often relegated to a lower status. This can discourage speakers of these languages from using or passing them on, reinforcing their marginalization. 

To combat the inequalities named above, the CEFR promotes plurilingual and intercultural education, which values all languages and cultures, and trains teachers and professionals on issues of diversity. Initiatives exist, such as language enrichment programs for immigrant students or intercomprehension courses in prisons, but they remain insufficient given the scale of the needs. The challenge, therefore, is to recognize and actively support this diversity to build a more inclusive society.

Plurilingualism develops through measures that strengthen linguistic and intercultural skills, which are essential in multicultural societies. It encourages the learning of multiple languages and fosters intercultural awareness to promote mutual understanding. By integrating diverse pedagogical approaches, pPlurilingual education becomes a tool for social inclusion. It adapts educational programs to the needs of learners from varied backgrounds, particularly in disadvantaged neighborhoods and marginalized contexts such as prisons or rural areas. Intercomprehension courses are offered there, often in partnership with associations and universities. Appropriate language policies are necessary to support these initiatives and allocate the required resources. Examples, such as programs for immigrant students in the Basque Country or integration measures in Quebec, demonstrate a commitment to a more equitable society. Raising awareness of linguistic diversity and protecting minority languages remain crucial to ensuring an environment that respects all cultures.

== History/Concept ==
If we take a European national context as an example, the foreign inmates in Spanish correctional facilities account for 29.5% and live in an environment characterized by linguistic and cultural diversity. This diversity gives rise to complex communication situations, where language poses a major obstacle in three distinct levels:

* An administrative setting, where Spanish is the dominant language, which can exclude non-Spanish-speaking inmates.
* Interactions among foreign inmates, whose diverse backgrounds involve languages belonging to different linguistic families, make communication difficult.
* Interactions between foreign and Spanish inmates in common areas, where linguistic and cultural differences can hinder mutual understanding.

Added to this linguistic diversity is cultural diversity, as each individual brings a unique linguistic and cultural background. Some inmates lack access to basic education or do not speak any language other than their L1. Thus, the language barrier is not the only challenge they face in their interpersonal relationships and their integration within the facility.

== Concepts / Practical Applications ==
A module of plurilingual intercomprehension was implemented in prisons in Spain as a 20-hour summer course offered by the Universidad Nacional de Educación a Distancia (National University of Distance Education), bringing together external students, inmates, and prison staff. The method combined oral materials (advertisements, films, TV shows) and written materials (authentic texts) in five Romance languages (Portuguese, Italian, French, Catalan, Romanian), with a strong cultural focus (gastronomy, native stories, plurilingual workshops).

The teaching team, composed of L1-speakers (Spanish, Italian, French, Portuguese), created an immediate linguistic immersion. The benefits include the development of active listening and textual analysis, enabling learners to acquire linguistic structures that can be reused in other contexts.

== Take Home messages ==

* Plurilingual and intercultural communication in the prison setting is part of an effort to adapt to contemporary trends in mobility and societal diversity.
* Prisons house a population characterized by significant linguistic and cultural diversity, making it essential to establish communication models tailored to this pluralistic reality.
* Taking this linguistic and cultural diversity into account within the prison setting is a fundamental challenge for ensuring effective and inclusive communication.
* The plurilingual approach promotes linguistic understanding and respect for sociocultural differences, ensuring inmates’ equitable access to their rights and instilling fundamental values such as respect and tolerance.
* This approach strengthens inmates’ autonomy and cooperation while reducing their dependence on institutions. It fosters values such as tolerance and linguistic and cultural mutual understanding, and develops their skills for more effective and appropriate communication.

== Self-assessment ==

# How can effective and inclusive communication be ensured in prisons?
# What types of communication models could be implemented in prisons to address linguistic and cultural diversity, and how do they differ from traditional methods?
# How does a plurilingual  approach promote equity in prisoners’ access to their rights?
# What concrete impacts does this approach have on inmates’ autonomy, cooperation, and institutional dependence?

== Further reading ==

* Benucci, A. (2007). ''Italiano libera-mente. L’insegnamento dell’italiano a stranieri in carcere''. Guerra Edizioni.
* Benucci, A. &amp; Monaci, V. (2025). ''Questioni di genere, lingue e culture in carcere. Tutela della differenze contrasto all’emarginazione e all’esclusione in carcere''. Edizioni Ca’ Foscari.
* Gómez Fernández, A. (2023). Educación y centros penitenciarios. Comunicación plurilingüe e intercultural como modo de inclusión. In J. L. Muñoz de Baena Simón &amp; J. M. Enríquez Sánchez (Eds.), ''Vigilar y Educar: buenas prácticas formativas en centros penitenciarios'' (pp. 207 - 227). Tirant lo Blanch, Colección Márgenes.
* Pacini Volpe, P. (2021).  ''L’enseignement universitaire en milieu carcéral. Expériences comparées entre la France et l’Italie''. Champ social éditions.

= References =
Audras, I. (2014). Impact de séances d’éveil aux langues au sein d’un atelier parents-enfants dans une “Maison pour tous” : entretiens avec les acteurs de la structure. In C. Troncy (Ed.), ''Didactique du plurilinguisme. Approches plurielles des langues et des cultures. Autour de Michel Candelier'' (pp. 355-361). Presses Universitaires de Rennes.

Béacco, J-C. (2000). ''Les dimensions culturelles des enseignements de langue : des mots aux discours''. Hachette.

Conseil de l’Europe (2011). ''Cadre européen commun de référence pour les langues : apprendre, enseigner, évaluer'' – Volume complémentaire. Conseil de l’Europe.

Coste, D. (2010). Diversité des plurilinguismes et formes de l’éducation plurilingue et interculturelle. ''Recherches en didactique des langues et des cultures''.  [online], ''Les Cahiers de l’Acedle'', 7-1. https://doi.org/10.4000/rdlc.2031


==Credits==
This resource has been created by [[User:Projet PEP|Projet PEP]] ([[User talk:Projet PEP|discuss]] • [[Special:Contributions/Projet PEP|contribs]]) (Erasmus+ project, co-financed by the European Commission) : 

* Filomena Capucho (Universidade Católica Portuguesa)
* Araceli Gomez Fernandez (UNED) 

[[Portal: Plurilingual education]]

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      <text bytes="49872" sha1="493apuy08j5hkoez31v3vkmluevh7l2" xml:space="preserve">==Background==

===Richard II===


===William Lambarde===
Lambarde was the royal archivist and had been instructed to bring the queen a calendar of manuscripts stored at the Tower.

===Elizabeth I===

====Essex's rebellion====
Seven months before Lambarde's discussion with Elizabeth.{{sfn|Orgel|2011|pp=11–43}}

==IaR2==
Lambarde's account of his interview with the queen in August 1601 is well known.{{sfn|Orgel|2011|pp=11–43}}


===Political anxiety and tension===

===Censorship===

==Consequences and aftermath==

== Notes ==
{{reflist|group=note}}

==References==
{{Reflist|20em}}

==Bibliography==
{{refbegin|30em|indent=yes}}
* {{Cite book|url=https://doi.org/10.1057/9780230307261_2|title=Prologue: I am Richard II|last=Orgel|first=Stephen|date=2011|publisher=Palgrave Macmillan UK|isbn=978-0-230-30726-1|editor-last=Petrina|editor-first=Alessandra|location=London|pages=11–43|language=en|doi=10.1057/9780230307261_2|editor-last2=Tosi|editor-first2=Laura}}
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* [https://www.academia.edu/5321346/Looking_Back_Shakespeare_s_Indebtedness_to_Chaucer_and_the_Representations_of_Chivalry_in_King_Richard_II_The_Two_Noble_Kinsmen_and_the_Knight_s_Tale Looking Back: Shakespeare’s Indebtedness to Chaucer and the Representations of Chivalry in King Richard II, The Two Noble Kinsmen and the Knight’s Tale]
* [https://www.academia.edu/92423912/The_Good_Usurper_in_the_eyes_of_God_and_the_people_An_analysis_of_the_role_of_the_usurper_in_Shakespeares_Richard_II_and_Henry_IV_Part_1 The Good Usurper in the eyes of God and the people: An analysis of the role of the usurper in Shakespeare's Richard II and Henry IV Part 1]
* [https://www.academia.edu/91789871/The_Bacon_Shakespeare_Manuscript_Hitherto_known_as_the_Northumberland_Manuscript_which_originally_Contained_Copies_of_his_Shakespeare_Plays_Richard_II_and_Richard_III THE BACON-SHAKESPEARE MANUSCRIPT (HITHERTO KNOWN AS THE NORTHUMBERLAND MANUSCRIPT) WHICH ORIGINALLY CONTAINED COPIES OF HIS SHAKESPEARE PLAYS RICHARD II AND RICHARD III]
* [https://www.google.co.uk/books/edition/Richard_II/f4gGCAAAQBAJ?hl=en&amp;gbpv=1&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;pg=PA88&amp;printsec=frontcover Richard II: Critical Essays]
* [https://www.google.co.uk/books/edition/A_Companion_to_Shakespeare_s_Works_Volum/JlDNEAAAQBAJ?hl=en&amp;gbpv=1&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;pg=PA400&amp;printsec=frontcover A Companion to Shakespeare's Works, Volume II]
* [https://www.google.co.uk/books/edition/Shakespeare_Reread/Z6JhDwAAQBAJ?hl=en&amp;gbpv=1&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;pg=PA239&amp;printsec=frontcover Shakespeare Reread]
* [https://www.google.co.uk/books/edition/Shakespeare_in_the_Present/yTKWEAAAQBAJ?hl=en&amp;gbpv=1&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;pg=PT8&amp;printsec=frontcover Shakespeare in the Present]
* [https://www.google.co.uk/books/edition/Studying_Shakespeare/1N4FBAAAQBAJ?hl=en&amp;gbpv=1&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;pg=PA129&amp;printsec=frontcover Studying Shakespeare A Practical Introduction]
* [https://www.google.co.uk/books/edition/Critical_Essays_on_William_Faulkner/kAGBEAAAQBAJ?hl=en&amp;gbpv=1&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;pg=PT178&amp;printsec=frontcover Critical Essays on William Faulkner]
* [https://www.google.co.uk/books/edition/Shakespeare/QY6aEQAAQBAJ?hl=en&amp;gbpv=1&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;pg=PA216&amp;printsec=frontcover Shakespeare: An Anthology of Criticism and Theory 1945-2000]
* [https://www.google.co.uk/books/edition/The_Reign_of_Richard_II/cv0WAQAAIAAJ?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 Dodd, G., R2]
* [https://www.google.co.uk/books/edition/Shakespeare_and_the_Institution_of_Theat/Nkt9DAAAQBAJ?hl=en&amp;gbpv=1&amp;dq=%22I+am+Richard+II,+know+ye+not+that%3F%22&amp;pg=PA108&amp;printsec=frontcover Shakespeare and the Institution of Theatre]
* [https://www.google.co.uk/books/edition/The_Theory_of_the_King_s_Two_Bodies_in_t/8SRXAAAAYAAJ?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 Theory of the King's Two Bodies in the Age of Shakespeare]
* [https://www.google.co.uk/books/edition/Stealing_the_Story/RaJlAAAAMAAJ?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 Stealing the Story: Shakespeare's Self-Conscious Use of the Mimetic Tradition in the Tragedies]
* [https://www.google.co.uk/books/edition/The_Greenwood_Companion_to_Shakespeare_O/JkcgAQAAIAAJ?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 Greenwood Companion to Shakespeare: Overviews and the history plays]
* [https://www.google.co.uk/books/edition/British_and_Irish_Literature_and_Its_Tim/xH4jAQAAIAAJ?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 British and Irish Literature and Its Times]
* [https://www.google.co.uk/books/edition/Shakespeare_s_Philosophy_of_History_Reve/1jwgAQAAIAAJ?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 Philosophy of History Revealed in a Detailed Analysis of Henry V and Examined in Other History Plays]
* [https://www.google.co.uk/books/edition/Shakespeare_and_His_Contemporaries/7em7coOthxQC?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 Contemporaries]
* [https://www.google.co.uk/books/edition/The_Politics_of_the_Public_Sphere_in_Ear/0sGHAAAAMAAJ?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 Politics of the Public Sphere in Early Modern England]
* [https://www.google.co.uk/books/edition/Shakespeare_and_the_Catholic_Religion/Fl4gAQAAIAAJ?hl=en&amp;gbpv=0&amp;bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 Shakespeare and the Catholic religion]
* [https://www.google.co.uk/books/edition/Proceedings_of_the_British_Academy_Volum/8CsoAQAAMAAJ?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 Lectures]
* [https://www.google.co.uk/books/edition/Slander_and_Censorship_in_Late_Sixteenth/wb_oAvuJgbAC?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 Slander and Censorship in Late Sixteenth Century Literature]
* [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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      <text bytes="49895" sha1="4dvd0t11z8nwawdx3vddsu4mniwrnt7" xml:space="preserve">==Background==

===Richard II===


===William Lambarde===
Lambarde was the royal archivist and had been instructed to bring the queen a calendar of manuscripts stored at [[wp:Tower of London|the Tower]].

===Elizabeth I===

====Essex's rebellion====
Seven months before Lambarde's discussion with Elizabeth.{{sfn|Orgel|2011|pp=11–43}}

==IaR2==
Lambarde's account of his interview with the queen in August 1601 is well known.{{sfn|Orgel|2011|pp=11–43}}


===Political anxiety and tension===

===Censorship===

==Consequences and aftermath==

== Notes ==
{{reflist|group=note}}

==References==
{{Reflist|20em}}

==Bibliography==
{{refbegin|30em|indent=yes}}
* {{Cite book|url=https://doi.org/10.1057/9780230307261_2|title=Prologue: I am Richard II|last=Orgel|first=Stephen|date=2011|publisher=Palgrave Macmillan UK|isbn=978-0-230-30726-1|editor-last=Petrina|editor-first=Alessandra|location=London|pages=11–43|language=en|doi=10.1057/9780230307261_2|editor-last2=Tosi|editor-first2=Laura}}
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* [https://www.google.co.uk/books/edition/Shakespeare_s_Philosophy_of_History_Reve/1jwgAQAAIAAJ?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 Philosophy of History Revealed in a Detailed Analysis of Henry V and Examined in Other History Plays]
* [https://www.google.co.uk/books/edition/Shakespeare_and_His_Contemporaries/7em7coOthxQC?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 Contemporaries]
* [https://www.google.co.uk/books/edition/The_Politics_of_the_Public_Sphere_in_Ear/0sGHAAAAMAAJ?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 Politics of the Public Sphere in Early Modern England]
* [https://www.google.co.uk/books/edition/Shakespeare_and_the_Catholic_Religion/Fl4gAQAAIAAJ?hl=en&amp;gbpv=0&amp;bsq=%22I%20am%20Richard%20II,%20know%20ye%20not%20that?%22 Shakespeare and the Catholic religion]
* [https://www.google.co.uk/books/edition/Proceedings_of_the_British_Academy_Volum/8CsoAQAAMAAJ?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 Lectures]
* [https://www.google.co.uk/books/edition/Slander_and_Censorship_in_Late_Sixteenth/wb_oAvuJgbAC?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 Slander and Censorship in Late Sixteenth Century Literature]
* [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?]
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* [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]
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* [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]
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* [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]
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* [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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      <text bytes="49924" sha1="svjdgsyw8rq1w8ug9lnsf8bj2q9t6mb" xml:space="preserve">==Background==

===Richard II===


===William Lambarde===
Lambarde was the royal archivist and had been instructed to bring the queen a calendar of manuscripts stored at [[WP:Tower of London|the Tower]].{{sfn|Orgel|2011|pp=11–43}}

===Elizabeth I===

====Essex's rebellion====
Seven months before Lambarde's discussion with Elizabeth.{{sfn|Orgel|2011|pp=11–43}}

==IaR2==
Lambarde's account of his interview with the queen in August 1601 is well known.{{sfn|Orgel|2011|pp=11–43}}


===Political anxiety and tension===

===Censorship===

==Consequences and aftermath==

== Notes ==
{{reflist|group=note}}

==References==
{{Reflist|20em}}

==Bibliography==
{{refbegin|30em|indent=yes}}
* {{Cite book|url=https://doi.org/10.1057/9780230307261_2|title=Prologue: I am Richard II|last=Orgel|first=Stephen|date=2011|publisher=Palgrave Macmillan UK|isbn=978-0-230-30726-1|editor-last=Petrina|editor-first=Alessandra|location=London|pages=11–43|language=en|doi=10.1057/9780230307261_2|editor-last2=Tosi|editor-first2=Laura}}
* [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]
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* [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/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]
{{refend}}</text>
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  <page>
    <title>User:Ijustwantotalk34/Libby's Top Class Education on Equestrian</title>
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      <timestamp>2026-07-19T15:34:11Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>Atcovi moved page [[Libby's Top Class Education on Equestrian]] to [[User:Ijustwantotalk34/Libby's Top Class Education on Equestrian]] without leaving a redirect: moving under userspace</comment>
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      <text bytes="3526" sha1="ivooqa0fz0fsmz2oh1a81h2l7l743ey" xml:space="preserve">&lt;blockquote&gt;
=== Writers Introduction: Greetings, I am Libby a passionate 11-year-old equestrian aspire from NSW now moved to South Australia who's love and education for horses has expanded over the course of 2 years. and I'm going to be honest with you, I've only had ever 1 horse riding lesson and rides at carnivals but don't please don't underestimate me 2 years of complete obsession and having a experienced mother and grandmother in the show horse world when younger really gets you somewhere in learning how to ride without riding a horse, a lot of people including Olympians say you can't learn how to ride with only reading and learning from social media but I deny that dearly my first lesson was in 2026 Jan with a welcoming women, Jackie. Below explains what I think to be a lot of what I know. I've stuck to only writing 2 parts of how to ride but maybe (Means when I feel like doing it) I'll add more such as the canter, bottom seat etc. Anyway the lesson was so fun Jackie said I was ready for a canter, in her 20+ years of teaching she said &quot;I think you're ready for a canter&quot; little did I know that me and Sparkles broke a stables record I was the first person ever In her entire lesson program to canter on their very first horse riding lesson there-and that was my first lesson ever so I am still very proud of myself I know that Almighty God gave me this present even though I never get to ride I'm being very excitingly patient whilst I wait for the next opportunity I get to ride so YAYYYYYY! a bit of a niche quote from me: &quot;Patience brings records to your name&quot; - Libby C. ===
&lt;/blockquote&gt;

== '''Upper seat while pony is in walking gait''': ==
You need your back comfortably straight or else you'll look like a sloth and it won't be very good for your other natural riding aids needed to make your pony excellent and happy, once you’ve achieved that it's now time to put your shoulders back and relaxed to be aligned with you're ears, elbows hip and heel-Arms by the side of your body with your elbows bent enough so that your arms are in line with your pony's ears your middle and index fingers holding the reins with your thumbs on top.

Chin up eyes above the horses ears-not looking down or directly on the horses ears, that will cause a distraction. 

Move your hips with the horses motion for example; if the horse is moving forward your hips go forward with the horses motion and the same when the horse is using its hindlegs to make a backward motion.

(Make sure you lock your thighs comfortably without squeezing so that your legs don’t move while you're doing this) 

== Upper seat while pony is in trotting gait: ==
Make sure when you're doing your rising trot that you don’t slam your bottom back down to the saddle and slouch so that it dosent upset your pony and distract him. whilst your bringing your body down, you don’t fully sit back down in the saddle when rising, EVER! you should only brush your bottom on the saddle slightly like a paint brush

It's also important that you don’t just go up and down like a straight line that’s a big No-No; from when I was speaking about only brushing the saddle slightly like a paint brush I was saying instead of just going up , down of a straight line you have to rise like a half circle on the saddle let me explain- when you first go up pretend there is a little bridge there where you have to put your hip area over it and once you’ve crossed it make sure your hip area hasn’t gone on the pommel of the saddle or over.</text>
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  </page>
  <page>
    <title>User talk:Ijustwantotalk34</title>
    <ns>3</ns>
    <id>330640</id>
    <revision>
      <id>2818536</id>
      <timestamp>2026-07-19T15:34:46Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* Welcome */ new section</comment>
      <origin>2818536</origin>
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      <format>text/x-wiki</format>
      <text bytes="2612" sha1="o8yxprmtrq6ua2sge7tmsvp80tp0ah3" xml:space="preserve">==Welcome==
{{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]],  Ijustwantotalk34!'''|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:Atcovi|me personally]] if you would like some [[Help:Contents|help]].

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See you around Wikiversity! --—[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:34, 19 July 2026 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}

Hi Libby. I've moved your new page under: [[User:Ijustwantotalk34/Libby's Top Class Education on Equestrian]] so you're able to work on it with no disruptions. Thanks, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:34, 19 July 2026 (UTC)</text>
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  <page>
    <title>File:LCal.9A.Recursion.20260707.pdf</title>
    <ns>6</ns>
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    <revision>
      <id>2818544</id>
      <timestamp>2026-07-19T18:10:12Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=LCal.9A: Recursion (20260707 - 20260706)
|Source={{own|Young1lim}}
|Date=2026-07-20
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}</comment>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="263" sha1="f09zdc5hhqt02nwmxyy2ckqrma8jabf" xml:space="preserve">== Summary ==
{{Information
|Description=LCal.9A: Recursion (20260707 - 20260706)
|Source={{own|Young1lim}}
|Date=2026-07-20
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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  <page>
    <title>File:LCal.9A.Recursion.20260713.pdf</title>
    <ns>6</ns>
    <id>330642</id>
    <revision>
      <id>2818546</id>
      <timestamp>2026-07-19T18:11:08Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=LCal.9A: Recursion (20260713 - 20260707)
|Source={{own|Young1lim}}
|Date=2026-07-20
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}</comment>
      <origin>2818546</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
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{{Information
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Please see [[Wikiversity:What is Wikiversity?]]. The content on your userpage has been deleted for the following reason(s): &quot;'''Solicitation''' for products, services, companies, events, people, or other things with no educational merit or which generate direct financial benefit to the contributor.&quot; Please be aware that such contributions can get you blocked on the English Wikiversity. Thanks, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:59, 19 July 2026 (UTC)</text>
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* [[User:Atcovi/OGM &amp; Suicide/The Paper]] - ''[moved]''</text>
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* [[Intuitive Calculus]]
* [[User:Atcovi/OGM &amp; Suicide/The Paper]] - ''[moved]''</text>
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* [[Intuitive Calculus]]
* [[User:Atcovi/OGM &amp; Suicide/The Paper]] - ''[moved]''

[[Category:Atcovi's Work]]</text>
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